Cauchy Problems for Semilinear Parabolic Equations in Grand Herz Spaces

Abstract

In this paper, we study Cauchy problems for the semilinear parabolic equations ${\partial }_{t}u-▵u=G\left(u\right)$ with initial data in grand Herz spaces. We extend previous results established for classical Herz spaces to the broader framework of grand Herz spaces. The existence, uniqueness and stablity of solutions, as well as for their behaviour at small time are obtained by empolying heat kernel estimates, fixed-point theorems and some functional space theory.

1. Introduction

The present paper studies the Cauchy problem for semilinear parabolic equations. Let G be a suitable function; the Cauchy problem for semilinear parabolic equations on ${\mathbb{R}}^n$ takes the following form:

$${\displaystyle \frac{\partial u}{\partial t}}(t,x)=\mathsf{\Delta }u(t,x)+G\left(u(t,x)\right),(t,x)\in (0,\infty )\times {\mathbb{R}}^n$$

(1)

subject to the initial value condition $u(0,x)=u_0\left(x\right)$ on ${\mathbb{R}}^n$.

It is well known that many classical equations can be written as such evolution equations, examples of which are as follows:

1.

The semilinear heat equation:

$${\displaystyle \frac{\partial u}{\partial t}}(t,x)=\mathsf{\Delta }u(t,x)+u{\left|u\right|}^{\beta -1},(t,x)\in (0,\infty )\times {\mathbb{R}}^n,\beta >1.$$

2.

The Burgers viscous equation:

$${\displaystyle \frac{\partial u}{\partial t}}(t,x)=\mathsf{\Delta }u(t,x)+{\partial }_x{\left(\right|u|}^{\beta }),(t,x)\in (0,\infty )\times {\mathbb{R}}^n,\beta >1.$$

Weissler, in refs. [1,2] investigated the semilinear heat equation with singular data in Lebesgue spaces $L^p$; Giga, in ref. [3] derived that the solution belongs to $L^q([0,T),L^p)$ with ${\displaystyle \frac{1}{q}}=$${\displaystyle \frac{n}{2}}({\displaystyle \frac{1}{p_{ϵ}}}-{\displaystyle \frac{1}{p}}),p,q>p_{ϵ}$ and $q>\mu .$ Brezis and Cazenave, in ref. [4], considered the case of $1<p<p_c$, they proved that there exist some non-negative initial data in $L^p$ for which there is no non-negative solution for any positive time $T>0$. For further results, such as the well-posedness of the Cauchy problem of semilinear heat equations and in the framework of fractional Sobolev spaces, we can see the work of refs. [5,6]. In 2004, Miao and Zhang, in ref. [7], established the local well-posedness and small global well-posedness of problem (1) in the critical Besov spaces. For more information about this topic, the reader can refer to refs. [8,9,10,11,12] and the references therein.

For convenience, we set $B_k=B(0,2^k),{\overline{B}}_k=\{x\in {\mathbb{R}}^n:\left|x\right|\le 2^k\}$ and $C_k=B_k\setminus B_{k-1}$, if $E\subset {\mathbb{R}}^n$ is a measurable set, then ${\chi }_E$ denotes its characteristic function and ${\chi }_k={\chi }_{C_k}$ and $k\in \mathbb{Z}.$

Definition 1 ([13]).

Let $1<p,q<\infty$ and $\alpha \in \mathbb{R}$. The homogeneous Herz space ${\dot{K}}_{p,q}^{\alpha }$ is defined as the set of all $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n\setminus \left\{0\right\})$ such that

$${\parallel f\parallel }_{{\dot{K}}_{p,q}^{\alpha }}={\left(\sum _{k\in \mathbb{Z}}{2}^{k\alpha q}{\parallel f{\chi }_k\parallel }_p^q\right)}^{1/q}<\infty .$$

In ref. [14], Douadi investigated Equation (1) with

$$|G\left(x\right)-G\left(y\right)|\le |x-y|\left(\right|x{|}^{\mu -1}+\left|y{|}^{\mu -1}\right),x,y\in \mathbb{R},\mu \ge 1,G\left(0\right)=0$$

(2)

and initial data in Herz spaces ${\dot{K}}_{p,q}^{\alpha }.$ Douadi considered (1) via the corresponding integral equation

$$u(t,x)=e^{t\mathsf{\Delta }}{u}_0\left(x\right)+{\int }_0^t{e}^{(t-\tau )\mathsf{\Delta }}G\left(u\right)(\tau ,x)\mathrm{d}\tau .$$

(3)

Let $G:\mathbb{R}\to \mathbb{R}$ be a function satisfying (2). Under some suitable assumptions on ${\alpha }_0,\alpha ,p$ and r, Douadi proved the existence of solution u of (3) on $[0,T_0),0<T_0\le T$ such that

$$t^{{\displaystyle \frac{1}{r}}}u\in BC([0,T_0);{\dot{K}}_{p,q}^{\alpha })$$

(4)

and

$$\underset{t\to 0}{lim}{t}^{{\displaystyle \frac{1}{r}}}{∥u(t,·)∥}_{{\dot{K}}_{p,q}^{\alpha }}=0,p_0\le r_0\le p,r>\mu \mathrm{and}{p}_0={\displaystyle \frac{n}{\frac{2}{\mu -1}-{\alpha }_0}}>1.$$

(5)

If $r_0=p_0$ and $\mu <r<\infty$, then there exists a solution u of (3) on $[0,T_0),T_0$ arbitrary, such that (4) and (5) hold.

If $r_0=p_0$ and $r=\infty$, then there exists a solution u of (3) on $[0,T_0),T_0$ arbitrary, such that

$$u\in BC([0,T_0);{\dot{K}}_{p_0,q}^{{\alpha }_0}),$$

where the notation $BC$ stands for bounded and continuous spaces. Also, some results in ref. [3] were also improved, if we assume that $1<p<p_0\le r_0<\infty$, if $r_0>p_0$ and $\mu <r\le \infty$, then there is $T_0,0<T_0\le T$ and a solution u of (3) on $[0,T_0)$ such that (4) holds and

$$\underset{t\to 0}{lim}{t}^{{\displaystyle \frac{1}{r}}}{∥u(t,·)∥}_{{\dot{K}}_{p,q}^{\alpha }}=0,p<p_0.$$

(6)

If $r_0=p_0$ and $\mu <r<\infty$, then there exists a solution u of (3) on $[0,T_0),T_0$ arbitrary, such that (4) and (6) hold. The results for $1<p<p_0\le r_0<\infty$ are possible, since there exists a hypothesis in addition that

$$\alpha +{\displaystyle \frac{n}{p}}-{\displaystyle \frac{n}{r_0}}\le {\alpha }_0<n-{\displaystyle \frac{n}{r_0}}.$$

In addition, the grand Lebesgue space has been widely studied in modern analysis, which was introduced in refs. [15,16]. The necessity to investigate these spaces emerged from their rather essential role in various fields, in particular, in the integrability problem of Jacobian under minimal hypotheses. In fact, It turns out that in theory of PDEs, generalized grand Lebesgue spaces are appropriate for treating the existence and uniqueness, as well as the regularity problems, of various non-linear differential equations [17,18,19]. The space defined on bounded domains in $R^n$ was introduced by L. Greco, T. Iwaniec and C. Sbordone [15] when they studied existence and uniqueness of solutions to non-homogeneous n-harmonic equation $divA(x,\nabla u)=\mu$. Various operators of harmonic analysis have been intensively studied in recent years, cf. [20,21,22,23] and the references therein.

In this paper, our main interest is to study Cauchy problems for semilinear parabolic equations with suitable initial data in grand Herz spaces. We first recall several related definitions of grand spaces.

Now, we recall the definition of grand Lebesgue sequence spaces [24].

Definition 2 ([24]).

Let $1\le p<\infty$ and $\theta >0$. Then the grand Lebesgue sequence space $l^{p),\theta }$ is defined by

$${\parallel X\parallel }_{l^{p),\theta }\left(\mathbb{X}\right)}=\underset{ϵ>0}{sup}{\left({ϵ}^{\theta }\sum _{k\in \mathbb{X}}{\left|x_k\right|}^{p(1+ϵ)}\right)}^{{\displaystyle \frac{1}{p(1+ϵ)}}}=\underset{ϵ>0}{sup}{ϵ}^{{\displaystyle \frac{\theta }{p(1+ϵ)}}}{\parallel X\parallel }_{l^{p(1+ϵ)}\left(\mathbb{X}\right)}<\infty ,$$

where $X={\left\{x_k\right\}}_{k\in \mathbb{X}}$ and $\mathbb{X}$ represents one of sets $\mathbb{Z}$ and $\mathbb{N}$.

Note that the following nesting properties hold:

$$l^{p(1-\epsilon )}\hookrightarrow l^p\hookrightarrow l^{p),{\theta }_1}\hookrightarrow l^{p),{\theta }_2}\hookrightarrow l^{p(1+\delta )}$$

for $0<\epsilon <{\displaystyle \frac{1}{p^{\prime }}}$, $\delta >0$ and $0<{\theta }_1\le {\theta }_2$.

The homogeneous grand Herz spaces are given as follows.

Definition 3 ([25]).

Let $1<p,q<\infty$, $\alpha \in \mathbb{R}$ and $\theta >0$. The homogeneous grand Herz space ${\dot{K}}_{p,q}^{\alpha ),\theta }$ is defined as the set of all $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n\setminus \left\{0\right\})$ such that

$${\parallel f\parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta }}=\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k\in \mathbb{Z}}{2}^{k\alpha q(1+\epsilon )}{\parallel f{\chi }_k\parallel }_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}<\infty .$$

Remark 1.

(1) It is easy to see that the relation of implication between Herz space ${\dot{K}}_{p,q}^{\alpha }$ and grand Herz space ${\dot{K}}_{p,q}^{\alpha ),\theta }$ is that

$${\dot{K}}_{p,q}^{\alpha }\subset {\dot{K}}_{p,q}^{\alpha ),\theta },1<p,q<\infty ,\alpha \in \mathbb{R},\theta >0.$$

(2) For some ${\epsilon }_0\in (0,\infty )$, we have

$${\dot{K}}_{p,q}^{\alpha ),\theta }={\dot{K}}_{p,q}^{\alpha ),\theta ,{\epsilon }_0}$$

and

$${\parallel f\parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta ,{\epsilon }_0}}\le {\parallel f\parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta }}\le {\left({\displaystyle \frac{{\epsilon }_0^{-\frac{1}{1+{\epsilon }_0}}}{eW(1/e)}}\right)}^{{\displaystyle \frac{\theta }{p}}}{\parallel f\parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta ,{\epsilon }_0}},$$

where

$${\parallel f\parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta ,{\epsilon }_0}}=\underset{0<\epsilon <{\epsilon }_0}{sup}{\left({\epsilon }^{\theta }\sum _{k\in \mathbb{Z}}{2}^{k\alpha q(1+\epsilon )}{\parallel f{\chi }_k\parallel }_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}<\infty ,$$

$W\left(t\right)$ is the Lambert function, inverse to the function $t\to t{e}^t,t>0$ (see, e.g., ref. [26] and the references therein).

In this paper, our main purpose is to generalize the results in ref. [14] to grand Herz spaces ${\dot{K}}_{p,q}^{\alpha ),\theta }$.

Recall that ${\zeta }_{R,m}\left(x\right)=R^n{(1+R|x\left|\right)}^{-N}$ for any $x\in {\mathbb{R}}^n$ and $N,R>0$. Note that ${\zeta }_{R,N}\in L^1\left({\mathbb{R}}^n\right)$ when $N>n$ and that $\parallel {\zeta }_{R,N}{\parallel }_1=C_N$ is independent of R, where this type of function was introduced in ref. [27].

Let

$$\mathcal{M}\left(f\right)\left(x\right)=\underset{Q\subset {\mathbb{R}}^n}{sup}{\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q\left|f\left(y\right)\right|dy,f\in L_{\mathrm{loc}}^1\left({\mathbb{R}}^n\right),$$

where the supremum is taken over all cubes Q in ${\mathbb{R}}^n$ with sides parallel to the axis and $x\in Q.$

Throughout this paper, the notation $f\lesssim g$ means that $f\le Cg$ for some independent constant C, and $f\approx g$ means $f\lesssim g\lesssim f.$ For $x\in {\mathbb{R}}^n$ and $r>0$, we denote by $B(x,r)$ the open ball in ${\mathbb{R}}^n$ with center x and radius r. For any $r>0$, we set $C\left(r\right)=\{x\in {\mathbb{R}}^n:{\displaystyle \frac{r}{2}}<|x|\le r\}$. If 1 $\le p\le \infty$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p^{\prime }}}=1$, then $p^{\prime }$ is called the conjugate exponent of p.

The paper is organized as follows: Section 2 establishes several necessary lemmas which play key roles in proofs of the main results, and Section 3 gives the main results and proofs. We would like to remark that the main ideas of our proofs are taken from ref. [14].

2. Preliminaries

Hammad [28] obtained the boundedness of the maximal function $\mathcal{M}\left(f\right)$ on grand Herz spaces ${\dot{K}}_{p,q}^{\alpha ),\theta }$.

Lemma 1 ([28]).

Let $1<p,q<\infty$, $\alpha \in \mathbb{R}$ and $\theta >0$. Let $f\in {\dot{K}}_{p,q}^{\alpha ),\theta }$ with $-{\displaystyle \frac{n}{p}}<\alpha <{\displaystyle \frac{n}{p^{\prime }}}$. Then

$${\parallel \mathcal{M}\left(f\right)\parallel }_{K_{p,q}^{\alpha ),\theta }}\lesssim {\parallel f\parallel }_{K_{p,q}^{\alpha ),\theta }}.$$

Let $t>0,x\in {\mathbb{R}}^n$ and $f\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right)$,

$${\mathrm{e}}^{t\mathsf{\Delta }}f\left(x\right)={\mathcal{F}}^{-1}(exp(-{t\left|\xi \right|}^2\left)\mathcal{F}f\right)\left(x\right),$$

where $\mathcal{F}$ is the Fourier transform.

Recall that

$$g\left(x\right)={\mathcal{F}}^{-1}(exp(-{t\left|\xi \right|}^2\left)\right)\left(x\right)={\left(4\pi t\right)}^{-{\displaystyle \frac{n}{2}}}exp(-4t^{-1}{\left|x\right|}^2),x\in {\mathbb{R}}^n.$$

Next, we will establish several key estimates of heat kernel $e^{t\mathsf{\Delta }}$, which play important roles in the proofs of main results.

Lemma 2.

Let ${\alpha }_1,{\alpha }_2\in \mathbb{R},0<t,\theta <\infty$ and $l<p,q,r,\tilde{r}<\infty$. Suppose that $1<q\le p<\infty$ and $-{\displaystyle \frac{n}{p}}<{\alpha }_1\le {\alpha }_2<{\displaystyle \frac{n}{q^{\prime }}}$. If $f\in {\dot{K}}_{q,s}^{{\alpha }_2),\tilde{\theta }}$, then

$$\parallel e^{t\mathsf{\Delta }}{f\parallel }_{{\dot{K}}_{p,r}^{{\alpha }_1),\theta }}\lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{\parallel f\parallel }_{{\dot{K}}_{q,s}^{{\alpha }_2),\tilde{\theta }}},$$

where

$$\left\{\begin{array}{cc}s=r,\hfill & \mathit{if}{\alpha }_2={\alpha }_1,\hfill \\ s>\tilde{r},\hfill & \mathit{if}{\alpha }_2>{\alpha }_1.\hfill \end{array}\right.\tilde{\theta }=\left\{\begin{array}{cc}\theta ,\hfill & \mathit{if}r=s,\hfill \\ {\displaystyle \frac{s\theta }{r}},\hfill & \mathit{if}r\ne s.\hfill \end{array}\right.$$

Proof. 

Let $f\in {\dot{K}}_{q,s}^{{\alpha }_2,\tilde{\theta }}$, for any $\epsilon >0$, we have the following decomposition:

$$\begin{array}{cc}\hfill & {\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}{\left(\sum _{k=-\infty }^{\infty }{2}^{k{\alpha }_{1}r(1+\epsilon )}{\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k\parallel }_p^{r(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{r(1+\epsilon )}}}\hfill \\ \hfill & \le {\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k=-\infty }^{\infty }{2}^{k{\alpha }_1}{\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k\parallel }_p\hfill \\ \hfill & ={\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^k\le t^{{\displaystyle \frac{1}{2}}}}{2}^{k{\alpha }_1}\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k{\parallel }_p+{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{2}^{k{\alpha }_1}{\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k\parallel }_p\hfill \\ \hfill & =I_1+I_2.\hfill \end{array}$$

Therefore, to obtain estimate of $I_1$, we first establish the following inequality,

$$\underset{x\in B(0,{\displaystyle \frac{1}{H}})}{sup}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\left|e^{t\mathsf{\Delta }}f\left(x\right)\right|\lesssim t^{-{\displaystyle \frac{n}{2}}}{H}^{{\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}}{∥f∥}_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}},\forall {\alpha }_2<{\displaystyle \frac{n}{q^{\prime }}},1<s<\infty ,0<t\le H^{-2}.$$

Note that

$$\left|e^{t\mathsf{\Delta }}f\left(x\right)\right|\lesssim \left|f\right|\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right),\forall x\in B(0,{\displaystyle \frac{1}{H}}),N>0.$$

From this, we need only to show that

$$\underset{x\in B(0,{\displaystyle \frac{1}{H}})}{sup}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\left|f\right|\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)\lesssim t^{-{\displaystyle \frac{n}{2}}}{H}^{{\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}}{\parallel f\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}.$$

Next, we turn to estimate $\left|f\right|\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)$, which can be divided as follows

$$\begin{array}{cc}\hfill & \left|f\right|{\chi }_{\overline{B}(0,{\displaystyle \frac{4}{H}})}\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)+\left|f\right|{\chi }_{{\overline{B}}^c(0,{\displaystyle \frac{4}{H}})}\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)\hfill \\ \hfill & =\sum _{j=0}^{\infty }\left|f\right|{\chi }_{C\left({\displaystyle \frac{2^{2-j}}{H}}\right)}\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}}N}\left(x\right)+\sum _{j=0}^{\infty }\left|f\right|{\chi }_{C\left({\displaystyle \frac{2^{j+3}}{H}}\right)}\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)\hfill \\ \hfill & :=J_1(x,t)+J_2(x,t).\hfill \end{array}$$

For $J_1(x,t)$, Hölder’s inequality yields that

$$\begin{array}{cc}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}{J}_1(x,t)\hfill & \le t^{-{\displaystyle \frac{n}{2}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{j=0}^{\infty }{\parallel f{\chi }_{C\left({\displaystyle \frac{2^{2-j}}{H}}\right)}\parallel }_1\hfill \\ \hfill & \le t^{-{\displaystyle \frac{n}{2}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^{k-2}<{\displaystyle \frac{1}{H}}}{\parallel f{\chi }_{C_k}\parallel }_1\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{n}{2}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^{k-2}<{\displaystyle \frac{1}{H}}}{2}^{nk{\displaystyle \frac{1}{q^{\prime }}}}{\parallel f{\chi }_{C_k}\parallel }_q\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{n}{2}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^{k-2}<{\displaystyle \frac{1}{H}}}{2}^{k{\alpha }_2}{∥f{\chi }_{C_k}∥}_q{2}^{k({\displaystyle \frac{n}{q^{\prime }}}-{\alpha }_2)}\hfill \\ \hfill & \le t^{-{\displaystyle \frac{n}{2}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}{\left(\sum _{k=-\infty }^{\infty }{2}^{k{\alpha }_{2}s(1+\epsilon )}{∥f{\chi }_{C_k}∥}_q^{s(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{s(1+\epsilon )}}}{\left(\sum _{k\in \mathbb{Z},2^{k-2}<{\displaystyle \frac{1}{H}}}{2}^{k({\displaystyle \frac{n}{q^{\prime }}}-{\alpha }_2)\iota }\right)}^{1/\iota }\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{n}{2}}}{H}^{{\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}}{\left({\epsilon }^{{\displaystyle \frac{\tilde{\theta }}{s(1+\epsilon )}}}\sum _{k=-\infty }^{\infty }{2}^{k{\alpha }_{2}s(1+\epsilon )}{∥f{\chi }_{C_k}∥}_q^{s(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{s(1+\epsilon )}}}({\alpha }_2<{\displaystyle \frac{n}{q^{\prime }}})\hfill \\ \hfill & \le t^{-{\displaystyle \frac{n}{2}}}{H}^{{\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}}{∥f∥}_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}},\hfill \end{array}$$

where $C_k=\{x\in {\mathbb{R}}^n:2^{k-2}<\left|x\right|\le 2^{k+1}\}$ and $\iota ={\displaystyle \frac{s(1+\epsilon )}{s(1+\epsilon )-1}}$.

For $J_2(x,t)$, it is not hard to see that for any $y\in C\left({\displaystyle \frac{2^{3+j}}{H}}\right)$ and $x\in B(0,{\displaystyle \frac{1}{H}})$, we have

$${\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x-y\right)\le t^{-{\displaystyle \frac{n}{2}}}{\left({\displaystyle \frac{2^j{t}^{-\frac{1}{2}}}{H}}\right)}^{-N}\le 2^{-jN}{t}^{-{\displaystyle \frac{n}{2}}},t^{-{\displaystyle \frac{1}{2}}}\ge H.$$

Again, by using Hölder’s inequality we deduce that

$$\begin{array}{cc}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}{J}_2(x,t)\hfill & \le t^{-{\displaystyle \frac{n}{2}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{j=0}^{\infty }{2}^{-jN}{\parallel f{\chi }_{C\left({\displaystyle \frac{2^{3-j}}{H}}\right)}\parallel }_1\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{n}{2}}}{H}^{-N}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^{k-2}\ge {\displaystyle \frac{1}{H}}}{2}^{-kN}{\parallel f{\chi }_{C_k}\parallel }_1\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{n}{2}}}{H}^{-N}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^{k-2}\ge {\displaystyle \frac{1}{H}}}{2}^{k({\displaystyle \frac{n}{q^{\prime }}}-N)}{\parallel f{\chi }_{C_k}\parallel }_q\hfill \\ \hfill & \approx t^{-{\displaystyle \frac{n}{2}}}{H}^{{\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^{k-2}\ge {\displaystyle \frac{1}{H}}}{2}^{k{\alpha }_2}{∥f{\chi }_{C_k}∥}_q{\left(2^{k}H\right)}^{{\displaystyle \frac{n}{q^{\prime }}}-{\alpha }_2-N}\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{n}{2}}}{H}^{{\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}{\left(\sum _{k=-\infty }^{\infty }{2}^{k{\alpha }_{2}s(1+\epsilon )}{∥f{\chi }_{C_k}∥}_q^{s(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{s(1+\epsilon )}}}({\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}<N)\hfill \\ \hfill & \le t^{-{\displaystyle \frac{n}{2}}}{H}^{{\alpha }_2-{\displaystyle \frac{n}{q^{\prime }}}}{∥f∥}_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}.\hfill \end{array}$$

For $I_1$, we get

$$\begin{array}{cc}{I}_1\hfill & \le \underset{x\in B(0,t^{{\displaystyle \frac{1}{2}}})}{sup}{\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\left|e^{t\mathsf{\Delta }}f\left(x\right)\right|\sum _{k\in \mathbb{Z},2^k\le t^{{\displaystyle \frac{1}{2}}}}{2}^{k({\alpha }_1+{\displaystyle \frac{n}{p}})}\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{∥f∥}_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}.({\alpha }_1+{\displaystyle \frac{n}{p}}>0,{\alpha }_2<{\displaystyle \frac{n}{q^{\prime }}})\hfill \end{array}$$

For $I_2$, choose a suitable N such that $N>qmax\left({\displaystyle \frac{n}{q}},{\displaystyle \frac{n}{p}}-{\alpha }_2+{\alpha }_1+n,{\displaystyle \frac{n-{\alpha }_2}{q}}\right)$. Note that

$$\begin{array}{cc}\left|e^{t\mathsf{\Delta }}f\left(x\right)\right|\hfill & \lesssim \left(\right|f|\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}){\chi }_{B_{k-2}}\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)+\left(\right|f|\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}){\chi }_{{\overline{C}}_k}\ast {\zeta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)\hfill \\ \hfill & +\left(\right|f|\ast {\eta }_{t^{-{\displaystyle \frac{1}{2}}},N}){\chi }_{{\mathbb{R}}^n\setminus B_{k+2}}\ast {\eta }_{t^{-{\displaystyle \frac{1}{2}}},N}\left(x\right)\hfill \\ \hfill & :=I_{t,k}^1\left(x\right)+I_{t,k}^2\left(x\right)+I_{t,k}^3\left(x\right).\hfill \end{array}$$

It is easy to check that

$${\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}{I}_{t,k}^1\left(x\right)\lesssim t^{-{\displaystyle \frac{1}{2q}}(n-N)}{\left(2^k{t}^{-{\displaystyle \frac{1}{2}}}\right)}^n{2}^{-({\alpha }_2+{\displaystyle \frac{N}{q}})k}{\parallel f\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}.$$

From this, for $N>q({\displaystyle \frac{n}{p}}-{\alpha }_2+{\alpha }_1+n)$, we have

$${\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{2}^{k{\alpha }_1}\parallel I_{t,k}^1{\chi }_k{\parallel }_p\lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{\parallel f\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}.$$

For $N>n$, by Young’s inequality for convolution, we derive

$$\parallel I_{t,k}^2{\chi }_k{\parallel }_p\lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}})}\parallel \left(\right|f|\ast {\eta }_{t^{-{\displaystyle \frac{1}{2}}},N}{){\chi }_{C_k}\parallel }_q.$$

This, together with Lemma 1 and ${\alpha }_2>{\alpha }_1$, implies

$$\begin{array}{cc}\hfill & {\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{2}^{k{\alpha }_1}{\parallel I_{t,k}^2{\chi }_k\parallel }_p\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}\underset{k\in \mathbb{Z}}{sup}\left(2^{k{\alpha }_2}{\parallel (f\ast {\eta }_{t-{\displaystyle \frac{1}{2}},N}){\chi }_{C_k}\parallel }_q\right)\sum _{k\in {\mathbb{Z}}^k,2^k>t^{{\displaystyle \frac{1}{2}}}}{\left(2^k{t}^{-{\displaystyle \frac{1}{2}}}\right)}^{({\alpha }_1-{\alpha }_2)}\hfill \\ & \lesssim t^{-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+a_2-a_1\right)}{\parallel f\ast {\eta }_{t-{\displaystyle \frac{1}{2}},N}\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{\parallel f\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}.\hfill \end{array}$$

For the case of ${\alpha }_2={\alpha }_1$, a slight modification can be derived.

For $N>q(n-{\alpha }_2)$, it is not hard to see that

$${\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}{I}_{t,k}^3\left(x\right)\lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{N}{q}}+{\alpha }_2)}{\parallel f\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}.$$

Therefore, we have

$$\begin{array}{cc}\hfill & {\epsilon }^{{\displaystyle \frac{\theta }{r(1+\epsilon )}}}\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{2}^{k{\alpha }_1}{\parallel I_{t,k}^3{\chi }_k\parallel }_p\hfill \\ \hfill & \lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{\parallel f\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}}\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{\left(2^k{t}^{-{\displaystyle \frac{1}{2}}}\right)}^{({\displaystyle \frac{n}{p}}-{\displaystyle \frac{N}{q}}-{\alpha }_2+{\alpha }_1+n)}.\hfill \end{array}$$

The proof of Lemma 2 is finished. □

For the case of $p<q$, we also have the following.

Lemma 3.

Let ${\alpha }_1,{\alpha }_2\in \mathbb{R},0<t,\theta <\infty$ and $1<p,q,r,\tilde{r}<\infty$. Suppose that $1<p<q<\infty$, ${\alpha }_1+{\displaystyle \frac{n}{p}}>0$ and ${\alpha }_1+{\displaystyle \frac{n}{p}}-{\displaystyle \frac{n}{q}}\le {\alpha }_2<{\displaystyle \frac{n}{q^{\prime }}}$. If $f\in {\dot{K}}_{q,s}^{{\alpha }_2),\tilde{\theta }}$, then

$$\parallel e^{t\mathsf{\Delta }}{f\parallel }_{K_{p,r}^{{\alpha }_1),\theta }}\lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{\parallel f\parallel }_{K_{q,s}^{{\alpha }_2),\tilde{\theta }}},$$

where

$$s=\left\{\begin{array}{cc}r,\hfill & \mathit{if}{\alpha }_2={\alpha }_1+{\displaystyle \frac{n}{p}}-{\displaystyle \frac{n}{q}},\hfill \\ \tilde{r},\hfill & \mathit{if}{\alpha }_2>{\alpha }_1+{\displaystyle \frac{n}{p}}-{\displaystyle \frac{n}{q}}.\hfill \end{array}\right.\tilde{\theta }=\left\{\begin{array}{cc}\theta ,\hfill & \mathit{if}r=s,\hfill \\ {\displaystyle \frac{s\theta }{r}},\hfill & \mathit{if}r\ne s.\hfill \end{array}\right.$$

Proof. 

We employ the same notations as in Lemma 2, so it suffices to estimate $I_2$. Note that $p<q$ and ${\alpha }_2>{\alpha }_1+{\displaystyle \frac{n}{p}}-{\displaystyle \frac{n}{q}}$. We have

$$\begin{array}{cc}\hfill & {\epsilon }^{\theta }\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{2}^{kr{\alpha }_1(1+\epsilon )}{\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k\parallel }_p^{r(1+\epsilon )}\hfill \\ \hfill & \le {\epsilon }^{\theta }\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{2}^{kr({\alpha }_1+{\displaystyle \frac{n}{p}}-{\displaystyle \frac{n}{q}}-{\alpha }_2)(1+\epsilon )}{2}^{k{\alpha }_{2}r(1+\epsilon )}{\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k\parallel }_q^{r(1+\epsilon )}\hfill \\ \hfill & \le {\epsilon }^{\theta }\underset{k\in \mathbb{Z}}{sup}{2}^{k{\alpha }_{2}r(1+\epsilon )}{\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k\parallel }_q^{r(1+\epsilon )}\sum _{k\in \mathbb{Z},2^k>t^{{\displaystyle \frac{1}{2}}}}{2}^{kr({\alpha }_1+{\displaystyle \frac{n}{p}}-{\displaystyle \frac{n}{q}}-{\alpha }_2)(1+\epsilon )}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \lesssim t^{-{\displaystyle \frac{r}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)(1+\epsilon )}{\epsilon }^{\theta }\underset{k\in \mathbb{Z}}{sup}{2}^{k{\alpha }_{2}r(1+\epsilon )}{\parallel \left(e^{t\mathsf{\Delta }}f\right){\chi }_k\parallel }_q^{r(1+\epsilon )}.\hfill \end{array}$$

This, together with Lemma 1, gives that

$$I_2\lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{\parallel \mathcal{M}\left(f\right)\parallel }_{K_{q,\tilde{r}}^{{\alpha }_2),\theta }}\lesssim t^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n}{q}}-{\displaystyle \frac{n}{p}}+{\alpha }_2-{\alpha }_1)}{\parallel f\parallel }_{K_{q,\tilde{r}}^{{\alpha }_2),\theta }}.$$

The proof of Lemma 3 is completed. □

The main purpose of this article is to study the solution in grand Herz spaces of the integral Equation (3), which is usually defined as the mild solution of the Cauchy problem (1). We set

$$F\left(u\right)(t,x)={\int }_0^t{e}^{(t-\tau )\mathsf{\Delta }}G\left(u\right)(\tau ,x)\mathrm{d}\tau .$$

The following lemma is useful in the sequel.

Lemma 4.

Let $t,\theta >0,\mu \ge 1,1<{\displaystyle \frac{p}{\mu }}\le r_1<\infty ,-{\displaystyle \frac{n}{r_1}}<{\alpha }_1\le \mu \alpha <n-{\displaystyle \frac{n\mu }{p}},1<q_1,r,$$s<\infty$ and

$$s=\left\{\begin{array}{cc}{q}_1,\hfill & \mathit{if}\alpha ={\displaystyle \frac{{\alpha }_1}{\mu }},\hfill \\ r,\hfill & \mathit{if}\alpha >{\displaystyle \frac{{\alpha }_1}{\mu }}.\hfill \end{array}\right.$$

Let $u,\nu \in {\dot{K}}_{p,s}^{\alpha ),\theta }$. Then,

$${∥F\left(u\right)(t,·)-F\left(v\right)(t,·)∥}_{{\dot{K}}_{r_1,q_1}^{{\alpha }_1),\theta }}\lesssim {\int }_0^t{(t-\tau )}^{-{\displaystyle \frac{1}{2}}({\displaystyle \frac{n\mu }{p}}-{\displaystyle \frac{n}{r_1}}+\mu \alpha -{\alpha }_1)}\Omega (u(\tau ,·),\nu (\tau ,·))d\tau$$

holds, where

$$\Omega (u(\tau ,·),v(\tau ,·))={∥u(\tau ,·)-v(\tau ,·)∥}_{{\dot{K}}_{p,s}^{\alpha ),\theta }}\left({∥u(\tau ,·)∥}_{{\dot{K}}_{p,s}^{\alpha ),\theta }}^{\mu -1}+{∥v(\tau ,·)∥}_{{\dot{K}}_{p,s}^{\alpha ),\theta }}^{\mu -1}\right),0<\tau \le t.$$

Proof. 

By adopting the same ideas as in ref. [14], together with embedding ${\dot{K}}_{{\displaystyle \frac{p}{\mu }},{\displaystyle \frac{s}{\mu }}}^{\mu \alpha ),\theta }\hookrightarrow {\dot{K}}_{{\displaystyle \frac{p}{\mu }},s}^{\mu \alpha ),\theta }$, Minkowski’s inequality and Hölder’s inequality in grand Herz spaces, the proof of Lemma 4 is just a simple imitation of Lemma 3.3 in ref. [14]. □

To obtain the solution in grand Herz spaces of the integral Equation (3), we also need the following lemma.

Lemma 5.

Let $\theta >0$, $\alpha \in \mathbb{R}$, $1<p<\infty$, $1\le q<\infty$ and $\alpha >-{\displaystyle \frac{n}{p}}$. Then $C_c$ is dense in ${\dot{K}}_{p,q}^{\alpha ),\theta }$.

Proof. 

It is not hard to see that $C_c\subset {\dot{K}}_{p,q}^{\alpha ),\theta }$ if and only if $\alpha >-{\displaystyle \frac{n}{p}}$. Indeed, let $f\in C_c$ be such that $f\left(x\right)=1,x\in B(0,2^N),N\in \mathbb{Z}$. We get

$$\begin{array}{cc}{\parallel f\parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta }}\hfill & =\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k\in \mathbb{Z}}{2}^{k\alpha q(1+\epsilon )}{\parallel f{\chi }_k\parallel }_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}\hfill \\ \hfill & ⩾\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k=-\infty }^N{2}^{k\alpha q(1+\epsilon )}{\parallel {\chi }_{C_k\cap B(0,2^N)}\parallel }_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}\hfill \\ \hfill & =\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k=-\infty }^N{2}^{k\alpha q(1+\epsilon )}{\parallel {\chi }_{C_k}\parallel }_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \approx \underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k=-\infty }^N{2}^{kq(\alpha +{\displaystyle \frac{n}{p}})(1+\epsilon )}\right)}^{1/q(1+\epsilon )}\hfill \end{array}$$

and this series is convergent if $\alpha >-{\displaystyle \frac{n}{p}}$.

It is clear that $C_c\subset {\dot{K}}_{p,q}^{\alpha ),\theta }$ whenever $\alpha >-{\displaystyle \frac{n}{p}}$. Let ${\dot{K}}_{p,q,c}^{\alpha ),\theta }$ be the set of all $g\in {\dot{K}}_{p,q}^{\alpha ),\theta }$ such that $g=0$ outside a compact A. As in ref. [29] (Proposition 3.1), we derive that ${\dot{K}}_{p,q,c}^{\alpha ),\theta }$ is dense in ${\dot{K}}_{p,q}^{\alpha ),\theta }$. Therefore, we need to prove the density of $C_c$ in ${\dot{K}}_{p,q,c}^{\alpha ),\theta }$.

Let $f\in {\dot{K}}_{p,q}^{\alpha ),\theta }$ with $f\left(x\right)=0$ if $x\notin A$ compact. As in ref. [29] (Theorem 2.19), the proof can be restricted to cases where f is real-valued and non-negative. Since f is measurable, there exists a monotonically increasing sequence ${\left\{u_i\right\}}_{i\in \mathbb{N}}$ of non-negative simple functions converging pointwise to f and

$$0⩽{\phi }_i⩽f,i\in \mathbb{N}.$$

Since

$$0⩽f-{\phi }_i⩽f,i\in \mathbb{N},$$

by the dominated convergence theorem, ${\left\{{\phi }_i\right\}}_{i\in \mathbb{N}}$ converge to f in ${\dot{K}}_{p,q}^{\alpha ),\theta }$. Therefore, we find an $\phi \in {\left\{{\phi }_i\right\}}_{i\in \mathbb{N}}$ such that

$${\parallel f-\phi \parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta }}\lesssim {\displaystyle \frac{\eta }{2}},\eta >0.$$

Since $0⩽\phi ⩽f$, $\mathrm{supp}\phi \subset A$. Let $\theta >0$ be such that $max(0,-{\displaystyle \frac{\alpha p}{n}})<\lambda <1$. Assume that $A\subset V\subset \overline{V}$ with $\overline{V}$ compact. We set

$$E=\sum _{k\in \mathbb{Z},C_k\cap \overline{V}\ne \varnothing }{2}^{k\alpha q(1+\epsilon )}{\left|C_k\right|}^{{\displaystyle \frac{\lambda q(1+\epsilon )}{p}}}.$$

By Lusin’s theorem we can find that $\psi \in C_c$ such that

$$\left|\psi \left(x\right)\right|⩽{\parallel \phi \parallel }_{\infty }$$

for $\mathrm{supp}\phi \subset \overline{V}$ and

$$\left|H\right|⩽{\left({\displaystyle \frac{\eta }{{4\parallel u\parallel }_{\infty }{\left(E{\epsilon }^{\theta }\right)}^{\frac{1}{q(1+\epsilon )}}}}\right)}^{{\displaystyle \frac{p}{1-\lambda }}},$$

where $H=\{x:\psi (x)\ne \phi (x\left)\right\}$.

We set $B=\{x\in \overline{V}:\psi \left(x\right)\ne \phi \left(x\right)\}$. Observe that $H=B$. We have

$$\begin{array}{cc}{\parallel \phi -\psi \parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta }}\hfill & =\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{\infty }{2}^{k\alpha q(1+\epsilon )}{∥(\phi -\psi ){\chi }_k∥}_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}\hfill \\ \hfill & =\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{\infty }{2}^{k\alpha q(1+\epsilon )}{∥(\phi -\psi ){\chi }_{C_k\cap H}∥}_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}\hfill \\ \hfill & =\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{\infty }{2}^{k\alpha q(1+\epsilon )}{∥(\phi -\psi ){\chi }_{C_k\cap B}∥}_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}\hfill \\ \hfill & =\underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k\in \mathbb{Z}:C_k\cap B}{2}^{k\alpha q(1+\epsilon )}{∥(\phi -\psi ){\chi }_{C_k\cap B}∥}_p^{q(1+\epsilon )}\right)}^{1/q(1+\epsilon )}.\hfill \end{array}$$

Let $k\in \mathbb{Z}$ be such that $C_k\cap B\ne \varnothing$. Then,

$$\begin{array}{cc}{∥{(\phi -\psi )}_{{\chi }_{C_k\cap B}}∥}_p\hfill & ⩽{2\parallel u\parallel }_{\infty }{∥{\chi }_{C_k\cap B}∥}_p\hfill \\ \hfill & ={2\parallel u\parallel }_{\infty }{∥{\chi }_{C_k\cap B}∥}_p^{\lambda }{∥{\chi }_{C_k\cap B}∥}_p^{1-\lambda }\hfill \\ \hfill & ⩽{2\parallel u\parallel }_{\infty }{∥{\chi }_{C_k}∥}_p^{\lambda }{∥{\chi }_B∥}_p^{1-\lambda }\hfill \\ \hfill & ⩽2{\left|C_k\right|}^{{\displaystyle \frac{\lambda }{p}}}{\parallel u\parallel }_{\infty }{\left|B\right|}^{{\displaystyle \frac{1-\lambda }{p}}}.\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}{\parallel \phi -\psi \parallel }_{{\dot{K}}_{p,q}^{\alpha ),\theta }}\hfill & \lesssim \underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }\sum _{k\in \mathbb{Z}:C_k\cap B}{2}^{k\alpha q(1+\epsilon )}{\left|C_k\right|}^{{\displaystyle \frac{\lambda q(1+\epsilon )}{p}}}{\parallel u\parallel }_{\infty }^{q(1+\epsilon )}{\left|B\right|}^{{\displaystyle \frac{(1-\lambda )q(1+\epsilon )}{p}}}\right)}^{1/q(1+\epsilon )}\hfill \\ \hfill & \le \underset{\epsilon >0}{sup}{\left({\epsilon }^{\theta }E{\parallel u\parallel }_{\infty }^{q(1+\epsilon )}{\left|B\right|}^{{\displaystyle \frac{(1-\lambda )q(1+\epsilon )}{p}}}\right)}^{1/q(1+\epsilon )}\hfill \\ \hfill & \lesssim \eta .\hfill \end{array}$$

The proof of Lemma 5 is completed. □

3. Main Result

Our main result can be stated as follows:

Theorem 1.

Let $\tilde{\theta },{\tilde{\theta }}_0>0$, $1<p,q,q_0,r_0,s<\infty ,\mu >1,{\displaystyle \frac{2}{\mu -1}}-n<{\alpha }_0<{\displaystyle \frac{2}{\mu -1}}$. Assume that $1<p_0\le r_0\le p$ and $-{\displaystyle \frac{n}{p}}<\alpha \le {\alpha }_0<n-{\displaystyle \frac{n}{r_0}}$. Let $u_0\in {\dot{K}}_{r_0,q_0}^{{\alpha }_0),{\tilde{\theta }}_0}$,

$$\begin{array}{ccc}& q_0=\left\{\begin{array}{cc}q,\hfill & \mathit{if}{\alpha }_0=\alpha ,\hfill \\ s,\hfill & \mathit{if}{\alpha }_0>\alpha ,\hfill \end{array}\right.,\hfill & {\displaystyle \frac{1}{r}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{n}{r_0}}-{\displaystyle \frac{n}{p}}+{\alpha }_0-\alpha \right)\ge 0,\hfill \end{array}$$

and $p_0={\displaystyle \frac{n}{\frac{2}{\mu -1}-{\alpha }_0}}>1$.

(i) 

Let $0\le \alpha <{\displaystyle \frac{n}{\mu }}-{\displaystyle \frac{n}{p}}$ and $r>\mu$. If $r_0>p_0$ and $\mu <r\le \infty$, there is $T_0$, $0<T_0\le T$ and a solution u of (3) on $[0,T_0)$ such that

$$t^{{\displaystyle \frac{1}{r}}}u\in BC([0,T_0);{\dot{K}}_{p,q}^{\alpha ),\tilde{\theta }}),\underset{t\to 0}{lim}{t}^{{\displaystyle \frac{1}{r}}}{∥u(t,·)∥}_{{\dot{K}}_{r,q}^{\alpha ),\tilde{\theta }}}=0,r_0\le p.$$

(7)

If $r_0=p_0$ and $\mu <r<\infty$, then there exists a solution u of (3) on $[0,T_0)$, $T_0$ arbitrary, such that the above conclusions (7) hold.

(ii) 

Let $0\le {\alpha }_0<{\displaystyle \frac{n}{u}}-{\displaystyle \frac{n}{\mu p_0}},r_0=p_0$ and $r=\infty .$ There exists a solution u of (3) on $[0,T_0)$, $T_0$ arbitrary, such that

$$u\in BC([0,T_0);{\dot{K}}_{p_0,q_0}^{{\alpha }_0),{\tilde{\theta }}_0}).$$

(8)

(iii) 

Let $0\le \alpha <{\displaystyle \frac{n}{\mu }}-{\displaystyle \frac{n}{p}}$ and $r>\mu$. If $r_0>p_0$ and $\mu <r\le \infty$, then

$$T\gtrsim \parallel u_0{(0,·)\parallel }_{{\dot{K}}_{r_0,q_0}^{{\alpha }_0),{\tilde{\theta }}_0}}^{-{\displaystyle \frac{1-\mu }{1-\frac{\mu -1}{2}\left(\frac{n}{r_0}+{\alpha }_0\right)}}}.$$

(iv) 

There exists a positive constant ε such that if $\parallel u_0{(0,·)\parallel }_{{\dot{K}}_{r_0,q_0}^{{\alpha }_0),\theta }}<\epsilon$, then

$$t^{{\displaystyle \frac{1}{r}}}{∥u(t,·)∥}_{{\dot{K}}_{r,q}^{\alpha ,){\tilde{\theta }}_0}}\lesssim 1,0<t<\infty .$$

(v) 

The solutions of (3) in (i) and (ii) satisfying (7) and (8) are unique.

Remark 2.

According to Remark 1, our result represents certain extension of ref. [14] (Theorem 3.4).

Proof. 

By virtue of Lemmas 1–5 and the fixed point theorem, the proof of Theorem 1 is only an imitation of ref. [14] (Theorem 3.4). Therefore, we omit the details here. □

Finally, we give an example of Theorem 1. In fact, once again we can obtain the solution in Herz spaces.

Example 1.

In Definition 3, if we take some

$$\tilde{\epsilon }=1\mathit{and}\theta \equiv 0,$$

which combine the results of Theorem 1, then we obtain the solution in Herz spaces ${\dot{K}}_{p,2q}^{\alpha }$ of the integral Equation (3).

4. Conclusions

The main work of this paper is to generalize Cauchy problems for semilinear parabolic equations

$${\partial }_{t}u-▵u=G\left(u\right)$$

with initial data in Herz spaces by adopting similar methods to grand Herz spaces. In future work, we will study the related solutions of semilinear parabolic equations in grand Herz–Morrey spaces.