Weakly Coupled Systems of Semi-Linear Fractional σ–Evolution Equations with Different Power Nonlinearities

Abstract

The study of small data Sobolev solutions to the Cauchy problem for weakly coupled systems of semi-linear fractional $\sigma –$evolution equations with different power nonlinearities is of interest to us in this research. These solutions must exist globally (in time). We explain the relationships between the admissible range of exponents $p_1$ and $p_2$ symmetrically in our main modeland the regularity assumptions for the data by using $L^m-L^q$ estimates of Sobolev solutions to related linear models with a vanishing right-hand side and some fixed point argument. This allows us to prove the global (in time) existence of small data Sobolev solutions.

1. Introduction

Fractional derivatives are a generalization of the traditional integer-order derivatives, allowing for the differentiation of a function to a non-integer order. This extension of the classical derivative concept has proven to be a powerful mathematical tool with a wide range of applications. As a result, fractional calculus has found applications in diverse fields such as viscoelasticity, rheology, control theory, signal processing and anomalous diffusion models, providing a more accurate and flexible framework for analyzing complex dynamics—see, e.g., [1,2,3,4] to illustrate some applications.

The Riemann–Liouville fractional derivative is an important concept in the field of fractional calculus, which is defined by

$$D_t^{\alpha }\left(f\right)={\partial }_t\left(I_t^{1-\alpha }f\right)$$

with

$$I_t^{\beta }f=\frac{1}{\mathrm{\Gamma }\left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}f\left(s\right)ds,$$

where $0<\alpha ,\beta <1$, $I_t^{\beta }f$ is the fractional Riemann–Liouville integral of f in $[0,t]$ and $\mathrm{\Gamma }$ is the Euler Gamma function defined by

$$\mathrm{\Gamma }\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt.$$

For the classical Cauchy problem for the semi-linear wave equation,

$$u_{tt}-∆u={\left|u\right|}^p,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right).$$

The authors in [5] proved that for $n=3$ dimensions, the critical exponent p is defined as the positive root of the quadratic equation

$$(n-1)p^2-(n+1)p-2=0.$$

In 2017, in [6], the authors considered a semi-linear fractional wave equation with a Riemann–Liouville fractional derivative. The equation has the form

$${\partial }_t^{1+\lambda }u-∆u={\left|u\right|}^p,u(0,x)=u_0\left(x\right),u_t(0,x)=0,$$

(1)

where $\lambda \in (0,1).$ They determine the critical exponent for the global existence of small data solutions in low space dimensions. The Caputo fractional order and the existence of non-zero initial conditions were studied in [7].

In another related work [8], the authors proved the global existence of small data solutions for semi-linear fractional $\theta –$evolution equations with mass or power nonlinearities. They also considered in [9] a similar problem with a memory term instead of a power nonlinearity.

In [10], the authors proved non-existence theorems for evolution equations involving Caputo and Riemann–Liouville fractional derivatives. Additionally, they proved existence results for subdiffusive equations in that study. In [11,12], the authors studied the Cauchy-type problem for certain multi-term fractional partial differential equations involving Caputo derivatives. They provided asymptotic decay estimates for the solution and applied those results to studying global existence for the corresponding semilinear problem with a power nonlinearity of the form ${\left|u\right|}^p$. Additionally, the authors presented a detailed review of the existing literature on fractional (in time) partial differential equations.

Since the fractional wave equation with the mentioned conditions plays the role of an interpolation between the heat equation and the wave equation, let us start by considering some previous results on weakly coupled systems of heat or wave equations. Considering the system of heat equations,

$$\begin{array}{c}{u}_t-∆u={\left|v\right|}^{p_1},u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),\hfill \\ v_t-∆v={\left|u\right|}^{p_2},v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),\hfill \end{array}$$

where $t\in [0,\infty ),x\in {\mathbb{R}}^d$ and $p_1,p_2>1$. In the paper by Escobedo et al. [13], it was shown that the exponents $p_1$ and $p_2$ satisfying the following equation are critical:

$$\frac{d}{2}=\frac{max\{p_1,p_2\}+1}{p_1{p}_2-1}.$$

This means the following apply:

• For $\frac{d}{2}>\frac{max\{p_1,p_2\}+1}{p_1{p}_2-1}$, the solutions exist globally.
• For $\frac{d}{2}\le \frac{max\{p_1,p_2\}+1}{p_1{p}_2-1}$, the solutions blow up.

For more information on the system of damped wave equations and semi-linear heat equations, the reader may consult the following references [14,15,16,17].

In [18,19,20], the authors considered weakly coupled systems of semilinear classical damped wave equations with power nonlinearities. Considering time-dependent dissipation terms, the authors in the works of Djaouti et al. [21,22,23] proved the global (in time) existence of small data solutions under certain conditions. These conditions describe the interplay between the exponents of the power nonlinearities. In the paper [24], the authors investigated a weakly coupled system where fractional derivatives are incorporated into the equations, considering special Cauchy data. In [24], the authors studied a weakly coupled system where the fractional derivative involves the equations with special Cauchy data.

The weakly coupled system of semi-linear fractional $\sigma –$evolution equations with different power nonlinearities is the subject of this paper. We are interested in the global existence of small data solutions to the following Cauchy problem:

$$\left\{\begin{array}{c}{\partial }_t^{1+{\alpha }_1}u+{(-∆)}^{{\sigma }_1}u={\left|v\right|}^{p_1},\hfill \\ {\partial }_t^{1+{\alpha }_2}v+{(-∆)}^{{\sigma }_2}v={\left|u\right|}^{p_2},,\hfill \\ u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right),u_t(0,x)=v_t(0,x)=0,\hfill \end{array}\right.$$

(2)

where $(t,x)\in [0,\infty )\times {\mathbb{R}}^n$, for $i=1,2$, ${\alpha }_i\in (0,1)$, ${\sigma }_i\ge 1$, ${\partial }_t^{1+{\alpha }_i}u=D_t^{{\alpha }_i}\left(u_t\right).$

Our main results establish the global (in time) existence of small data Sobolev solutions. These are given in the next section.

The paper is organized as follows: After this introduction, the main results are presented in Section 1, which include six theorems covering all cases of the data. The necessary preliminaries are then shown in Section 2. Finally, the proofs of the main results are provided in Section 3.

2. Main Results

2.1. The Case $n\ge max\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$

Theorem 1.

Let $0<{\alpha }_1,{\alpha }_2<1$; ${\sigma }_1,{\sigma }_2\ge 1$; and $m_1,m_2\ge 1$. We assume that $n\ge max\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$. Moreover, for all $ϵ>0$, the exponents $p_1$ and $p_2$ satisfy the conditions

$$p_1>p_{{\alpha }_2,{\sigma }_2}^{m_1,m_2}(n,ϵ):=max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{1}{1-{\alpha }_2},\frac{m_2}{m_1}-ϵ\right\}$$

and

$$p_2>p_{{\alpha }_1,{\sigma }_1}^{m_1,m_2}(n,ϵ):=max\left\{p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right);\frac{1}{1-{\alpha }_1},\frac{m_1}{m_2}-ϵ\right\},$$

where

$$\begin{array}{c}\hfill p_{\alpha ,\sigma }^r\left(n\right):=1+\frac{\left(n\right(r-1)+2\sigma r)(1+\alpha )}{(n-2\sigma r)(1+\alpha )+2\sigma r}.\end{array}$$

Then, there exists a positive constant ε such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}:=\left(L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times \left(L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right),$$

and with $\parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$, we have a unique global (in time) Sobolev solution

$$(u,v)\in C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)$$

to the Cauchy problem (2). Moreover, the solution satisfies the following decay estimate for any $s\ge 0$ and for all sufficiently small $\delta >0$:

$$\begin{array}{cc}\hfill {\parallel u(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\alpha }_1}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}for allq\in [m_1,\infty ],\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\parallel v(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }+{\alpha }_2}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}for allq\in [m_2,\infty ],\hfill \end{array}$$

(4)

where

$${\beta }_{\alpha ,q,\sigma }^{r,\delta }:=min\left\{\frac{n(1+\alpha )}{2\sigma }\left(\frac{1}{r}-\frac{1}{q}\right);1-\delta \right\}.$$

Example 1.

If we take in Theorem 1, $n=2$, ${\alpha }_1={\alpha }_2=\frac{1}{2}$, ${\sigma }_1={\sigma }_2=1$ and $m_1=m_2=1$. Then, the admissible range of global existence is

$$p_1>\frac{5}{2}and{p}_2>\frac{5}{2}.$$

Theorem 2.

(Existence of loss of decay) Let $0<{\alpha }_1,{\alpha }_2<1$; ${\sigma }_1,{\sigma }_2\ge 1$; and $m_1,m_2\ge 1$. We assume that $n\ge max\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$. Moreover, for all $ϵ>0$, the exponents $p_1$ and $p_2$ satisfy the following conditions:

$$\begin{array}{cc}\hfill p_1& <p_{{\alpha }_2,{\sigma }_2}^{m_1,m_2}(n,ϵ):=max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{1}{1-{\alpha }_2}\right\}\hfill \\ & p_1>max\{\frac{({\alpha }_1+1)(-n+2{\sigma }_1{m}_1)}{2{\sigma }_1{m}_1(1-{\alpha }_2)};\frac{{\alpha }_1}{1-{\alpha }_2};\frac{2{\sigma }_2{m}_2{\alpha }_1+m_{2}n(1+{\alpha }_2)}{n(1+{\alpha }_2)-2{\sigma }_2{m}_2};\hfill & \\ & m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1(1{\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2};\frac{m_2}{m_1}-ϵ;1\}\hfill \end{array}$$

and

$$p_2>p_{{\alpha }_1,{\sigma }_1}^{m_1,m_2}(n,ϵ):=max\left\{{\overline{p}}_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{(m_1,m_2)}(n,p_1);\frac{1}{1-{\alpha }_1},\frac{m_1}{m_2}-ϵ\right\},$$

where

$$\begin{array}{c}\hfill p_{\alpha ,\sigma }^r\left(n\right):=1+\frac{\left(n\right(r-1)+2\sigma r)(1+\alpha )}{(n-2\sigma r)(1+\alpha )+2\sigma r}\end{array}$$

and

$$\begin{array}{ccc}\hfill & {\overline{p}}_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{(m_1,m_2)}(n,p_1)=max\{\frac{2{\sigma }_2{m}_2}{n({\alpha }_1+1)(p_1-m_2)-2{\sigma }_2{m}_2({\alpha }_1+p_1{\alpha }_2)};\hfill & \\ & \frac{1}{p_1(1-{\alpha }_2)-{\alpha }_1};\frac{2{\sigma }_1{m}_1+n({\alpha }_1+1)m_1}{({\alpha }_1+1)(n-2{\sigma }_1{m}_1)+2p_1{\sigma }_1{m}_1(1-{\alpha }_2)};\hfill \\ & {\sigma }_2{m}_2\frac{2{\sigma }_1{m}_1+n{m}_1(1+{\alpha }_1)}{n[{\sigma }_2{m}_2({\alpha }_1+1)+{\sigma }_1{m}_1({\alpha }_2+1)(p_1-m_2)]-2{\sigma }_2{m}_2{\sigma }_1{m}_1[({\alpha }_1+1)+p_1{\alpha }_2)]}\}.\hfill \end{array}$$

Then, there exists a positive constant ε such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}:=\left(L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times \left(L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right),$$

and with $\parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$, we have a unique global (in time) Sobolev solution

$$(u,v)\in C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)$$

to the Cauchy problem (2). Moreover, the solution satisfies the following decay estimate for any $s\ge 0$ and for all sufficiently small $\delta >0$:

$$\begin{array}{cc}\hfill {\parallel u(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}for allq\in [m_1,\infty ],\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\parallel v(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }+{\alpha }_2}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}for allq\in [m_2,\infty ],\hfill \end{array}$$

(6)

where

$${\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }=1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)$$

and

$${\beta }_{\alpha ,q,\sigma }^{r,\delta }:=min\left\{\frac{n(1+\alpha )}{2\sigma }\left(\frac{1}{r}-\frac{1}{q}\right);1-\delta \right\}.$$

Example 2.

If we take in Theorem 2, $n=2$, ${\alpha }_1=\frac{1}{2}$, ${\alpha }_2=\frac{9}{10}$, ${\sigma }_1={\sigma }_2=1$ and $m_1=m_2=1$. Then, the admissible range of global existence is

$$5<p_1<10and{p}_2>\frac{10}{p_1-5}.$$

2.2. The Case $n<min\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$

Theorem 3.

Let $0<{\alpha }_1,{\alpha }_2<1$, $1\le {\sigma }_1<\frac{1+{\alpha }_1}{2{\alpha }_1}$, $1\le {\sigma }_2<\frac{1+{\alpha }_2}{2{\alpha }_2}$, $1\le m_1<\frac{1+{\alpha }_1}{2{\alpha }_1{\sigma }_1}$ and $1\le m_2<\frac{1+{\alpha }_2}{2{\alpha }_2{\sigma }_2}$. We assume that $1\le n<min\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$. Moreover, for all $ϵ>0$, the exponents $p_1$ and $p_2$ satisfy the conditions

$$p_1>p_{{\alpha }_2,{\sigma }_2}^{m_1,m_2}\left(n\right):=max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{m_2}{m_1}-ϵ\right\}$$

and

$$p_2>p_{{\alpha }_1,{\sigma }_1}^{m_1,m_2}\left(n\right):=max\left\{p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right);\frac{m_1}{m_2}-ϵ\right\},$$

where

$$\begin{array}{c}\hfill p_{\alpha ,\sigma }^r\left(n\right):=1+\frac{\left(n\right(r-1)+2\sigma r)(1+\alpha )}{(n-2\sigma r)(1+\alpha )+2\sigma r}.\end{array}$$

Then, there exists a positive constant ε such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}:=\left(L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times \left(L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right),$$

and with $\parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$, we have a unique global (in time) Sobolev solution

$$(u,v)\in C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)$$

to the Cauchy problem (2). Moreover, the solution satisfies the following decay estimate for any $s\ge 0$ and for all sufficiently small $\delta >0$:

$$\begin{array}{cc}\hfill {\parallel u(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}+{\alpha }_1}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}for allq\in [m_1,\infty ],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\parallel v(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}+{\alpha }_2}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}for allq\in [m_2,\infty ],\hfill \end{array}$$

(8)

where

$${\beta }_{\alpha ,q,\sigma }^r:=\frac{n(1+\alpha )}{2\sigma }\left(\frac{1}{r}-\frac{1}{q}\right).$$

Example 3.

If we take in Theorem 3, $n=1$, ${\alpha }_1={\alpha }_2=\frac{1}{2}$, ${\sigma }_1={\sigma }_2=1$ and $m_1=m_2=1$. Then, the admissible range of global existence is $p_1,p_2>7.$

Theorem 4.

(Existence of loss of decay) Let $0<{\alpha }_1,{\alpha }_2<1$, $1\le {\sigma }_1\le \frac{1+{\alpha }_1}{2{\alpha }_1}$, $1\le {\sigma }_2<\frac{1+{\alpha }_2}{2{\alpha }_2}$,$1\le m_1\le \frac{1+{\alpha }_1}{2{\alpha }_1{\sigma }_1}$ and $1\le m_2<\frac{1+{\alpha }_2}{2{\alpha }_2{\sigma }_2}$. We assume that $1\le n<min\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$. Moreover, for all $ϵ>0$, the exponents $p_1$ and $p_2$ satisfy the conditions

$$max\left\{1;m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1({\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2)};\frac{m_2}{m_1}-ϵ\right\}<p_1<p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right)$$

and

$$p_2>p_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{m_1,m_2}\left(n\right):=max\left\{{\overline{p}}_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{(m_1,m_2)}(n,p_1);\frac{m_1}{m_2}-ϵ\right\},$$

where

$$\begin{array}{c}\hfill p_{\alpha ,\sigma }^r\left(n\right):=1+\frac{\left(n\right(r-1)+2\sigma r)(1+\alpha )}{(n-2\sigma r)(1+\alpha )+2\sigma r}\end{array}$$

and

$$\begin{array}{cc}\hfill & {\overline{p}}_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{(m_1,m_2)}(n,p_1)=\hfill \\ & {\sigma }_2{m}_2\frac{2{\sigma }_1{m}_1+n{m}_1({\alpha }_1+1)}{n[{\sigma }_2{m}_2({\alpha }_1+1)+{\sigma }_1{m}_1({\alpha }_2+1)(p_1-m_2)]-2{\sigma }_2{m}_2{\sigma }_1{m}_1[({\alpha }_1+1)+p_1{\alpha }_2)]}.\hfill \end{array}$$

Then, there exists a positive constant ε such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}:=\left(L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times \left(L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right),$$

and with $\parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$, we have a unique global (in time) Sobolev solution

$$(u,v)\in C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)$$

to the Cauchy problem (2). Moreover, the solution satisfies the following decay estimate for any $s\ge 0$ and for all sufficiently small $\delta >0$:

$$\begin{array}{cc}\hfill {\parallel u(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}+{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}for allq\in [m_1,\infty ],\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\parallel v(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}+{\alpha }_2}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}for allq\in [m_2,\infty ],\hfill \end{array}$$

(10)

where

$${\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}=1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)$$

and

$${\beta }_{\alpha ,q,\sigma }^r:=\frac{n(1+\alpha )}{2\sigma }\left(\frac{1}{r}-\frac{1}{q}\right).$$

Example 4.

If we take in Theorem 4, $n=1$, ${\alpha }_1={\alpha }_2=\frac{1}{2}$, ${\sigma }_1={\sigma }_2=1$ and $m_1=m_2=1$. Then, the admissible range of global existence is $6<p_1<7$ and $p_2>\frac{7}{p_1-6}$.

2.3. The Case $min\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}\le n<max\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$

In the next section, we suppose that $\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1}<\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}$.

Theorem 5.

Let $0<{\alpha }_1,{\alpha }_2<1$, ${\sigma }_1\ge 1$, $1\le {\sigma }_2<\frac{1+{\alpha }_2}{2{\alpha }_2}$, $m_1\ge 1$ and $1\le m_2<\frac{1+{\alpha }_2}{2{\alpha }_2{\sigma }_2}$. We assume that $\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1}\le n<\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}.$ Moreover, for all $ϵ>0$, the exponents $p_1$ and $p_2$ satisfy the conditions

$$p_1>p_{{\alpha }_2,{\sigma }_2}^{m_1,m_2}(n,ϵ):=max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{m_2}{m_1}-ϵ\right\}$$

and

$$p_2>p_{{\alpha }_1,{\sigma }_1}^{m_1,m_2}(n,ϵ):=max\left\{p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right);\frac{1}{1-{\alpha }_1};\frac{m_1}{m_2}-ϵ\right\},$$

where

$$\begin{array}{c}\hfill p_{\alpha ,\sigma }^r\left(n\right):=1+\frac{\left(n\right(r-1)+2\sigma r)(1+\alpha )}{(n-2\sigma r)(1+\alpha )+2\sigma r}.\end{array}$$

Then, there exists a positive constant ε such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}:=\left(L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times \left(L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right),$$

and with $\parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$, we have a unique global (in time) Sobolev solution

$$(u,v)\in C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)$$

to the Cauchy problem (2). Moreover, the solution satisfies the following decay estimate for any $s\ge 0$ and for all sufficiently small $\delta >0$:

$$\begin{array}{cc}\hfill {\parallel u(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\alpha }_1}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}for allq\in [m_1,\infty ],\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\parallel v(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}+{\alpha }_2}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}for allq\in [m_2,\infty ],\hfill \end{array}$$

(12)

where

$${\beta }_{\alpha ,q,\sigma }^{r,\delta }:=min\left\{\frac{n(1+\alpha )}{2\sigma }\left(\frac{1}{r}-\frac{1}{q}\right);1-\delta \right\}$$

and

$${\beta }_{\alpha ,q,\sigma }^r:=\frac{n(1+\alpha )}{2\sigma }\left(\frac{1}{r}-\frac{1}{q}\right).$$

Example 5.

If we take in Theorem 5, $n=4$, ${\alpha }_1=\frac{1}{2}$, ${\alpha }_2=\frac{1}{8}$, ${\sigma }_1=1$, ${\sigma }_2=2$, $m_1=1$ and $m_2=2$. Then, the admissible range of global existence is $p_1>\frac{34}{4}$ and $p_2>\frac{8}{5}$.

Theorem 6.

(Existence of loss of decay) Let $0<{\alpha }_1,{\alpha }_2<1$,${\sigma }_1\ge 1$, $1\le {\sigma }_2<\frac{1+{\alpha }_2}{2{\alpha }_2}$ and $1\le m_2<\frac{1+{\alpha }_2}{2{\alpha }_2{\sigma }_2}$. We assume that $\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1}\le n<\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}$. Moreover, for all $ϵ>0$, the exponents $p_1$ and $p_2$ satisfy the conditions

$$\begin{array}{ccc}\hfill & p_1<p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right)\hfill & \\ & p_1>max\{\frac{2{\sigma }_2{m}_2{\alpha }_1+n(1+{\alpha }_2)m_2}{n(1+{\alpha }_2)-2{\sigma }_2{m}_2{\alpha }_2};\frac{m_2}{m_1}-ϵ,1\hfill & \\ & m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1({\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2)}\},\hfill \end{array}$$

and

$$\begin{array}{c}\hfill p_2>p_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{m_1,m_2}\left(n\right):=max\left\{{\overline{p}}_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{(m_1,m_2)}(n,p_1);\frac{m_1}{m_2}-ϵ\right\},\end{array}$$

where

$$\begin{array}{c}\hfill p_{\alpha ,\sigma }^r\left(n\right):=1+\frac{\left(n\right(r-1)+2\sigma r)(1+\alpha )}{(n-2\sigma r)(1+\alpha )+2\sigma r}\end{array}$$

and

$$\begin{array}{ccc}\hfill & {\overline{p}}_{({\alpha }_1,{\alpha }_2),\left({\sigma }_1{\sigma }_2\right)}^{(m_1,m_2)}(n,p_1)=max\{\frac{2{\sigma }_2{m}_2}{n({\alpha }_2+1)(p_1-m_2)-2{\sigma }_2{m}_2({\alpha }_1+p_1{\alpha }_2)};\hfill & \\ & {\sigma }_2{m}_2\frac{2{\sigma }_1{m}_1+n{m}_1({\alpha }_1+1)}{n[{\sigma }_2{m}_2({\alpha }_1+1)+{\sigma }_1{m}_1({\alpha }_2+1)(p_1-m_2)]-2{\sigma }_2{m}_2{\sigma }_1{m}_1[({\alpha }_1+1)+p_1{\alpha }_2)]}\}.\hfill \end{array}$$

Then, there exists a positive constant ε such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}:=\left(L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times \left(L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right),$$

and with $\parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$, we have a unique global (in time) Sobolev solution

$$(u,v)\in C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)$$

to the Cauchy problem (2). Moreover, the solution satisfies the following decay estimate for any $s\ge 0$ and for all sufficiently small $\delta >0$:

$$\begin{array}{cc}\hfill {\parallel u(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}for allq\in [m_1,\infty ],\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\parallel v(s,·)\parallel }_{L^q}& \lesssim {(1+s)}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}+{\alpha }_2}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}for allq\in [m_2,\infty ],\hfill \end{array}$$

(14)

where

$${\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}=1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)$$

$${\beta }_{\alpha ,q,\sigma }^{r,\delta }:=min\left\{\frac{n(1+\alpha )}{2\sigma }\left(\frac{1}{r}-\frac{1}{q}\right);1-\delta \right\}.$$

Example 6.

If we take in Theorem 6, $n=4$, ${\alpha }_1=\frac{1}{2}$, ${\alpha }_2=\frac{1}{8}$, ${\sigma }_1=1$, ${\sigma }_2=2$, $m_1=1$ and $m_2=2$. Then, the admissible range of global existence is $\frac{26}{7}<p_1<\frac{34}{4}$ and $p_2>\frac{16}{7p_1-26}$.

The notation $f\lesssim g$, which denotes the existence of a constant $C\ge 0$ such that $f\le Cg$, is used in the sections that follow.

3. Some Preliminaries

Let us consider the single Cauchy problem

$$\begin{array}{c}{\partial }_t^{1+\lambda }\varphi +{(-∆)}^{\sigma }\varphi ={\left|\varphi \right|}^p,\hfill \\ \varphi (0,x)={\varphi }_0\left(x\right),{\varphi }_t(0,x)=0,\hfill \end{array}$$

(15)

with $\lambda \in (0,1)$, $\sigma \ge 1$ and $p>1$. Under the data condition ${\varphi }_t(0,x)=0$, it can be formally converted to an integral equation and its solution is given by

$$\begin{array}{c}\hfill \varphi (t,x)=S_{\alpha }^{\sigma }(t,x){\ast }_{\left(x\right)}{\varphi }_0\left(x\right)+N_{\alpha }^{\sigma }\left(\varphi \right)(t,x)\end{array}$$

(16)

with

$$\begin{array}{cc}\hfill S_{\lambda }^{\sigma }(t,x)& {\displaystyle ={\int }_{{\mathbb{R}}^n}{e}^{ix·\xi }{E}_{\lambda +1}\left(-t^{\lambda +1}{\left|\xi \right|}^{2\sigma }\right)d\xi ,}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill N_{\lambda }^{\sigma }\left(\varphi \right)(t,x)& ={\int }_0^t{S}_{\lambda }^{\sigma }(t-s){*}_{\left(x\right)}{I}_s^{\lambda }{\left(\right|\varphi |}^p)(t,s,x)ds,\hfill \end{array}$$

(18)

where ${\left\{S_{\lambda }^{\sigma }(t,·)\right\}}_{t\ge 0}$ denotes the semigroup of operators, which is defined via Fourier transform by

$$\widehat{\left(S_{\lambda }^{\sigma }(t,·)f\right)}(t,\xi )=E_{\lambda +1}\left(-t^{\lambda +1}{\left|\xi \right|}^{2\sigma }\right)\widehat{f}\left(\xi \right).$$

Here, $E_{\gamma }\left(z\right)={\displaystyle \sum _{k=0}^{\infty }}\frac{z^k}{\mathrm{\Gamma }(\gamma k+1)}$ denotes the Mittag-Leffler function (see [25]).

As noted in [8], $\varphi (t,x)=S_{\lambda }^{\sigma }(t,x){\ast }_{\left(x\right)}{\varphi }_0(t,x)$ is a representation of solutions of the linear Cauchy problem associated with (15) with a vanishing right-hand side. In [8], the authors proved the following results.

Proposition 1

(see [8]). Let ${\varphi }_0\in L^m\cap L^{\infty }$, $m\ge 1$ and $\lambda \in (0,1)$. Then, the solution of the linear Cauchy problem

$$\begin{array}{c}{\partial }_t^{1+\lambda }\varphi +{(-∆)}^{\sigma }\varphi =0,\hfill \\ \varphi (0,x)={\varphi }_0\left(x\right),{\varphi }_t(0,x)=0:\hfill \end{array}$$

(19)

satisfies the following $L^m-L^q$ estimates:

$${\parallel \varphi (s,·)\parallel }_{L^q\left({\mathbb{R}}^n\right)}\lesssim {(1+s)}^{-{\beta }_{\lambda ,q,\sigma }^{m,\delta }}{\parallel {\varphi }_0\parallel }_{L^m\cap L^q}for allq\in [m,\infty ],$$

(20)

where

$${\beta }_{\lambda ,q,\sigma }^{m,\delta }:=min\left\{\frac{n(1+\lambda )}{2\sigma }\left(\frac{1}{m}-\frac{1}{q}\right);1-\delta \right\}.$$

Philosophy of the Approach and Proofs

The decay estimates for solutions to

$$\left\{\begin{array}{c}{\partial }_t^{1+{\alpha }_1}u+{(-∆)}^{{\sigma }_1}u=0,u(0,x)=u_0\left(x\right)u_t(0,x)=0,\hfill \\ {\partial }_t^{1+{\alpha }_2}v+{(-∆)}^{{\sigma }_2}v=0,v(0,x)=v_0\left(x\right),v_t(0,x)=0,\hfill \end{array}\right.$$

(21)

will be used to show the global (in time) existence of small data Sobolev solutions to the weakly coupled systems (2). We write their solutions in the following form:

$$\begin{array}{c}{u}^{ln}(s,x):=S_{{\alpha }_1}^{{\sigma }_1}(s,x){\ast }_{\left(x\right)}{u}_0\left(x\right),v^{ln}(s,x):=S_{{\alpha }_2}^{{\sigma }_2}(s,x){\ast }_{\left(x\right)}{v}_0\left(x\right).\hfill \end{array}$$

(22)

Proposition 2.

Let $u_0\in L^{m_1}\cap L^{\infty }$, $v_0\in L^{m_2}\cap L^{\infty }$ and $m_1,m_2\ge 1$. Then, the solution of the linear Cauchy problem (21) satisfies the following $L^m-L^q$ estimates with $m=m_1$ or $m=m_2$:

$$\begin{array}{cc}\hfill & \parallel u^{ln}(s,·){\parallel }_{L^q}\lesssim {(1+s)}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\parallel u_0\parallel }_{L^{m_1}\cap L^q}forallq\in [m_1,\infty ],\hfill \\ & \parallel v^{ln}(s,·){\parallel }_{L^q}\lesssim {(1+s)}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }}{\parallel v_0\parallel }_{L^{m_2}\cap L^q}forallq\in [m_2,\infty ].\hfill \end{array}$$

(23)

Applying Duhamel’s principle and some fixed point argument give the formal integral formulation of solutions to (2) as follows:

$$\begin{array}{c}u(s,x):=u^{ln}(s,x)+{\int }_0^s{S}_{{\alpha }_1}^{{\sigma }_1}(s-\rho ,·){*}_{\left(x\right)}{\left|v(\rho ,·)\right|}^{p_1}d\rho =(u^{ln}+u^{nl})(s,x),\hfill \\ v(s,x):=v^{ln}(s,x)+{\int }_0^s{S}_{{\alpha }_2}^{{\sigma }_2}(s-\rho ,·){*}_{\left(x\right)}{\left|u(\rho ,·)\right|}^{p_2}d\rho =(v^{ln}+v^{nl})(s,x).\hfill \end{array}$$

(24)

4. Proof of Main Results

Let us recall the lemma from [26] before presenting our proofs.

Lemma 1.

Suppose that $a,b\ge 0$ and $\theta \in [0,1)$. Then, there exists a constant $K=K(a,b,\theta )>0$ such that for all $\tau >0$ the following estimate holds:

$$\begin{array}{c}\hfill \begin{array}{c}{\int }_0^{\tau }{(\tau -\rho )}^{-\theta }{(1+\tau -\rho )}^{-a}{(1+\rho )}^{-b}d\rho \hfill \\ \le \left\{\begin{array}{cc}K{(1+\tau )}^{-min\{\theta +a,b\}}\hfill & \mathit{if}max\{\theta +a,b\}>1,\hfill \\ K{(1+\tau )}^{-min\{\theta +a,b\}}ln(2+t)\hfill & \mathit{if}max\{\theta +a,b\}=1,\hfill \\ K{(1+\tau )}^{1-\theta -a-b}\hfill & \mathit{if}max\{\theta +a,b\}<1.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(25)

4.1. Proof of Main Results fo the Case $n\ge max\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$

4.1.1. Proof of Theorem 1

Let $i=1;2$. For any $n\ge \frac{2{\sigma }_i{m}_i}{1+{\alpha }_i}$, and if $\delta \in (0,1)$ is sufficiently small, there exists a parameter $\overline{q_i}=\overline{q_i}\left(\delta \right)\in (m_i,\infty )$ such that

$$\frac{n(1+{\alpha }_i)}{2{\sigma }_i}\left(\frac{1}{m_i}-\frac{1}{\overline{q_i}}\right)=1-\delta .$$

(26)

Let $T>0$. We define the space $\mathcal{X}\left(T\right)$ as follows:

$$\begin{array}{c}\hfill \mathcal{X}\left(T\right):=C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\end{array}$$

equipped with the norm

$$\begin{array}{cc}\hfill {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}:=& \underset{0⩽s⩽T}{sup}\left\{{\mathcal{R}}_1(s,u)+{\mathcal{R}}_2(s,v)\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}& {\mathcal{R}}_1(s,u):={(1+s)}^{-{\alpha }_1}{\parallel u(s,·)\parallel }_{L^{m_1}}+{(1+s)}^{1-\delta -{\alpha }_1}\left({\parallel u(s,·)\parallel }_{L^{\overline{q_1}}}+{\parallel u(s,·)\parallel }_{L^{\infty }}\right),\hfill \\ & {\mathcal{R}}_2(s,v)={(1+s)}^{-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{m_2}}+{(1+s)}^{1-\delta -{\alpha }_2}\left({\parallel v(s,·)\parallel }_{L^{\overline{q_2}}}+{\parallel v(s,·)\parallel }_{L^{\infty }}\right),\hfill \end{array}$$

and the operator P by

$$P:(u,v)\in \mathcal{X}\left(T\right)\to P(u,v)=P(u,v)(s,x):={(u,v)}^{ln}(s,x)+{(u,v)}^{nl}(s,x).$$

To obtain the global (in time) existence and uniqueness of Sobolev solutions in $\mathcal{X}\left(T\right)$, we can consider a global (in time) Sobolev solution to (2) as a fixed point of the operator P. We will prove that P satisfies, for any $(u,v)$, $(\tilde{u},\overline{v})\in \mathcal{X}\left(T\right)$, the next inequalities:

$$\begin{array}{ccc}\hfill & & {\parallel P(u,v)\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}+\sum _{i=1}^2{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_i},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}& & \parallel P(u,v)-P(\tilde{u},\overline{v}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)-(\tilde{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\sum _{i=1}^2\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}+{\parallel (\tilde{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}\right),\hfill \end{array}$$

(28)

A global (in time) well-posedness result for small data Sobolev solutions is also obtained from the estimates (27) and (28).

By applying the definition of the norm in $\mathcal{X}\left(T\right)$ and Proposition 2, we may conclude

$${\parallel (u,v)}^{ln}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u_0,v_0)\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}.$$

Hence, it is reasonable to show the following inequality to complete the proof of (27)

$${\parallel (u,v)}^{nl}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

If $(u,v)\in \mathcal{X}\left(T\right)$, then by interpolation we derive for all $s\in [0,T]$

$$\begin{array}{cc}\hfill & {\parallel u(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_1,\infty ],\hfill \\ & {\parallel v(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_2,\infty ].\hfill \end{array}$$

On the other hand, we have

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|u(s,·)\right|}^{p_2}{\parallel }_{L^q}\hfill & \le {\parallel u(s,·)\parallel }_{L^{p_{2}q}}^{p_2}\lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1,\delta }-{\alpha }_1)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}\hfill \\ & \lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}\end{array}$$

(29)

for any q such that $p_{2}q\in [m_1,\infty ]$ and due to ${\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1,\delta }\ge {\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }$.

Also,

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}\hfill & \le {\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}\lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \\ & \lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \end{array}\end{array}$$

(30)

for any q such that $p_{1}q\in [m_2,\infty ]$ and due to ${\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2,\delta }\ge {\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }$. Thanks to (30), we have for $q\in [m_1,\infty ]$ the estimates

$$\begin{array}{cc}\hfill \parallel u^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}dsd\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}dsd\rho }\hfill \\ & \lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}{\mathcal{J}}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{1}q\in [m_2,\infty ],\hfill \end{array}$$

where

$${\displaystyle {\mathcal{J}}_1\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}dsd\rho .}$$

(31)

The right side of (31) is what we are interested in estimating. We apply Lemma 1 for this. We put

$${\displaystyle \omega \left(\rho \right)={\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}ds.}$$

By applying Lemma 1, we obtain $\omega \left(\rho \right)\lesssim {(1+\rho )}^{{\alpha }_1-1}$ if we assume that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)>1$. We notice that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)>1$ if and only if

$$p_1>max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{1}{1-\delta -{\alpha }_2}\right\}.$$

On the other hand, the conditions $q\in [m_1,\infty ]$ and $p_{1}q\in [m_2,\infty ]$ imply $p_1\ge \frac{m_2}{m_1}$.

Hence,

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{J}}_1\left(t\right)\lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}\omega \left(\rho \right)d\rho \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{(1+\rho )}^{{\alpha }_1-1}d\rho .}\end{array}$$

(32)

Once more, we apply Lemma 1 to (32) to obtain ${\mathcal{J}}_1\left(t\right)\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}.$

Hence,

$$\parallel (u^{nl},0){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}.$$

Additionally, we have for $q\in [m_2,\infty ]$

$$\begin{array}{c}\hfill \parallel v^{nl}{(t,·)\parallel }_{L^q}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}{\mathcal{J}}_2\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{2}q\in [m_1,\infty ],\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{J}}_2\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_2-1}{(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1)}dsd\rho .}\end{array}$$

(33)

If $p_2>max\left\{p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right);\frac{1}{1-\delta -{\alpha }_1}\right\}$, then ${\mathcal{J}}_2\left(t\right)\lesssim {(1+t)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }}$.

Hence,

$$\parallel ((0,v^{nl}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

Also, the conditions $q\in [m_2,\infty ]$ and $p_{2}q\in [m_1,\infty ]$ imply $p_2\ge \frac{m_1}{m_2}$.

To prove (28), we assume that $(u,v)$ and $(\tilde{u},\overline{v})$ are two vector functions belonging to $\mathcal{X}\left(T\right)$. Then, we have

$$\begin{array}{cc}\hfill & P(u,v)-P(\tilde{u},\overline{v})\hfill \\ & =({\int }_0^t{S}_{{\alpha }_1}^{{\sigma }_1}(t-s){\ast }_{\left(x\right)}{I}_s^{{\alpha }_1}{\left(\right|v(s,·)|}^{p_1}-|\overline{v}(s,·){|}^{p_1})(t,s,x)ds,\hfill \\ & {\int }_0^t{G}_{{\alpha }_2}^{{\sigma }_2}(t-s){\ast }_{\left(x\right)}{I}_s^{{\alpha }_2}{\left(\right|u(s,·)|}^{p_2}-|\tilde{u}(s,·){|}^{p_2})(t,s,x)ds).\hfill \end{array}$$

We have for $q\in [m_1,\infty ]$

$$\begin{array}{cc}\hfill \parallel {\int }_0^t& S_{{\alpha }_1}^{{\sigma }_1}(t-s){\ast }_{\left(x\right)}{I}_s^{{\alpha }_1}{\left(\right|v(s,·)|}^{p_1}-|\overline{v}(s,·){|}^{p_1})(t,s,·)ds{\parallel }_{L^q}\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel \left|v(s,·)\right|}^{p_1}-{\left|\overline{v}(s,·)\right|}^{p_1}{\parallel }_{L^q}dsd\rho .}\hfill \end{array}$$

By using Hölder’s inequality, we obtain

$$\begin{array}{cc}\hfill {\parallel \left|v(s,·)\right|}^{p_1}& -|\overline{v}{(s,·)|}^{p_1}{\parallel }_{L^q}\hfill \\ & \lesssim \parallel v(s,·)-\overline{v}(s,·){\parallel }_{L^{q{p}_1}}\left({\parallel v(s,·)\parallel }_{L^{q{p}_1}}^{p_1-1}+{\parallel \overline{v}(s,·)\parallel }_{L^{q{p}_1}}^{p_1-1}\right).\hfill \end{array}$$

By using the definition of the norm of the solution space $\mathcal{X}\left(T\right)$ we obtain for $p_1\ge \frac{m_2}{m_1}$ and $0⩽s⩽t$ the following estimates:

$$\begin{array}{ccc}& & \parallel v(s,·)-\overline{v}(s,·){\parallel }_{L^{q{p}_1}}\lesssim {(1+s)}^{{\alpha }_2-{\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }}{\mathcal{R}}_2(s,v-\overline{v}),\hfill \\ & & {\parallel v(s,·)\parallel }_{L^{q{p}_1}}^{p_1-1}\lesssim {(1+s)}^{(p_1-1)({\alpha }_2-{\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta })}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p-1},\hfill \\ & & \parallel \overline{v}(s,·){\parallel }_{L^{q{p}_1}}^{p_1-1}\lesssim {(1+s)}^{(p_1-1)({\alpha }_2-{\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta })}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p-1}.\hfill \end{array}$$

Hence, we obtain

$$\begin{array}{cc}\hfill \parallel |& {v(s,·)|}^{p_1}-{\left|\overline{v}(s,·)\right|}^{p_1}{\parallel }_{L^q}\hfill \\ & \lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}{\mathcal{R}}_2(s,v-\overline{v})\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_1-1}\right)\hfill \\ & \lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}{\parallel (u,v)-(\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_1-1}\right).\hfill \end{array}$$

By the same arguments, we obtain for $p_2\ge \frac{m_1}{m_2}$ and $0⩽s⩽t$ the following estimate:

$$\begin{array}{cc}\hfill \parallel |& {u(s,·)|}^{p_2}-{\left|\overline{u}(s,·)\right|}^{p_2}{\parallel }_{L^q}\hfill \\ & \lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1)}{\parallel (u,v)-(\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_2-1}\right).\hfill \end{array}$$

So, for

$$p_1>max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{1}{1-\delta -{\alpha }_2};\frac{m_2}{m_1}-ϵ\right\}$$

and

$$p_2>max\left\{p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right);\frac{1}{1-\delta -{\alpha }_1};\frac{m_1}{m_2}-ϵ\right\}$$

we obtain the desired estimate (28).

Notice that

$$p_1>max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{1}{1-\delta -{\alpha }_2};\frac{m_2}{m_1}-ϵ\right\}$$

and

$$p_2>max\left\{p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right);\frac{1}{1-\delta -{\alpha }_1};\frac{m_1}{m_2}-ϵ\right\}$$

for all $\delta >0$ if and only if

$$p_1>p_{{\alpha }_2,{\sigma }_2}^{m_1,m_2}(n,ϵ):=max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{1}{1-{\alpha }_2};\frac{m_2}{m_1}-ϵ\right\}$$

and

$$p_2>p_{{\alpha }_1,{\sigma }_1}^{m_1,m_2}(n,ϵ):=max\left\{p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right);\frac{1}{1-{\alpha }_1};\frac{m_1}{m_2}-ϵ\right\}.$$

Remark 1.

All estimates (27) and (28) are uniform with respect to $T\in (0,\infty )$ if $p_1>p_{{\alpha }_2,{\sigma }_2}^{m_1,m_2}(n,ϵ)$ and $p_2>p_{{\alpha }_1,{\sigma }_1}^{m_1,m_2}(n,ϵ)$ for all $ϵ>0$.

It follows from (27) that for any T and small data, P maps $\mathcal{X}\left(T\right)$ into itself. The estimates (27) and (28) result in the existence of a single fixed point for $(u,v)=P(u,v)$ by conventional contraction arguments. As a result, we obtain Sobolev solutions to (2) that are well-posed and satisfy the required decay estimates. Since all constants are independent of T, we establish a global (in time) existence result for small data Sobolev solutions to (2) by letting T tend to ∞. The proof is now complete.

4.1.2. Proof of Theorem 2

Let $T>0$. We define the space $\mathcal{X}\left(T\right)$ as follows:

$$\begin{array}{c}\hfill \mathcal{X}\left(T\right):=C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\end{array}$$

equipped with the norm

$$\begin{array}{cc}\hfill {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}:=& \underset{0⩽s⩽T}{sup}\left\{{\mathcal{R}}_1(s,u)+{\mathcal{R}}_2(s,v)\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}& {\mathcal{R}}_1(s,u)={(1+s)}^{-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }}{\parallel u(s,·)\parallel }_{L^{m_1}}+{(1+s)}^{1-\delta -{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }}\hfill \\ & \times \left({\parallel u(s,·)\parallel }_{L^{\overline{q_1}}}+{\parallel u(s,·)\parallel }_{L^{\infty }}\right),\hfill \\ & {\mathcal{R}}_2(s,v)={(1+s)}^{-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{m_2}}+{(1+s)}^{1-\delta -{\alpha }_2}\left({\parallel v(s,·)\parallel }_{L^{\overline{q_2}}}+{\parallel v(s,·)\parallel }_{L^{\infty }}\right),\hfill \end{array}$$

where $\overline{q_1}$ and $\overline{q_2}$ are defined as in the proof of Theorem 1. Finally, the operator P is defined by

$$P:(u,v)\in \mathcal{X}\left(T\right)\to P(u,v)=P(u,v)(s,x):={(u,v)}^{ln}(s,x)+{(u,v)}^{nl}(s,x).$$

We will prove that P satisfies, for any $(u,v)$ and $(\overline{u},\overline{v})\in \mathcal{X}\left(T\right)$, the next two inequalities

$$\begin{array}{ccc}\hfill & & {\parallel P(u,v)\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}+\parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}$$

(34)

$$\begin{array}{ccc}& & \parallel P(u,v)-P(\overline{u},\overline{v}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)-(\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\sum _{i=1}^2\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}\right),\hfill \end{array}$$

(35)

By applying the definition of the norm in $\mathcal{X}\left(T\right)$ and Proposition 2, we may conclude

$${\parallel (u,v)}^{ln}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u_0,v_0)\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}.$$

Hence, it is reasonable to show the following inequality to complete the proof of (34)

$${\parallel (u,v)}^{nl}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

If $(u,v)\in \mathcal{X}\left(T\right)$, then we derive for all $s\in [0,T]$

$$\begin{array}{cc}\hfill & {\parallel u(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_1,\infty ],\hfill \\ & {\parallel v(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_2,\infty ].\hfill \end{array}$$

On the other hand, we have

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|u(s,·)\right|}^{p_2}{\parallel }_{L^q}\hfill & \le {\parallel u(t,·)\parallel }_{L^{p_{2}q}}^{p_2}\lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta })}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}\hfill \\ & \lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta })}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}\end{array}$$

(36)

for any q such that $p_{2}q\in [m_1,\infty ]$ and due to ${\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1,\delta }\ge {\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }$.

Also,

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}\hfill & \le {\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}\lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \\ & \lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \end{array}\end{array}$$

(37)

for any q such that $p_{1}q\in [m_2,\infty ]$ and due to ${\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2,\delta }\ge {\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }$. Thanks to (37), we have for $q\in [m_1,\infty ]$ the estimates

$$\begin{array}{cc}\hfill \parallel u^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}dsd\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}dsd\rho }\hfill \\ & \lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}{\mathcal{J}}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{1}q\in [m_2,\infty ],\hfill \end{array}$$

where

$${\displaystyle {\mathcal{J}}_1\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}dsd\rho .}$$

(38)

The right side of (38) is what we are interested in estimating. We apply Lemma 1 for this. We put

$${\displaystyle \omega \left(\rho \right)={\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}ds.}$$

By applying Lemma 1, we obtain $\omega \left(\rho \right)\lesssim {(1+\rho )}^{{\alpha }_1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}$ if we assume that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)<1$. We notice that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)<1$ if and only if

$$p_1<max\left\{p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right);\frac{1}{1-\delta -{\alpha }_2}\right\}.$$

On the other hand, the conditions $q\in [m_1,\infty ]$ and $p_{1}q\in [m_2,\infty ]$ imply $p_1\ge \frac{m_2}{m_1}$.

Hence,

$$\begin{array}{cc}\hfill {\mathcal{J}}_1\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}\omega \left(\rho \right)d\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{(1+\rho )}^{{\alpha }_1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}d\rho .}\hfill \end{array}$$

(39)

Once more, we apply Lemma 1 to (39) to obtain

$${\mathcal{J}}_1\left(t\right)\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2)}.$$

Hence,

$$\parallel (u^{nl},0){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}.$$

Additionally, we have for $q\in [m_2,\infty ]$

$$\begin{array}{c}\hfill \parallel v^{nl}{(t,·)\parallel }_{L^q}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}{\mathcal{J}}_2\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{2}q\in [m_1,\infty ],\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{J}}_2\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_2-1}{(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-1+p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2))}dsd\rho .}\end{array}$$

(40)

If

$$p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-1+p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2))>1,$$

then ${\mathcal{J}}_2\left(t\right)\lesssim {(1+t)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }}$.

Hence,

$$\parallel ((0,v^{nl}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

The condition

$$p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-1+p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2,\delta }-{\alpha }_2))>1,$$

is equivalent to

$$\begin{array}{cc}\hfill & p_2>max\{\frac{2{\sigma }_2{m}_2}{n({\alpha }_1+1)(p_1-m_2)-2{\sigma }_2{m}_2(\delta +{\alpha }_1+p_1{\alpha }_2)};\frac{1}{-\delta -{\alpha }_1+p_1(1-\delta -{\alpha }_2)}\hfill \\ & ;{\sigma }_2{m}_2\frac{2{\sigma }_1{m}_1+n{m}_1({\alpha }_1+1)}{n[{\sigma }_2{m}_2({\alpha }_1+1)+{\sigma }_1{m}_1({\alpha }_2+1)(p_1-m_2)]-2{\sigma }_2{m}_2{\sigma }_1{m}_1[({\alpha }_1+1)+p_1{\alpha }_2)]};\hfill \\ & \frac{2{\sigma }_1{m}_1+n({\alpha }_1+1)m_1}{({\alpha }_1+1)(n-2{\sigma }_1{m}_1)+2p_1{\sigma }_1{m}_1(1-\delta -{\alpha }_2)}\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill p_1>max\{& \frac{({\alpha }_1+1)(-n+2{\sigma }_1{m}_1)}{2{\sigma }_1{m}_1(1-\delta -{\alpha }_2)};\frac{\delta +{\alpha }_1}{1-\delta -{\alpha }_2};\frac{2{\sigma }_2{m}_2(\delta +{\alpha }_1)+m_{2}n(1+{\alpha }_2)}{n(1+{\alpha }_2)-2{\sigma }_2{m}_2};\hfill \\ & m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1({\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2}\}.\hfill \end{array}$$

Also, the conditions $q\in [m_2,\infty ]$ and $p_{2}q\in [m_1,\infty ]$ imply $p_2\ge \frac{m_1}{m_2}$.

The proof of (35) is similar to the proof of (28) of the proof to Theorem 1. This completes the proof.

4.2. Proof Main Results for the Case $n<min\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$

4.2.1. Proof of Theorem 3

If $1\le n<min\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$, then for all $i=1;2$ and for all $q\in [m_i,\infty ]$ we obtain

$$\frac{n(1+{\alpha }_i)}{2{\sigma }_i}\left(\frac{1}{m_i}-\frac{1}{q}\right)<1-\frac{n(1+{\alpha }_i)}{2q{\sigma }_i}\le 1.$$

(41)

Hence, we can choose a positive $\delta$ such that there does not exist any $\overline{q}\in [m_i,\infty ]$ that satisfies (26). For this reason,

$${\beta }_{{\alpha }_i,q,{\sigma }_i}^{m_i,\delta }={\beta }_{{\alpha }_i,q,{\sigma }_i}^{m_i}:=\frac{n(1+{\alpha }_i)}{2{\sigma }_i}\left(\frac{1}{m_i}-\frac{1}{q}\right).$$

Let $T>0$. We define the space $\mathcal{X}\left(T\right)$ as follows:

$$\begin{array}{c}\hfill \mathcal{X}\left(T\right):=C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\end{array}$$

equipped with the norm

$$\begin{array}{cc}\hfill {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}:=& \underset{0⩽s⩽T}{sup}\left\{{\mathcal{R}}_1(s,u)+{\mathcal{R}}_2(s,v)\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}& {\mathcal{R}}_1(s,u):={(1+s)}^{-{\alpha }_1}{\parallel u(s,·)\parallel }_{L^{m_1}}+{(1+s)}^{{\beta }_{{\alpha }_1,\infty ,{\sigma }_1}^{m_1}-{\alpha }_1}{\parallel u(s,·)\parallel }_{L^{\infty }},\hfill \\ & {\mathcal{R}}_2(s,v)={(1+s)}^{-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{m_2}}+{(1+s)}^{{\beta }_{{\alpha }_2,\infty ,{\sigma }_2}^{m_2}-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{\infty }},\hfill \end{array}$$

where ${\beta }_{\alpha ,\infty ,\sigma }^r=\frac{n(1+\alpha )}{2\sigma r}$. For any $(u,v)\in \mathcal{X}\left(T\right)$, the operator P

$$P:(u,v)\in \mathcal{X}\left(T\right)\to P(u,v)=P(u,v)(s,x):={(u,v)}^{ln}(s,x)+{(u,v)}^{nl}(s,x).$$

We will prove that P satisfies, for any $(u,v)$ and $(\overline{u},\overline{v})\in \mathcal{X}\left(T\right)$, the next two inequalities

$$\begin{array}{ccc}\hfill & & {\parallel P(u,v)\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}+\parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & \parallel P(u,v)-P(\overline{u},\overline{v}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)-(\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\sum _{i=1}^2\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}\right).\hfill \end{array}$$

(43)

By applying the definition of the norm in $\mathcal{X}\left(T\right)$ and Proposition 2, we may conclude

$${\parallel (u,v)}^{ln}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u_0,v_0)\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}.$$

Hence, it is reasonable to show the following inequality to complete the proof of (42),

$${\parallel (u,v)}^{nl}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

If $(u,v)\in \mathcal{X}\left(T\right)$, then we derive for all $s\ge 0$

$$\begin{array}{cc}\hfill & {\parallel u(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_1,\infty ],\hfill \\ & {\parallel v(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_2,\infty ].\hfill \end{array}$$

On the other hand, we have

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|u(s,·)\right|}^{p_2}{\parallel }_{L^q}\hfill & \le {\parallel u(s,·)\parallel }_{L^{p_{2}q}}^{p_2}\lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1}-{\alpha }_1)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}\hfill \\ & \lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1}-{\alpha }_1)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}\end{array}$$

(44)

for any q such that $p_{2}q\in [m_1,\infty ]$ and due to ${\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1}\ge {\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1}$.

Also,

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}\hfill & \le {\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}\lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2}-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \\ & \lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1},\hfill \end{array}\end{array}$$

(45)

for any q such that $p_{1}q\in [m_2,\infty ]$ and due to ${\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2}\ge {\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}$. Thanks to (45), we have for $q\in [m_1,\infty ]$ the estimates

$$\begin{array}{cc}\hfill \parallel u^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}dsd\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}dsd\rho }\hfill \\ & \lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}{\mathcal{J}}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{1}q\in [m_2,\infty ],\hfill \end{array}$$

where

$${\displaystyle {\mathcal{J}}_1\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}dsd\rho .}$$

(46)

The right side of (46) is what we are interested in estimating. We apply Lemma 1 for this. We put

$${\displaystyle \omega \left(\rho \right)={\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}ds.}$$

By applying Lemma 1, we obtain $\omega \left(\rho \right)\lesssim {(1+\rho )}^{{\alpha }_1-1}$ if we assume that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)>1$. We notice that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)>1$ if and only if

$$p_1>p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right):=1+\frac{(n(m_2-1)+2{\sigma }_2{m}_2)(1+{\alpha }_2)}{(n-2{\sigma }_2{m}_2)(1+{\alpha }_2)+2\sigma m_2}$$

under the assumptions $1\le {\sigma }_2<\frac{1+{\alpha }_2}{2{\alpha }_2}$ and $1\le m_2<\frac{1+{\alpha }_2}{2{\alpha }_2{\sigma }_2}$.

On other hand, the conditions $q\in [m_1,\infty ]$ and $p_{1}q\in [m_2,\infty ]$ imply $p_1\ge \frac{m_2}{m_1}$.

Hence,

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{J}}_1\left(t\right)\lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}\omega \left(\rho \right)d\rho \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{(1+\rho )}^{{\alpha }_1-1}d\rho .}\end{array}$$

(47)

Once more, we apply Lemma 1 to (47) to obtain ${\mathcal{J}}_1\left(t\right)\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}.$

Hence,

$$\parallel (u^{nl},0){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}.$$

Additionally, we have for $q\in [m_2,\infty ]$

$$\begin{array}{c}\hfill \parallel v^{nl}{(t,·)\parallel }_{L^q}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}{\mathcal{J}}_2\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{2}q\in [m_1,\infty ],\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{J}}_2\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_2-1}{(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1}-{\alpha }_1)}dsd\rho .}\end{array}$$

(48)

If

$$p_2>p_{{\alpha }_1,{\sigma }_1}^{m_1}\left(n\right):=1+\frac{(n(m_1-1)+2{\sigma }_1{m}_1)(1+{\alpha }_1)}{(n-2{\sigma }_1{m}_1)(1+{\alpha }_1)+2\sigma m_1},$$

then

$${\mathcal{J}}_2\left(t\right)\lesssim {(1+t)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}.$$

Hence,

$$\parallel ((0,v^{nl}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

Also, the conditions $q\in [m_2,\infty ]$ and $p_{2}q\in [m_1,\infty ]$ imply $p_2\ge \frac{m_1}{m_2}$.

The proof of (43) is similar to the proof of (28) of Theorem 1.

The proof is complete.

4.2.2. Proof of Theorem 4

Let $T>0$. We define the space $\mathcal{X}\left(T\right)$ as follows:

$$\begin{array}{c}\hfill \mathcal{X}\left(T\right):=C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\end{array}$$

equipped with the norm

$$\begin{array}{cc}\hfill {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}:=& \underset{0⩽s⩽T}{sup}\left\{{\mathcal{R}}_1(s,u)+{\mathcal{R}}_2(s,v)\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}& {\mathcal{R}}_1(s,u):={(1+s)}^{-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel u(s,·)\parallel }_{L^{m_1}}+{(1+s)}^{{\beta }_{{\alpha }_1,\infty ,{\sigma }_1}^{m_1}-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel u(s,·)\parallel }_{L^{\infty }},\hfill \\ & {\mathcal{R}}_2(s,v)={(1+s)}^{-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{m_2}}+{(1+s)}^{{\beta }_{{\alpha }_2,\infty ,{\sigma }_2}^{m_2}-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{\infty }},\hfill \end{array}$$

where ${\beta }_{\alpha ,\infty ,\sigma }^r=\frac{n(1+\alpha )}{2\sigma r}$ and ${\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}=1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)$.

For any $(u,v)\in \mathcal{X}\left(T\right)$, the operator P

$$p:(u,v)\in \mathcal{X}\left(T\right)\to P(u,v)=P(u,v)(s,x):={(u,v)}^{ln}(s,x)+{(u,v)}^{nl}(s,x).$$

We will prove that the operator P satisfies, for any $(u,v)$ and $(\overline{u},\overline{v})\in \mathcal{X}\left(T\right)$, the next two inequalities

$$\begin{array}{ccc}\hfill & & {\parallel P(u,v)\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}+\parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & \parallel P(u,v)-P(\overline{u},\overline{v}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)-(\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\sum _{i=1}^2\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}\right).\hfill \end{array}$$

(50)

By applying the definition of the norm in $\mathcal{X}\left(T\right)$ and Proposition 2, we may conclude

$${\parallel (u,v)}^{ln}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u_0,v_0)\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}.$$

Hence, it is reasonable to show the following inequality to complete the proof of (49)

$${\parallel (u,v)}^{nl}{\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

If $(u,v)\in \mathcal{X}\left(T\right)$, then we derive for all $s\in [0,T]$,

$$\begin{array}{cc}\hfill & {\parallel u(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}+{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_1,\infty ],\hfill \\ & {\parallel v(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_2,\infty ].\hfill \end{array}$$

On the other hand, we have

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|u(s,·)\right|}^{p_2}{\parallel }_{L^q}\hfill & \le {\parallel u(s,·)\parallel }_{L^{p_{2}q}}^{p_2}\lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1}-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2})}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}\hfill \\ & \lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1}-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2})}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}\end{array}$$

(51)

for any q such that $p_{2}q\in [m_1,\infty ]$ and due to ${\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1}\ge {\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1}$.

Also,

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}\hfill & \le {\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}\lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2}-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \\ & \lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \end{array}\end{array}$$

(52)

for any q such that $p_{1}q\in [m_2,\infty ]$ and due to ${\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2}\ge {\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}$. Thanks to (52), we have for $q\in [m_1,\infty ]$ the estimates

$$\begin{array}{cc}\hfill \parallel u^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}dsd\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}dsd\rho }\hfill \\ & \lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}{\mathcal{J}}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{1}q\in [m_2,\infty ],\hfill \end{array}$$

where

$${\displaystyle {\mathcal{J}}_1\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}dsd\rho .}$$

(53)

The right side of (53) is what we are interested in estimating. We apply Lemma 1 for this. We put

$${\displaystyle \omega \left(\rho \right)={\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}ds.}$$

By applying Lemma 1, we obtain $\omega \left(\rho \right)\lesssim {(1+\rho )}^{{\alpha }_1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}$ if we assume that

$$p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)<1.$$

We notice that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)<1$ if and only if

$$p_1<p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right):=1+\frac{(n(m_2-1)+2{\sigma }_2{m}_2)(1+{\alpha }_2)}{(n-2{\sigma }_2{m}_2)(1+{\alpha }_2)+2\sigma m_2},$$

under the assumptions $1\le {\sigma }_2<\frac{1+{\alpha }_2}{2{\alpha }_2}$ and $1\le m_2<\frac{1+{\alpha }_2}{2{\alpha }_2{\sigma }_2}$.

On the other hand, the conditions $q\in [m_1,\infty ]$ and $p_{1}q\in [m_2,\infty ]$ imply $p_1\ge \frac{m_2}{m_1}$.

Hence,

$$\begin{array}{cc}\hfill {\mathcal{J}}_1\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}\omega \left(\rho \right)d\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}}{(1+\rho )}^{{\alpha }_1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}d\rho .}\hfill \end{array}$$

(54)

Once more, we apply Lemma 1 to (54) to obtain

$${\mathcal{J}}_1\left(t\right)\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}+1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}+{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}.$$

Hence,

$$\parallel (u^{nl},0){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1}+{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}.$$

Additionally, we have for $q\in [m_2,\infty ]$

$$\begin{array}{c}\hfill \parallel v^{nl}{(t,·)\parallel }_{L^q}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}{\mathcal{J}}_2\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{2}q\in [m_1,\infty ],\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{J}}_2\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_2-1}{(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1}-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2})}dsd\rho .}\end{array}$$

(55)

If

$$\begin{array}{c}\hfill p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1}-{\alpha }_1-1+p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2))>1,\end{array}$$

(56)

then

$${\mathcal{J}}_2\left(t\right)\lesssim {(1+t)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}.$$

The condition (56) is equivalent to

$$\begin{array}{c}\hfill p_1>m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1({\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2)}\end{array}$$

and

$$\begin{array}{c}\hfill p_2>\frac{1}{2{\sigma }_1}\frac{2{\sigma }_1+n(1+{\alpha }_1)}{n[(1+{\alpha }_1){\sigma }_2{m}_2+{\sigma }_1{m}_1(1+{\alpha }_2)(p_1-m_2)]-2{\sigma }_1{m}_1{\sigma }_2{m}_2(p_1+{\alpha }_1+1)}.\end{array}$$

Hence,

$$\parallel ((0,v^{nl}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_2}.$$

Also, the conditions $q\in [m_2,\infty ]$ and $p_{2}q\in [m_1,\infty ]$ imply $p_2\ge \frac{m_1}{m_2}$.

The proof of (50) is similar to the proof of (28) of Theorem 1.

The proof is complete.

4.3. Proof Main Results for the Case $min\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}\le n<max\left\{\frac{2{\sigma }_1{m}_1}{1+{\alpha }_1},\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}\right\}$

4.3.1. Proof of Theorem 5

For any $n\ge \frac{2{\sigma }_1{m}_1}{1+{\alpha }_1}$ and when $\delta \in (0,1)$ is sufficiently small, there exists a parameter $\overline{q_1}=\overline{q_1}\left(\delta \right)\in (m_1,\infty )$ such that

$$\frac{n(1+{\alpha }_1)}{2{\sigma }_1}\left(\frac{1}{m_1}-\frac{1}{\overline{q_1}}\right)=1-\delta .$$

(57)

If $n<\frac{2{\sigma }_2{m}_2}{1+{\alpha }_2}$, then for all $q\in [m_2,\infty ]$ we obtain

$$\frac{n(1+{\alpha }_2)}{2{\sigma }_2}\left(\frac{1}{m_2}-\frac{1}{q}\right)<1-\frac{n(1+{\alpha }_2)}{2q{\sigma }_2}\le 1.$$

(58)

Hence, we can choose a positive $\delta$ such that there does not exist any $\overline{q}\in [m_2,\infty ]$ that satisfies (26). For this reason,

$${\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2,\delta }={\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}:=\frac{n(1+{\alpha }_2)}{2{\sigma }_2}\left(\frac{1}{m_2}-\frac{1}{q}\right).$$

Let $T>0$. We define the space $\mathcal{X}\left(T\right)$ as follows:

$$\begin{array}{c}\hfill \mathcal{X}\left(T\right):=C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\end{array}$$

equipped with the norm

$$\begin{array}{cc}\hfill {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}:=& \underset{0⩽s⩽T}{sup}\left\{{\mathcal{R}}_1(s,u)+{\mathcal{R}}_2(s,v)\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}& {\mathcal{R}}_1(s,u):={(1+s)}^{-{\alpha }_1}{\parallel u(s,·)\parallel }_{L^{m_1}}+{(1+s)}^{1-\delta -{\alpha }_1}\left({\parallel u(s,·)\parallel }_{L^{\overline{q_1}}}+{\parallel u(s,·)\parallel }_{L^{\infty }}\right),\hfill \\ & {\mathcal{R}}_2(s,v)={(1+s)}^{-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{m_2}}+{(1+s)}^{{\beta }_{{\alpha }_2,\infty ,{\sigma }_2}^{m_2}-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{\infty }},\hfill \end{array}$$

where ${\beta }_{{\alpha }_2,\infty ,{\sigma }_2}^r=\frac{n(1+{\alpha }_2)}{2\sigma m_2}$ and the operator P is defined by

$$P:(u,v)\in \mathcal{X}\left(T\right)\to p(u,v)=P(u,v)(s,x):={(u,v)}^{ln}(s,x)+{(u,v)}^{nl}(s,x).$$

We will prove that the operator P satisfies, for any $(u,v)$ and $(\overline{u},\overline{v})\in \mathcal{X}\left(T\right)$, the next two inequalities

$$\begin{array}{ccc}\hfill & & {\parallel P(u,v)\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}+\parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}$$

(59)

$$\begin{array}{ccc}& & \parallel P(u,v)-P(\overline{u},\overline{v}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)-(\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\sum _{i=1}^2\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}\right).\hfill \end{array}$$

(60)

The proof of (59) and (60) is similar to the proof in Theorems 1 and 3. This completes the proof.

4.3.2. Proofof Theorem 6

Let $T>0$. We define the space $\mathcal{X}\left(T\right)$ as follows:

$$\begin{array}{c}\hfill \mathcal{X}\left(T\right):=C\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\times C\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^n\right)\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)\right)\end{array}$$

equipped with the norm

$$\begin{array}{cc}\hfill {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}:=& \underset{0⩽s⩽T}{sup}\left\{{\mathcal{R}}_1(s,u)+{\mathcal{R}}_2(s,v)\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}& {\mathcal{R}}_1(s,u):={(1+s)}^{-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel u(s,·)\parallel }_{L^{m_1}}+{(1+s)}^{1-\delta -{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}\hfill \\ & \times \left({\parallel u(s,·)\parallel }_{L^{\overline{q_1}}}+{\parallel u(s,·)\parallel }_{L^{\infty }}\right),\hfill \\ & {\mathcal{R}}_2(s,v):={(1+s)}^{-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{m_2}}+{(1+s)}^{{\beta }_{{\alpha }_2,\infty ,{\sigma }_2}^{m_2}-{\alpha }_2}{\parallel v(s,·)\parallel }_{L^{\infty }},\hfill \end{array}$$

where ${\beta }_{{\alpha }_2,\infty ,{\sigma }_2}^r=\frac{n(1+{\alpha }_2)}{2\sigma m_2}$. The operator P is defined by

$$P:(u,v)\in \mathcal{X}\left(T\right)\to P(u,v)=P(u,v)(s,x):={(u,v)}^{ln}(s,x)+{(u,v)}^{nl}(s,x).$$

We will prove that the operator P satisfies, for any $(u,v)$ and $(\overline{u},\overline{v})\in \mathcal{X}\left(T\right)$, the next two inequalities

$$\begin{array}{ccc}\hfill & & {\parallel P(u,v)\parallel }_{\mathcal{X}\left(T\right)}\lesssim \parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}+\parallel (u,v){\parallel }_{\mathcal{X}\left(T\right)}^{p_1}+{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}$$

(61)

$$\begin{array}{ccc}& & \parallel P(u,v)-P(\overline{u},\overline{v}){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {\parallel (u,v)-(\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}\sum _{i=1}^2\left({\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}+{\parallel (\overline{u},\overline{v})\parallel }_{\mathcal{X}\left(T\right)}^{p_i-1}\right),\hfill \end{array}$$

(62)

If $(u,v)\in \mathcal{X}\left(T\right)$, then we derive for all $s\in [0,T]$

$$\begin{array}{cc}\hfill & {\parallel u(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_1,\infty ],\hfill \\ & {\parallel v(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_2,\infty ].\hfill \end{array}$$

On the other hand, we have

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|u(s,·)\right|}^{p_2}{\parallel }_{L^q}\hfill & \le {\parallel u(s,·)\parallel }_{L^{p_{2}q}}^{p_2}\lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2})}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}\hfill \\ & \lesssim {(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2})}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2},\hfill \end{array}\end{array}$$

(63)

for any q such that $p_{2}q\in [m_1,\infty ]$ and due to ${\beta }_{{\alpha }_1,p_{2}q,{\sigma }_1}^{m_1,\delta }\ge {\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }$.

Also,

$$\begin{array}{c}\hfill \begin{array}{cc}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}\hfill & \le {\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}\lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2}-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \\ & \lesssim {(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}\hfill \end{array}\end{array}$$

(64)

for any q such that $p_{1}q\in [m_2,\infty ]$ and due to ${\beta }_{{\alpha }_2,p_{1}q,{\sigma }_2}^{m_2}\ge {\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}$. Thanks to (64), we have for $q\in [m_1,\infty ]$ the estimates

$$\begin{array}{cc}\hfill \parallel u^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel \left|v(s,·)\right|}^{p_1}{\parallel }_{L^q}dsd\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{\parallel v(s,·)\parallel }_{L^{p_{1}q}}^{p_1}dsd\rho }\hfill \\ & \lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}{\mathcal{J}}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{1}q\in [m_2,\infty ],\hfill \end{array}$$

where

$${\displaystyle {\mathcal{J}}_1\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}dsd\rho .}$$

(65)

The right side of (65) is what we are interested in estimating. We apply Lemma 1 for this. We put

$${\displaystyle \omega \left(\rho \right)={\int }_0^{\rho }{(\rho -s)}^{{\alpha }_1-1}{(1+s)}^{-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}ds.}$$

By applying Lemma 1, we obtain $\omega \left(\rho \right)\lesssim {(1+\rho )}^{{\alpha }_1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}$ if we assume that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)<1$. We notice that $p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)<1$ if and only if

$$p_1<p_{{\alpha }_2,{\sigma }_2}^{m_2}\left(n\right):=1+\frac{(n(m_2-1)+2{\sigma }_2{m}_2)(1+{\alpha }_2)}{(n-2{\sigma }_2{m}_2)(1+{\alpha }_2)+2{\sigma }_2{m}_2}$$

under the assumptions $1\le {\sigma }_2<\frac{1+{\alpha }_2}{2{\alpha }_2}$ and $1\le m_2<\frac{1+{\alpha }_2}{2{\alpha }_2{\sigma }_2}$.

On other hand, the conditions $q\in [m_1,\infty ]$ and $p_{1}q\in [m_2,\infty ]$ imply $p_1\ge \frac{m_2}{m_1}$.

Hence,

$$\begin{array}{cc}\hfill {\mathcal{J}}_1\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}\omega \left(\rho \right)d\rho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }}{(1+\rho )}^{{\alpha }_1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}d\rho .}\hfill \end{array}$$

(66)

Once more, we apply Lemma 1 to (66) to obtain

$${\mathcal{J}}_1\left(t\right)\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+1-p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2)}\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}.$$

Hence,

$$\parallel (u^{nl},0){\parallel }_{\mathcal{X}\left(T\right)}\lesssim {(1+t)}^{{\alpha }_1-{\beta }_{{\alpha }_1,q,{\sigma }_1}^{m_1,\delta }+{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}}{\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_1}.$$

Additionally, we have for $q\in [m_2,\infty ]$

$$\begin{array}{c}\hfill \parallel v^{nl}{(t,·)\parallel }_{L^q}\lesssim {\parallel (u,v)\parallel }_{\mathcal{X}\left(T\right)}^{p_2}{\mathcal{J}}_2\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{2}q\in [m_1,\infty ],\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{J}}_2\left(t\right)={\int }_0^t{(1+t-\rho )}^{-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}{\int }_0^{\rho }{(\rho -s)}^{{\alpha }_2-1}{(1+s)}^{-p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-{\gamma }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2})}dsd\rho .}\end{array}$$

(67)

If

$$\begin{array}{c}\hfill p_2({\beta }_{{\alpha }_1,p_2,{\sigma }_1}^{m_1,\delta }-{\alpha }_1-1+p_1({\beta }_{{\alpha }_2,p_1,{\sigma }_2}^{m_2}-{\alpha }_2))>1,\end{array}$$

(68)

then

$${\mathcal{J}}_2\left(t\right)\lesssim {(1+t)}^{{\alpha }_2-{\beta }_{{\alpha }_2,q,{\sigma }_2}^{m_2}}.$$

The condition (68) is equivalent to

$$\begin{array}{ccc}\hfill & p_1>max\{\frac{2{\sigma }_2{m}_2(\delta +{\alpha }_1)+n(1+{\alpha }_2)m_2}{n(1+{\alpha }_2)-2{\sigma }_2{m}_2{\alpha }_2};\frac{m_2}{m_1}-ϵ,1,\hfill & \\ & m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1({\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2)}\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & p_2>max\{\frac{2{\sigma }_2{m}_2}{n({\alpha }_2+1)(p_1-m_2)-2{\sigma }_2{m}_2(\delta +{\alpha }_1+p_1{\alpha }_2)},\hfill \\ & {\sigma }_2{m}_2\frac{2{\sigma }_1{m}_1+n{m}_1({\alpha }_1+1)}{n[{\sigma }_2{m}_2({\alpha }_1+1)+{\sigma }_1{m}_1({\alpha }_2+1)(p_1-m_2)]-2{\sigma }_2{m}_2{\sigma }_1{m}_1[({\alpha }_1+1)+p_1{\alpha }_2)]}\}.\hfill \end{array}$$

Notice that the

$$\begin{array}{ccc}\hfill & p_1>max\{\frac{2{\sigma }_2{m}_2(\delta +{\alpha }_1)+n(1+{\alpha }_2)m_2}{n(1+{\alpha }_2)-2{\sigma }_2{m}_2{\alpha }_2};\frac{m_2}{m_1}-ϵ,1,\hfill & \\ & m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1({\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2)}\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & p_2>max\{\frac{2{\sigma }_2{m}_2}{n({\alpha }_2+1)(p_1-m_2)-2{\sigma }_2{m}_2(\delta +{\alpha }_1+p_1{\alpha }_2)},\hfill \\ & {\sigma }_2{m}_2\frac{2{\sigma }_1{m}_1+n{m}_1({\alpha }_1+1)}{n[{\sigma }_2{m}_2({\alpha }_1+1)+{\sigma }_1{m}_1({\alpha }_2+1)(p_1-m_2)]-2{\sigma }_2{m}_2{\sigma }_1{m}_1[({\alpha }_1+1)+p_1{\alpha }_2)]}\},\hfill \end{array}$$

for all $\delta >0$ if and only if

$$\begin{array}{cc}\hfill & p_1>max\{\frac{2{\sigma }_2{m}_2{\alpha }_1+n(1+{\alpha }_2)m_2}{n(1+{\alpha }_2)-2{\sigma }_2{m}_2{\alpha }_2};\frac{m_2}{m_1}-ϵ,1,\hfill \\ & m_2\frac{n[(1+{\alpha }_2){\sigma }_1{m}_1-(1+{\alpha }_1){\sigma }_2]+2{\sigma }_1{m}_1({\alpha }_1+1){\sigma }_2}{{\sigma }_1{m}_1(n(1+{\alpha }_2)-2{\alpha }_2{\sigma }_2{m}_2)}\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & p_2>max\{\frac{2{\sigma }_2{m}_2}{n({\alpha }_2+1)(p_1-m_2)-2{\sigma }_2{m}_2({\alpha }_1+p_1{\alpha }_2)},\hfill \\ & {\sigma }_2{m}_2\frac{2{\sigma }_1{m}_1+n{m}_1({\alpha }_1+1)}{n[{\sigma }_2{m}_2({\alpha }_1+1)+{\sigma }_1{m}_1({\alpha }_2+1)(p_1-m_2)]-2{\sigma }_2{m}_2{\sigma }_1{m}_1[({\alpha }_1+1)+p_1{\alpha }_2)]}\}.\hfill \end{array}$$

The proof of (62) is similar to the proof of (28) of Theorem 6.

5. Conclusions

In this paper, we have proven the global (in time) existence of small data Sobolev solutions to the weakly coupled system of semi-linear fractional $\sigma –$evolution equations with different nonlinearities. We proved the connection between the regularity assumptions for the data, the equation parameters and the allowable range of exponents $(p_1,p_2)$ in Equation (2). In an upcoming paper, we plan to investigate the blow-up results for (2).