Existence of Blow-Up Solution to the Cauchy Problem of Inhomogeneous Damped Wave Equation

Abstract

This paper is concerned with the non-existence of a global solution to the initial value problem of the inhomogeneous damped wave equation with a nonlinear memory term and a nonlinear gradient term. The critical exponent and formation of singularity of solution are closely related to symmetry in the study of blow-up dynamics for nonlinear wave equations, which provides a profound mathematical tool for analyzing the explosion of solutions within finite time. The proofs of blow-up results of solutions are based on the test function method, where the test function is variable separated. The influences of two types of damping terms, two types of nonlinearities, and an inhomogeneous term on exponents of the problem in blow-up cases are explained. It is worth pointing out that the inhomogeneous term in the problem is discussed with respect to the exponent $\sigma$ in three cases (namely, $\sigma =0$, $-1<\sigma <0$, and $\sigma >0$). As far as we know, the results in Theorems 1–4 are new.

1. Introduction

In the present paper, we aim to investigate the following Cauchy problem for inhomogeneous wave equation with damping terms and nonlinear terms

$$\left\{\begin{array}{c}{u}_{tt}-\Delta u+u_t-\Delta u_t={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{\left|u\right|}^{p}d\tau +{|\nabla u|}^q+t^{\sigma }w\left(x\right),\hfill \\ t>0,x\in {\mathbb{R}}^n,\hfill \\ u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in {\mathbb{R}}^n,\hfill \end{array}\right.$$

(1)

where $\Delta ={\displaystyle \sum _{i=1}^n}{\displaystyle \frac{{\partial }^2}{\partial x_i^2}}$ is the Laplacian operator, $1<p,q<\infty$. The term $u_t$ is a weak damping term. $\Delta u_t$ is a strong damping term. $f(u,u_t)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{\left|u\right|}^{p}d\tau$ stands for the memory term, where $\Gamma \left(\alpha \right)={\int }_0^{\infty }{t}^{\alpha -1}{e}^{-t}dt(0<\alpha <1)$ denotes the Gamma function. The nonlinear term ${|\nabla u|}^q$ is related to the derivative of spatial variables of solution u. $t^{\sigma }w\left(x\right)(-1<\sigma <\infty )$ represents the inhomogeneous term, where $w\left(x\right)\in L_{loc}^1\left({\mathbb{R}}^n\right)$, $w\left(x\right)\ge 0$, and $w\left(x\right)\not\equiv 0$. The initial values satisfy $u_0\left(x\right),u_1\left(x\right)\in L_{loc}^1\left({\mathbb{R}}^n\right)$ and $u_0\left(x\right),u_1\left(x\right)\ge 0$.

As we know, the wave equation is a type of hyperbolic equation that explains various wave phenomena in nature. The nonlinear wave equation plays an important role in fields such as physics and mathematics, and it is widely used in fluid mechanics, electromagnetics, and quantum mechanics. Symmetry is very useful in revealing the structure and conservation of solutions to nonlinear wave equations, such as translational symmetry or scale symmetry, and so on. The combination of symmetry analysis with modern dynamical systems will further promote the development of nonlinear wave theory. By revealing the inherent geometric structure and invariance of the equation, symmetry plays a crucial role in the study of the blow-up of solutions to wave equations, which provides a profound mathematical tool for analyzing the singularity formation of solutions, such as a derivative explosion or unbounded growth of solutions within a finite time. Symmetry provides a complete analytical framework for the study of solution rupture, for example, the critical exponent determination by revealing the inherent geometric structure of the wave equation, which gains a deeper understanding of the essence of singularity formation of solutions to nonlinear wave equations.

Damping is a physical phenomenon in which the vibration system is blocked so that energy is dissipated with time. When a ship or floating offshore structure vibrates in waves, the damping is mainly viscoelastic damping, friction damping between structural members, and internal damping of materials. Friction damping is a measure of energy dissipation of the system due to the fact that heat is generated by friction during friction movement, which is applied to mechanical manufacturing, civil engineering, and other fields. There are different forms of friction phenomena in daily life. Actually, many dynamic phenomena are affected by one or more variables and their past history. As a consequence, the nonlinear memory term in the small data Cauchy problem is of interest. A lower decay rate in the low-dimension case can pass through the special nonlinear structure. In a forced vibration problem, the inhomogeneous term represents an external force acting on it, which has wide applications in describing physical phenomena. Nonlinear external force could affect the propagation of waves. The inhomogeneous term $t^{\sigma }w\left(x\right)$ in this work, which depends on both time and space variables, could be viewed as the usual form of perturbation nonlinear term, which exhibits a polynomial-type growth condition as well. Inhomogeneous external force could affect the propagation of waves. It could cause the wave to become steep until it blows up in the process of propagation. The influence of different nonlinearities on the blow-up of the solution is different. Therefore, it is of significance and application value to study the effects of the damping term and nonlinear term on blow-up and estimation of the lifespan of the solution to the Cauchy problem of nonlinear wave Equation (1) with small initial values.

Firstly, let us review several historical backgrounds related to the Cauchy problem of the classical wave equation.

$$\left\{\begin{array}{c}{u}_{tt}-\Delta u=f(u,u_t),t>0,x\in {\mathbb{R}}^n,\hfill \\ u(0,x)=\epsilon u_0\left(x\right),u_t(0,x)=\epsilon u_1\left(x\right),x\in {\mathbb{R}}^n\hfill \end{array}\right.$$

(2)

(see detailed illustrations in [1,2,3,4,5,6,7,8,9,10,11,12,13]). Strauss [3] conjectures that the solution to problem (2) with power nonlinearity $f(u,u_t)={\left|u\right|}^p$ blows up in finite time when $1<p<p_S\left(n\right)$. While the problem admits a global (in time) solution when $p>p_S\left(n\right)$. It is worthwhile to stress that $p_S\left(n\right)$ stands for the Strauss exponent, which is the positive root of the quadratic equation

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

for $n\ge 2$. Notice that $p_S\left(1\right)=\infty$ in one spatial dimension. John [7] acquires blow-up dynamics of solution to problem (2) with $f(u,u_t)={\left|u\right|}^p$ when $1<p<p_S\left(3\right)=1+\sqrt{2}$ provided that the initial values are sufficiently smooth and compactly supported. Furthermore, the existence of a global solution to the problem is demonstrated in the case $p>p_S\left(3\right)$. Glassey [12] illustrates the blow-up phenomenon of the solution to the initial value problem (2) with $f(u,u_t)={\left|u\right|}^p$ when $1<p<p_S\left(n\right)(n\le 3)$. Existence of a global solution to the problem is shown when $p>p_S\left(n\right)(n\le 3)$ (see [14]). Lai et al. [15] establish local well-posedness and an upper-bound lifespan estimate of the solution to problem (2) with $f(u,u_t)={\left|u\right|}^p(1<p\le p_S\left(2\right))$ on the exterior domain. The method utilized in the proof is the test function approach. Jleli and Samet [8] discuss the blow-up result of the solution to the initial boundary value problem of the nonlinear wave equation with $f(u,u_t)={\left|u\right|}^p$ for $n\ge 2$ by taking advantage of the test function approach. Moreover, the existence of a global solution to the problem is proved when $n\ge 3$. Zhou and Han [16] study the non-existence of a global solution and the upper-bound lifespan estimate of the solution to the variable coefficient wave equation with $f(u,u_t)={\left|u\right|}^p$ on the exterior domain by applying the Kato lemma when $n\ge 3$. Kitamura et al. [17] derive sharp upper-bound lifespan estimates of solutions to the Cauchy problem (2) with $f(u,u_t)={(1+x^2)}^{-{\displaystyle \frac{1+a}{2}}}{\left|u\right|}^p$ in one space dimension. The technique is a combination of the Duhamel principle and the iteration method. The influence of the parameter in the weighted coefficient on the lifespan estimate of the solution is precisely illustrated. Chen and Palmieri [10] consider blow-up results of problem (2) with memory term $f(u,u_t)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{\left|u\right|}^{p}d\tau$. Upper-bound lifespan estimates of the solution to the problem are established by utilizing the iteration approach in the sub-critical case and the slicing method in the critical case. Lai et al. [1] investigate the local existence of a solution to the initial boundary value problem of the wave equation with a memory term on the exterior domain when $n=1$. Finite-time blow-up of the solution is obtained under certain positive assumptions on the initial values. The key tools employed in proof of results are iteration arguments and the Kato lemma. Blow-up results and the upper-bound lifespan estimates of classical solutions to quasilinear wave equations in one space dimension are established (see [18]). The proofs of the main results are derived by a combination of ordinary differential inequality and the iteration method on the lower bound of the weighted functional of the solution. The interested readers may refer to the references [15,17,18,19,20,21,22,23,24,25] for more details.

There are a lot of previous works in the literature concerning the investigation of the damped wave equation

$$\left\{\begin{array}{c}{u}_{tt}-\Delta u+h\left(u_t\right)=f(u,u_t),t>0,x\in {\mathbb{R}}^n,\hfill \\ u(0,x)=\epsilon u_0\left(x\right),u_t(0,x)=\epsilon u_1\left(x\right),x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(3)

The readers refer to the instructions in [8,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42] for more details. Todorova and Yordanov [39] derive the blow-up result of the solution to problem (3) with weak damping term $h\left(u_t\right)=u_t$ and $f(u,u_t)={\left|u\right|}^p(1<p<1+{\displaystyle \frac{2}{n}})$ when initial values satisfy certain conditions. In addition, the solution exists globally when $1+{\displaystyle \frac{2}{n}}<p<{\displaystyle \frac{n}{n-2}}(n\ge 3)$ and $1+{\displaystyle \frac{2}{n}}<p<\infty (n=1,2)$. Regarding the critical case $p_F\left(n\right)=1+{\displaystyle \frac{2}{n}}$, Zhang [41] proves the non-existence of a global solution to problem (3) with $h\left(u_t\right)=u_t$ and $f(u,u_t)={\left|u\right|}^p$ by applying the test function method. Here, $p_F\left(n\right)=1+{\displaystyle \frac{2}{n}}$ stands for the so-called Fujita exponent, which is the critical exponent of the parabolic equation $u_t-\Delta u={\left|u\right|}^p$ (see [43]). Chen and Reissig [44] show global well-posedness for problem (3) with $h\left(u_t\right)=u_t$ and $f(u,u_t)={\left|u\right|}^p$ in the Sobolev spaces of negative order. Lower- and upper-bound lifespan estimates of the solution to the problem are established by making use of the Banach fixed point argument and test function technique. Lai and Yin [45] study finite-time blow-up dynamics of solutions to the initial boundary value problem of the wave equation with the Dirichlet boundary condition when $h\left(u_t\right)=u_t$ and $f(u,u_t)={\left|u\right|}^p$. The method applied in the proof is the test function technique, which is connected with the Riemann–Liouville fractional derivative and contradiction argument. Zhou [42] illustrates the blow-up phenomenon of the local solution to problem (3) with $h\left(u_t\right)=u_t$ and $f(u,u_t)={\left|u\right|}^{p-1}u$ in the case $1<p<1+{\displaystyle \frac{2}{n}}(n\ge 2)$. The non-existence and the existence of a global solution to problem (3) with $h\left(u_t\right)=u_t$ and a non-trivial boundary condition are demonstrated on the exterior domain (see [8]). The strategy of proof is based on the test function technique.

In addition to the weak damping, many researchers are interested in investigating wave equations with other damping terms. Chen and Fino [27] study blow-up dynamics of solutions to the initial boundary value problem of the wave equation with a strong damping term $h\left(u_t\right)=-\Delta u_t$ and combined nonlinearities $f(u,u_t)=|u_t{|}^p+{\left|u\right|}^q$ by taking advantage of the test function method. Lian and Xu [46] consider the global existence of solutions to the initial boundary value problem of nonlinear wave equations with weak and strong damping terms as well as the logarithmic source term $f(u,u_t)=uln\left|u\right|$. The non-existence of a global solution and the upper-bound lifespan estimate of a solution to the problem (3) with damping terms $h\left(u_t\right)=u_t-\Delta u_t$ are discussed on the exterior domain by constructing auxiliary functional and utilizing the Levines concavity argument (see [40]). Therefore, it is of great theoretical and practical significance to consider the combined effects of friction damping and viscoelastic damping on the behavior of the solution. Liao [35] verifies the local existence and blow-up result of the solution to problem (3) with $h\left(u_t\right)={\int }_0^{t}g(t-s)\Delta u\left(s\right)ds-\Delta u_t$ and logarithmic nonlinearity $f(u,u_t)={\left|u\right|}^{p-2}uln\left|u\right|$ on the exterior domain. Upper-bound lifespan estimates of the solution to problem (3) with scattering damping term $h\left(u_t\right)={\displaystyle \frac{\mu }{{(1+t)}^{\beta }}}{u}_t(\mu >0,\beta >1)$ and $f(u,u_t)=|u_t{|}^p+{\left|u\right|}^q$ are established when the exponents in nonlinear terms satisfy certain assumptions (see [47]). The main strategy of proof is the iteration approach. Hamouda and Hamza [30] derive the upper-bound lifespan estimate of the solution to problem (3) with scale invariant damping term $h\left(u_t\right)={\displaystyle \frac{\mu }{1+t}}{u}_t$ and $f(u,u_t)=|u_t{|}^p+{\left|u\right|}^q$ by employing the Kato lemma.

Many scholars pay attention to the study of inhomogeneous damped wave equation

$$\left\{\begin{array}{c}{u}_{tt}-\Delta u+h\left(u_t\right)=f(u,u_t)+t^{\sigma }w\left(x\right),t>0,x\in {\mathbb{R}}^n,\hfill \\ u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(4)

In order to investigate the influence of the inhomogeneous term on the critical behavior of the damped wave equation, Jleli and Samet [48] obtain the critical exponent $p_c\left(n\right)=1+{\displaystyle \frac{2}{n-1}}(n\ge 3)$ and $p_c\left(n\right)=\infty (n=1,2)$ in the particular case $\sigma =0$, $h\left(u_t\right)=u_t$, and $f(u,u_t)={\left|u\right|}^p$. It is shown that the critical exponent for the problem jumps from $1+{\displaystyle \frac{2}{n}}$ (critical exponent for equation $u_{tt}-\Delta u+u_t={\left|u\right|}^p$) to the bigger exponent $1+{\displaystyle \frac{2}{n-2}}$ in the case $n\ge 3$. Samet [49] investigates blow-up results and the existence of a global solution to problem (4) when $h\left(u_t\right)=u_t$, $f(u,u_t)={\left|u\right|}^p+{|\nabla u|}^q$ in the case $\sigma =0$ with the Fujita critical exponent and second critical exponent. Jleli et al. [50] consider blow-up dynamics and the existence of a global solution to problem (4) with double damping and potential terms, where $h\left(u_t\right)=u_t-\Delta u_t$, $f(u,u_t)=V\left(x\right){\left|u\right|}^p$, $-1<\sigma \le 0$ and $n\ge 2$, and the inhomogeneous term is $t^{\sigma }w\left(x\right)$. The Fujita critical exponent and second critical exponent are derived. We see that the inhomogeneous term has a certain impact on the critical exponent of the equation. Han et al. [51] investigate continuous dependence on initial values and high energy blow-up time estimates for porous elastic systems, which are similar to coupled damped wave equations in the structure of equations. The lower bound of blow-up time at arbitrary positive initial energy for the nonlinear damping case is derived. Pang et al. [52] consider the Cauchy problem for general nonlinear wave equations with doubly dispersive over equations in one dimension. The initial values such that a solution blows up in finite time or exists globally are classified by making use of the potential well theory.

It is worthwhile to notice that there is a similar blow-up phenomenon of the solution to the Cauchy problem for inhomogeneous parabolic equation

$$\left\{\begin{array}{c}{u}_t-\Delta u=f(u,u_t)+t^{\sigma }w\left(x\right),t>0,x\in {\mathbb{R}}^n,\hfill \\ u(0,x)=u_0\left(x\right),x\in {\mathbb{R}}^n\hfill \end{array}\right.$$

(5)

(see [53,54,55,56,57,58,59]). Jleli et al. [54] illustrate the behaviors of sign-changing solutions to problem (5) with $f(u,u_t)={\left|u\right|}^p$ when $n\ge 2$. The non-existence and the existence of a global solution are established in the case $-1<\sigma <0$. While blow-up dynamics of the solution to the problem are derived when $\sigma >0$. When the function $w\left(x\right)$ satisfies certain conditions, critical behavior for a semilinear parabolic equation with inhomogeneous forcing term $t^{\sigma }w\left(x\right)$ is obtained (see [54]). Alqahtani et al. [55] verify the local existence of the solution to problem (5) with $f(u,u_t)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{\left|u\right|}^{p}d\tau$ and $\sigma =0$ by using the contraction mapping principle. Meanwhile, formation of singularity of the solution to the problem is demonstrated when $n\ge 1$ under certain assumptions on initial value. The main tools in the proof are based on the Riemann–Liouville fractional integral and test function technique. Jleli and Samet [53] consider the blow-up phenomenon of the solution to the initial boundary value problem of the parabolic equation with a nonlinear memory term and the inhomogeneous term $w\left(x\right)$ in the case $n\ge 2$. The non-existence and the existence of global solutions to the Cauchy problem for a coupled system of semilinear heat equations with space and time forcing inhomogeneous terms are discussed by taking advantage of the test function approach (see [57]). Zhang [60] invetigates the coupled system of heat equations with the inhomogeneous term in a non-compact complete Riemannian manifold. Jleli et al. [58] consider the formation of singularity of the solution to the Cauchy problem of the heat equation $u_t-\Delta u={\left|u\right|}^p+b{|\nabla u|}^q$. The result shows that the gradient term induces an interesting phenomenon of discontinuity of the critical Fujita exponent. For other studies related to inhomogeneous problems, the readers may refer to works [58,60,61,62,63] for more details and the references therein. Hence, the combined influence of the memory term and gradient nonlinear term on the blow-up of the solution can be considered.

Inspired by the works in [27,40,45,49,50,54,64], the main concern of this work is to verify the non-existence of a global solution to problem (1). To our knowledge, there is no result for the inhomogeneous wave equation with double damping terms, a nonlinear memory term, and a nonlinear gradient term. We present blow-up results of the solution to the problem with different conditions by employing the test function method. We note that Chen and Fino [27] derive the formation of singularity of the solution to the wave equation with a strong damping term $-\Delta u_t$ and combined nonlinearities $|u_t{|}^p+{\left|u\right|}^q$ on the exterior domain. Lai and Yin [45] investigate blow-up dynamics of the solution to the initial boundary value problem for the wave equation with weak damping term $u_t$ and power nonlinearity ${\left|u\right|}^p$ by making use of the test function method, which is related to the Riemann–Liouville fractional derivative and contradiction argument. We extend the problems considered in [27,45] to problem (1), which contains double damping terms, double nonlinear terms, and an inhomogeneous term. In addition, the the inhomogeneous term depends on both space and time variables. Jleli et al. [50] discuss the non-existence of a global solution to the inhomogeneous wave equation with two types of damping terms and a potential term when $-1<\sigma \le 0$. The blow-up result of the solution to the nonlinear wave equation with an inhomogeneous term that only depends on the space variable is verified, where nonlinear terms are presented in the form of power nonlinearity and a gradient term (see [49]). Behaviors of the sign-changing solution to the inhomogeneous parabolic equation without a damping term are obtained when $\sigma \ne 0$ and $n\ge 2$ (see [54]). We enhance the results obtained in [49,50,54] on blow-up of the solution for $n\ge 1$ by investigating different ranges of index $\sigma$ in the the inhomogeneous term $t^{\sigma }w\left(x\right)$. More precisely, we intend to consider $\sigma =0$, $-1<\sigma <0$, and $\sigma >0$, respectively. Cazenave et al. [64] demonstrate blow-up dynamics and the existence of a global solution to a parabolic equation with a nonlinear memory term $u_t-\Delta u={\int }_0^t{(t-s)}^{\alpha -1}{\left|u\right|}^{p-1}uds$ on ${\mathbb{R}}^n$ and a bounded domain by taking advantage of ordinary differential inequality and the contraction mapping principle, respectively. The problem studied in [64] is extended to the inhomogeneous wave equation with a double damping term and a nonlinear gradient term. We observe that the form of the memory term in this paper is ${\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{\left|u\right|}^{p}d\tau$, which is different from that in [64]. Furthermore, the test function method utilized in the proof of this paper is different from the method applied in [64]. Yang and Han [40] establish an upper-bound lifespan estimate of the solution to the wave equation with weak and strong damping terms and a general nonlinear term by constructing an auxiliary functional and employing the Levine concavity argument. The problem studied in [40] is generalized to the problem (1) with double nonlinearities and an inhomogeneous term. It is worth mentioning that we explain the common influences of the nonlinear memory term, nonlinear gradient term, and the inhomogeneous term on exponents of problem (1) in blow-up cases, which are associated with the Fujita-type critical exponent. To the best knowledge of the authors, the results in Theorems 1–4 are new.

We list some notations throughout the paper. $C_c^2((0,T)\times {\mathbb{R}}^n)$ denotes the space of $C^2$ real-valued functions compactly supported in $(0,T)\times {\mathbb{R}}^n$. For $n\ge 3$, $\sigma =0$, $0<\xi <n$, we set

$$I_{\xi }^{+}=\{w\in C\left({\mathbb{R}}^n\right)|w\ge {0,\left|x\right|}^{-\xi }=O\left(w\left(x\right)\right),as\left|x\right|\to \infty \}.$$

Our main blow-up results of the solution to problem (1) are illustrated as follows.

Theorem 1. 

Let $\sigma =0$ and

$${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0.$$

(6)

The solution u to problem (1) blows up in finite time in one of the following three cases.

• (1) $n=1$, $p,q>1$.
• (2) $n=2$, $p>1$, $1<q<2$.
• (3) $n\ge 3$, when $1<p<1+{\displaystyle \frac{2(\alpha +1)}{n-2}}$ or $1<q<1+{\displaystyle \frac{1}{n-1}}$.

Theorem 2. 

Let $-1<\sigma <0$ and

$${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0.$$

The solution u to problem (1) blows up in finite time in one of the following three cases.

• (1) $n=1$, $-1<\sigma <-{\displaystyle \frac{1}{2}}$, when $1<p<1-{\displaystyle \frac{2(\alpha +1)}{2\sigma +1}}$ or $p+\alpha +\sigma q(p-1)>0$.
• (2) $n=1$, $-{\displaystyle \frac{1}{2}}\le \sigma <0$, when $p>1$ or $p+\alpha +\sigma q(p-1)>0$.
• (3) $n\ge 2$, when $1<p<1+{\displaystyle \frac{2(\alpha +1)}{n-2\sigma -2}}$ or $(p+\alpha )\left(\right(n-1)q-n)-\sigma q(p-1)<0$.

Theorem 3. 

Let $\sigma =0$, $n\ge 3$, $p>1+{\displaystyle \frac{2(\alpha +1)}{n-2}}$, $q>1+{\displaystyle \frac{1}{n-1}}$, $w\left(x\right)\in I_{\xi }^{+}$. If

$$0<\xi <max\{{\displaystyle \frac{2(p+\alpha )}{p-1}},{\displaystyle \frac{q}{q-1}}\},$$

then the solution u to problem (1) blows up in finite time.

Theorem 4. 

Let $\sigma >0$, $n\ge 1$, $p>1$, $q>1$. If

$${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0,$$

then the solution u to problem (1) blows up in finite time.

Remark 1. 

When $\alpha \to 0^{+}$, the blow-up results in Theorems 1 and 3 coincide with the blow-up results obtained in [49], where the Cauchy problem of inhomogeneous wave equation with weak damping term, power nonlinearity, and nonlinear gradient term is considered.

Remark 2. 

We observe that Jleli et al. [50] discuss the Cauchy problem of the inhomogeneous wave equation with a double damping term, ${V\left(x\right)\left|u\right|}^p$ and $t^{\sigma }w\left(x\right)(-1<\sigma <0)$. The partial result in Theorem 2 is the same as the result derived in [50] in the case $V\left(x\right)=1$ when $\alpha \to 0^{+}$ in problem (1). It is worth mentioning that the result in Theorem 4 is new.

2. Preliminaries

Assume $0<\alpha <1$ and $f\left(t\right)\in C\left(\right[0,T\left]\right)$. We define

$$I_{0|t}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau ,0<t\le T,$$

$$I_{t|T}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_t^T{(\tau -t)}^{\alpha -1}f\left(\tau \right)d\tau ,0\le t<T.$$

Lemma 1 

([53]). Let $0<\alpha <1$, $f\left(t\right),g\left(t\right)\in C\left(\right[0,T\left]\right)$. It holds that

$${\int }_0^T\left(I_{0|t}^{\alpha }f\left(t\right)\right)g\left(t\right)dt={\int }_0^{T}f\left(t\right)\left(I_{t|T}^{\alpha }g\left(t\right)\right)dt.$$

(7)

According to Lemma 1, we calculate

$$\begin{array}{ccc}& & {\int }_0^T{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{\left|u\right|}^{p}d\tau \phi (t,x)dxdt\hfill \\ & & ={\int }_0^T{\int }_{{\mathbb{R}}^n}{I}_{0|t}^{\alpha }{\left(\right|u|}^p)\phi (t,x)dxdt\hfill \\ & & ={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\left(\phi (t,x)\right)dxdt.\hfill \end{array}$$

We present the definition of weak solution to problem (1) as follows.

Definition 1. 

The function $u(t,x)$ is called a weak solution to the Cauchy problem (1) if $(u,\nabla u)\in L_{loc}^p([0,T)\times {\mathbb{R}}^n)\times L_{loc}^q([0,T)\times {\mathbb{R}}^n)$ and

$$\begin{array}{c}L(u_0,u_1,\phi )+{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\left(\phi (t,x)\right)dxdt\hfill \\ +{\int }_0^T{\int }_{{\mathbb{R}}^n}{|\nabla u|}^q\phi (t,x)dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ ={\int }_0^T{\int }_{{\mathbb{R}}^n}u(t,x)({\phi }_{tt}-\Delta \phi -{(\Delta \phi )}_t-{\phi }_t)dxdt,\hfill \end{array}$$

(8)

where $\phi (t,x)\in C_c^2((0,T)\times {\mathbb{R}}^n)$ and

$$\begin{array}{ccc}& & L(u_0,u_1,\phi )={\int }_{{\mathbb{R}}^n}{u}_0\left(x\right)(\phi (0,x)-{\phi }_t(0,x)-\Delta \phi (0,x))dx\hfill \\ & & +{\int }_{{\mathbb{R}}^n}{u}_1\left(x\right)\phi (0,x)dx\hfill \\ & & <\infty .\hfill \end{array}$$

Due to the fact that $L(u_0,u_1,\phi )$ is a constant, without loss of generality, we assume $u_0=u_1=0$.

We define a cut-off function $\eta \left(s\right)\in C^{\infty }\left(\mathbb{R}\right)$, which satisfies

$$\eta \left(s\right)=\left\{\begin{array}{c} 1,0\le s\le 1,\hfill \\ decreasing,1<s<2,\hfill \\ 0,2\le s<\infty .\hfill \end{array}\right.$$

We are in the position to derive several key estimates that are used in the proofs of main theorems in this work.

Firstly, we suppose that u is the global weak solution to problem (1). Taking advantage of (8), we achieve

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\phi (t,x)dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ \le {\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_{tt}|dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||\Delta \phi |dxdt\hfill \\ +{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right||(\Delta \phi )}_t|dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_t|dxdt.\hfill \end{array}$$

(9)

Let ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p^{\prime }}}=1$. Utilizing the Young inequality, we deduce

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_{tt}|dxdt\hfill \\ ={\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right|{\left(I_{t|T}^{\alpha }\phi \right)}^{{\displaystyle \frac{1}{p}}}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p}}}\left|{\phi }_{tt}\right|dxdt\hfill \\ \le {\displaystyle \frac{1}{6}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\phi dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_{tt}\right|}^{p^{\prime }}dxdt,\hfill \end{array}$$

(10)

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||\Delta \phi |dxdt\hfill \\ \le {\displaystyle \frac{1}{6}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\phi dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta \phi |}^{p^{\prime }}dxdt,\hfill \end{array}$$

(11)

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right||(\Delta \phi )}_t|dxdt\hfill \\ \le {\displaystyle \frac{1}{6}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\phi dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{(\Delta \phi )}_t\right|}^{p^{\prime }}dxdt,\hfill \end{array}$$

(12)

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_t|dxdt\hfill \\ \le {\displaystyle \frac{1}{6}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\phi dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_t\right|}^{p^{\prime }}dxdt.\hfill \end{array}$$

(13)

Combining (9)–(13), we acquire

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ \le C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}\left(\right|{\phi }_{tt}{|}^{p^{\prime }}+{|\Delta \phi |}^{p^{\prime }}+|{(\Delta \phi )}_t{|}^{p^{\prime }}+|{\phi }_t{|}^{p^{\prime }})dxdt\hfill \\ =I_1+I_2+I_3+I_4.\hfill \end{array}$$

(14)

Here, the constant C is different on different lines.

On the other hand, taking advantage of (8), it yields

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\left(\phi (t,x)\right)dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}{|\nabla u|}^q\phi (t,x)dxdt\hfill \\ +{\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ \le {\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_{tt}|dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}|\nabla u|·|\nabla \phi |dxdt\hfill \\ +{\int }_0^T{\int }_{{\mathbb{R}}^n}{|\nabla u|·|(\nabla \phi )}_t|dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_t|dxdt.\hfill \end{array}$$

From straightforward computation, we come to

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_{tt}|dxdt\hfill \\ \le {\displaystyle \frac{1}{3}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\phi dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_{tt}\right|}^{p^{\prime }}dxdt,\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right||{\phi }_t|dxdt\hfill \\ \le {\displaystyle \frac{1}{3}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^p{I}_{t|T}^{\alpha }\phi dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_t\right|}^{p^{\prime }}dxdt,\hfill \end{array}$$

$$\begin{array}{ccc}& & {\int }_0^T{\int }_{{\mathbb{R}}^n}|\nabla u|·|\nabla \phi |dxdt\hfill \\ & & ={\int }_0^T{\int }_{{\mathbb{R}}^n}{|\nabla u|\left(\phi (t,x)\right)}^{{\displaystyle \frac{1}{q}}}{\left(\phi (t,x)\right)}^{-{\displaystyle \frac{1}{q}}}|\nabla \phi |dxdt\hfill \\ & & \le {\displaystyle \frac{1}{3}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{|\nabla u|}^q\phi (t,x)dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(\phi (t,x)\right)}^{-{\displaystyle \frac{1}{q-1}}}{|\nabla \phi |}^{q^{\prime }}dxdt,\hfill \end{array}$$

$$\begin{array}{ccc}& & {\int }_0^T{\int }_{{\mathbb{R}}^n}{|\nabla u|·|(\nabla \phi )}_t|dxdt\hfill \\ & & \le {\displaystyle \frac{1}{3}}{\int }_0^T{\int }_{{\mathbb{R}}^n}{|\nabla u|}^q\phi (t,x)dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(\phi (t,x)\right)}^{-{\displaystyle \frac{1}{q-1}}}{\left|{(\nabla \phi )}_t\right|}^{q^{\prime }}dxdt.\hfill \end{array}$$

Therefore, we have

$$\begin{array}{c}{\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ \le C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}|{\phi }_{tt}{|}^{p^{\prime }}dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_t\right|}^{p^{\prime }}dxdt\hfill \\ +C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{|\nabla \phi |}^{q^{\prime }}dxdt+C{\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{\left|{(\nabla \phi )}_t\right|}^{q^{\prime }}dxdt\hfill \\ =I_1+I_4+I_5+I_6.\hfill \end{array}$$

(15)

We take $\phi (t,x)=a_T\left(t\right)b_T\left(x\right)$ as the test function in (8), where

$$\begin{array}{c}\hfill a_T\left(t\right)=T^{-\lambda }{(T-t)}^{\lambda }(\lambda \gg 1),b_T\left(x\right)={\eta }^k{\left(\right|x|}^2{T}^{-2\rho })(k\ge 2,\rho >0).\end{array}$$

From direct calculation, we observe

$$\begin{array}{ccc}& & I_1={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_{tt}\right|}^{p^{\prime }}dxdt\hfill \\ & & ={\int }_0^T{\left(I_{t|T}^{\alpha }{a}_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\partial }_t^2{a}_T\left(t\right)\right|}^{p^{\prime }}dt{\int }_{{\mathbb{R}}^n}{b}_T\left(x\right)dx.\hfill \end{array}$$

Using variable transformation $y={\displaystyle \frac{s-t}{T-t}}$, we conclude

$$\begin{array}{ccc}& & I_{t|T}^{\alpha }{a}_T\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_t^T{(s-t)}^{\alpha -1}{T}^{-\lambda }{(T-s)}^{\lambda }ds\hfill \\ & & ={\displaystyle \frac{T^{-\lambda }}{\Gamma \left(\alpha \right)}}{(T-t)}^{\alpha +\lambda }{\int }_0^1{y}^{\alpha -1}{(1-y)}^{\lambda }dy\hfill \\ & & ={\displaystyle \frac{\Gamma (\lambda +1)}{\Gamma (\alpha +\lambda +1)}}{T}^{-\lambda }{(T-t)}^{\alpha +\lambda }.\hfill \end{array}$$

Thus, we derive

$$\begin{array}{c}{\int }_0^T{\left(I_{t|T}^{\alpha }{a}_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\partial }_t^2{a}_T\left(t\right)\right|}^{p^{\prime }}dt\hfill \\ =C{\int }_0^T|T^{-\lambda }{(T-t)}^{\alpha +\lambda }{|}^{-{\displaystyle \frac{1}{p-1}}}{\left|\lambda (\lambda -1)T^{-\lambda }{(T-t)}^{\lambda -2}\right|}^{p^{\prime }}dt\hfill \\ =C{T}^{-\lambda }{\int }_0^T{(T-t)}^{-{\displaystyle \frac{\alpha +\lambda }{p-1}}+{\displaystyle \frac{(\lambda -2)p}{p-1}}}dt\hfill \\ =C{T}^{-1-{\displaystyle \frac{2+\alpha }{p-1}}}.\hfill \end{array}$$

(16)

Making use of variable transformation $x=T^{\rho }z$, we arrive at

$$\begin{array}{ccc}\hfill & & {\int }_{{\mathbb{R}}^n}{b}_T\left(x\right)dx={\int }_{\left\{x\right|\left|x\right|<\sqrt{2}{T}^{\rho }\}}{\eta }^k{\left(\right|x|}^2{T}^{-2\rho })dx\hfill \\ & & ={\int }_{\left\{z\right|\left|z\right|<\sqrt{2}\}}{\eta }^k{\left(\right|z|}^2)T^{n\rho }dz\hfill \\ & & \le C{T}^{n\rho }.\hfill \end{array}$$

(17)

From (16) and (17), we have

$$\begin{array}{c}\hfill I_1={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_{tt}\right|}^{p^{\prime }}dxdt\le C{T}^{n\rho -1-{\displaystyle \frac{2+\alpha }{p-1}}}.\end{array}$$

(18)

In a similar way, we deduce

$$\begin{array}{ccc}& & I_4={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_t\right|}^{p^{\prime }}dxdt\hfill \\ & & ={\int }_0^T{\left(I_{t|T}^{\alpha }{a}_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\partial }_t{a}_T\left(t\right)\right|}^{p^{\prime }}dt{\int }_{{\mathbb{R}}^n}{b}_T\left(x\right)dx.\hfill \end{array}$$

It follows from some calculations that

$$\begin{array}{ccc}\hfill & & {\int }_0^T{\left(I_{t|T}^{\alpha }{a}_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\partial }_t{a}_T\left(t\right)\right|}^{p^{\prime }}dt\hfill \\ & & =C{\int }_0^T|T^{-\lambda }{(T-t)}^{\alpha +\lambda }{|}^{-{\displaystyle \frac{1}{p-1}}}{\left|\lambda T^{-\lambda }{(T-t)}^{\lambda -1}\right|}^{p^{\prime }}dt\hfill \\ & & =C{T}^{-\lambda }{\int }_0^T{(T-t)}^{-{\displaystyle \frac{\alpha +\lambda }{p-1}}+{\displaystyle \frac{(\lambda -1)p}{p-1}}}dt\hfill \\ & & =C{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}.\hfill \end{array}$$

(19)

Employing (17) and (19), we derive

$$\begin{array}{c}\hfill I_4={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_t\right|}^{p^{\prime }}dxdt\le C{T}^{n\rho -{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(20)

Direct computation shows

$$\begin{array}{ccc}\hfill & & I_2={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta \phi |}^{p^{\prime }}dxdt\hfill \\ & & ={\int }_0^T{\left(I_{t|T}^{\alpha }{a}_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}|a_T{\left(t\right)|}^{p^{\prime }}dt{\int }_{{\mathbb{R}}^n}{\left(b_T\left(x\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta b_T\left(x\right)|}^{p^{\prime }}dx.\hfill \end{array}$$

(21)

It holds that

$$\begin{array}{c}\hfill {\int }_0^T{\left(I_{t|T}^{\alpha }{a}_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|a_T\left(t\right)\right|}^{p^{\prime }}dt\le C{T}^{1-{\displaystyle \frac{\alpha }{p-1}}}.\end{array}$$

(22)

Let $\overline{z}(t,x)={\displaystyle \frac{{\left|x\right|}^2}{T^{2\rho }}}$. We deduce

$$\begin{array}{ccc}& & \Delta b_T\left(x\right)=k(k-1){\eta }^{k-2}\left(\overline{z}\right){\left({\eta }^{\prime }\left(\overline{z}\right)\right)}^2{\displaystyle \frac{{4\left|x\right|}^2}{T^{4\rho }}}\hfill \\ & & +k{\eta }^{k-1}\left(\overline{z}\right){\eta }^{\prime \prime }\left(\overline{z}\right){\displaystyle \frac{{4\left|x\right|}^2}{T^{4\rho }}}+k{\eta }^{k-1}\left(\overline{z}\right){\eta }^{\prime }\left(\overline{z}\right){\displaystyle \frac{2n}{T^{2\rho }}}\hfill \\ & & =2k{T}^{-2\rho }{\eta }^{k-2}\left(\overline{z}\right)\theta \left(x\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}& & \theta \left(x\right)=2(k-1)\overline{z}{\left({\eta }^{\prime }\left(\overline{z}\right)\right)}^2+2\overline{z}\eta \left(\overline{z}\right){\eta }^{\prime \prime }\left(\overline{z}\right)+n\eta \left(\overline{z}\right){\eta }^{\prime }\left(\overline{z}\right).\hfill \end{array}$$

Notice that

$$\left\{\begin{array}{c} \theta \left(x\right)=0,0\le \left|x\right|\le T^{\rho },\hfill \\ \left|\theta \left(x\right)\right|\le C,T^{\rho }<\left|x\right|<\sqrt{2}{T}^{\rho },\hfill \\ \theta \left(x\right)=0,\left|x\right|\ge \sqrt{2}{T}^{\rho }.\hfill \end{array}\right.$$

As a result, we acquire

$$\begin{array}{c}\hfill \Delta b_T\left(x\right)\le C{T}^{-2\rho }{\eta }^{k-2}{\left(\right|x|}^2{T}^{-2\rho }),T^{\rho }<\left|x\right|<\sqrt{2}{T}^{\rho }.\end{array}$$

When $k>2p^{\prime }$, it follows that

$$\begin{array}{ccc}\hfill & & {\int }_{{\mathbb{R}}^n}{\left(b_T\left(x\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta b_T\left(x\right)|}^{p^{\prime }}dx\hfill \\ & & \le C{T}^{-2\rho p^{\prime }}{\int }_{\left\{x\right|T^{\rho }<\left|x\right|<\sqrt{2}{T}^{\rho }\}}{\left(\eta \right(\left|x\right|}^2{T}^{-2\rho }{\left)\right)}^{k-2p^{\prime }}dx\hfill \\ & & =C{T}^{-2\rho p^{\prime }}{\int }_{\left\{z\right|1<\left|z\right|<\sqrt{2}\}}{\left(\eta \right(\left|z\right|}^2{\left)\right)}^{k-2p^{\prime }}{T}^{n\rho }dz\hfill \\ & & \le C{T}^{n\rho -2\rho p^{\prime }}.\hfill \end{array}$$

(23)

Applying (21)–(23), we derive

$$\begin{array}{c}\hfill I_2={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta \phi |}^{p^{\prime }}dxdt\le C{T}^{n\rho -2\rho p^{\prime }+1-{\displaystyle \frac{\alpha }{p-1}}}.\end{array}$$

(24)

Taking advantage of (19) and (23), we conclude

$$\begin{array}{ccc}\hfill & & I_3={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{(\Delta \phi )}_t\right|}^{p^{\prime }}dxdt\hfill \\ & & ={\int }_0^T{\left(I_{t|T}^{\alpha }{a}_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}|{\partial }_t{a}_T{\left(t\right)|}^{p^{\prime }}dt{\int }_{{\mathbb{R}}^n}{\left(b_T\left(x\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta b_T\left(x\right)|}^{p^{\prime }}dx\hfill \\ & & \le C{T}^{n\rho -2\rho p^{\prime }-{\displaystyle \frac{\alpha +1}{p-1}}}.\hfill \end{array}$$

(25)

Combining (14), (18), (20) and (24)–(25), we achieve

$$\begin{array}{ccc}& & {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ & & \le C{T}^{n\rho -1-{\displaystyle \frac{2+\alpha }{p-1}}}+C{T}^{n\rho -2\rho p^{\prime }+1-{\displaystyle \frac{\alpha }{p-1}}}+C{T}^{n\rho -2\rho p^{\prime }-{\displaystyle \frac{\alpha +1}{p-1}}}+C{T}^{n\rho -{\displaystyle \frac{\alpha +1}{p-1}}}\hfill \\ & & \le C{T}^{n\rho -2\rho p^{\prime }+1-{\displaystyle \frac{\alpha }{p-1}}}+C{T}^{n\rho -{\displaystyle \frac{\alpha +1}{p-1}}}.\hfill \end{array}$$

Taking $\rho ={\displaystyle \frac{1}{2}}$, we arrive at

$$\begin{array}{c}\hfill {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\le C{T}^{{\displaystyle \frac{n}{2}}-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(26)

An elementary computation leads to

$$\begin{array}{ccc}\hfill & & I_5={\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{|\nabla \phi |}^{q^{\prime }}dxdt\hfill \\ & & =({\int }_0^T{a}_T\left(t\right)dt)({\int }_{{\mathbb{R}}^n}{\left(b_T\left(x\right)\right)}^{-{\displaystyle \frac{1}{q-1}}}|\nabla b_T\left(x\right){|}^{q^{\prime }}dx)\hfill \end{array}$$

(27)

and

$$\begin{array}{c}\hfill {\int }_0^T{a}_T\left(t\right)dt=T^{-\lambda }{\int }_0^T{(T-t)}^{\lambda }dt={\displaystyle \frac{T}{\lambda +1}}.\end{array}$$

(28)

In the case $T^{\rho }<\left|x\right|<\sqrt{2}{T}^{\rho }$, it holds that

$$\begin{array}{ccc}& & |\nabla b_T\left(x\right)|=2k|x|T^{-2\rho }{\eta }^{k-1}{\left(\right|x|}^2{T}^{-2\rho }\left)\right|{\eta }^{\prime }{\left(\right|x|}^2{T}^{-2\rho }\left)\right|\hfill \\ & & \le 2\sqrt{2}k{T}^{-\rho }{\eta }^{k-1}{\left(\right|x|}^2{T}^{-2\rho }\left)\right|{\eta }^{\prime }{\left(\right|x|}^2{T}^{-2\rho }\left)\right|\hfill \\ & & \le C{T}^{-\rho }{\eta }^{k-1}{\left(\right|x|}^2{T}^{-2\rho }).\hfill \end{array}$$

Hence, we derive

$$\begin{array}{ccc}\hfill & & {\int }_{{\mathbb{R}}^n}{\left(b_T\left(x\right)\right)}^{-{\displaystyle \frac{1}{q-1}}}{|\nabla b_T\left(x\right)|}^{q^{\prime }}dx\hfill \\ & & \le C{T}^{-\rho q^{\prime }}{\int }_{\left\{x\right|T^{\rho }<\left|x\right|<\sqrt{2}{T}^{\rho }\}}{\left(\eta \right(\left|x\right|}^2{T}^{-2\rho }{\left)\right)}^{k-q^{\prime }}dx\hfill \\ & & \le C{T}^{n\rho -\rho q^{\prime }}.\hfill \end{array}$$

(29)

Utilizing (27)–(29), we observe

$$\begin{array}{c}\hfill I_5={\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{|\nabla \phi |}^{q^{\prime }}dxdt\le C{T}^{n\rho +1-\rho q^{\prime }}.\end{array}$$

(30)

Similarly, we achieve

$$\begin{array}{ccc}\hfill & & I_6={\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{\left|{(\nabla \phi )}_t\right|}^{q^{\prime }}dxdt\hfill \\ & & ={\int }_0^T{\left(a_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{q-1}}}|{\partial }_t{a}_T{\left(t\right)|}^{q^{\prime }}dt{\int }_{{\mathbb{R}}^n}{\left(b_T\left(x\right)\right)}^{-{\displaystyle \frac{1}{q-1}}}{|\nabla b_T\left(x\right)|}^{q^{\prime }}dx.\hfill \end{array}$$

(31)

A simple calculation gives rise to

$$\begin{array}{ccc}\hfill & & {\int }_0^T{\left(a_T\left(t\right)\right)}^{-{\displaystyle \frac{1}{q-1}}}{\left|{\partial }_t{a}_T\left(t\right)\right|}^{q^{\prime }}dt=C{T}^{-\lambda }{\int }_0^T{(T-t)}^{-{\displaystyle \frac{\lambda }{q-1}}+(\lambda -1)q^{\prime }}dt\hfill \\ & & =C{T}^{1-q^{\prime }}.\hfill \end{array}$$

(32)

Applying (29), (31) and (32), we derive

$$\begin{array}{c}\hfill I_6={\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{\left|{(\nabla \phi )}_t\right|}^{q^{\prime }}dxdt\le C{T}^{n\rho +1-(\rho +1)q^{\prime }}.\end{array}$$

(33)

Using (15), (18), (20), (30) and (33), we conclude

$$\begin{array}{ccc}& & {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ & & \le C{T}^{n\rho -1-{\displaystyle \frac{2+\alpha }{p-1}}}+C{T}^{n\rho -{\displaystyle \frac{\alpha +1}{p-1}}}+C{T}^{n\rho +1-\rho q^{\prime }}+C{T}^{n\rho +1-(\rho +1)q^{\prime }}\hfill \\ & & \le C{T}^{n\rho -{\displaystyle \frac{\alpha +1}{p-1}}}+C{T}^{n\rho +1-\rho q^{\prime }}.\hfill \end{array}$$

When $\rho ={\displaystyle \frac{(p+\alpha )(q-1)}{(p-1)q}}$, we obtain

$$\begin{array}{c}\hfill {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\le C{T}^{{\displaystyle \frac{n(p+\alpha )(q-1)}{(p-1)q}}-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(34)

3. Proof of Theorem 1

We present the proof by making use of the contradiction arguments.

Proof. 

When $\sigma =0$, we achieve

$$\begin{array}{ccc}& & {\int }_0^T{\int }_{{\mathbb{R}}^n}w\left(x\right)\phi (t,x)dxdt={\int }_0^T{T}^{-\lambda }{(T-t)}^{\lambda }dt{\int }_{{\mathbb{R}}^n}w\left(x\right){\eta }^k{\left(\right|x|}^2{T}^{-2\rho })dx.\hfill \end{array}$$

We bear in mind that

$$\begin{array}{c}\hfill \underset{T\to \infty }{lim}{\int }_{{\mathbb{R}}^n}w\left(x\right){\eta }^k{\left(\right|x|}^2{T}^{-2\rho })dx={\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\end{array}$$

For sufficient large T, we have

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right){\eta }^k{\left(\right|x|}^2{T}^{-2\rho })dx\ge {\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\end{array}$$

Consequently, we calculate

$$\begin{array}{c}\hfill {\int }_0^T{\int }_{{\mathbb{R}}^n}w\left(x\right)\phi (t,x)dxdt={\displaystyle \frac{T}{2(\lambda +1)}}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx\ge CT{\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\end{array}$$

(35)

It is deduced from (26) and (35) that

$$\begin{array}{c}\hfill CT{\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{{\displaystyle \frac{n}{2}}-{\displaystyle \frac{\alpha +1}{p-1}}},\end{array}$$

which results in

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{{\displaystyle \frac{n}{2}}-{\displaystyle \frac{\alpha +1}{p-1}}-1}.\end{array}$$

(36)

(1) When $n=1$, we have

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}-{\displaystyle \frac{1}{2}}}.\end{array}$$

(37)

Passing to the limit as $T\to \infty$ in (37), we derive ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0$, which contradicts the assumption (6).

(2) When $n=2$, we rewrite (36) as

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(38)

Taking $T\to \infty$ in (38), we come to ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0$, which contradicts the assumption (6).

(3) When $n\ge 3$, $1<p<1+{\displaystyle \frac{2(\alpha +1)}{n-2}}$, passing to the limit as $T\to \infty$ in (36), one has

$${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0,$$

which contradicts the assumption (6).

According to (34) and (35), we acquire

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{{\displaystyle \frac{n(p+\alpha )(q-1)}{(p-1)q}}-{\displaystyle \frac{\alpha +1}{p-1}}-1}.\end{array}$$

(39)

(1) When $n=1$, we reformulate (39) as

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{-{\displaystyle \frac{p+\alpha }{(p-1)q}}}.\end{array}$$

(40)

Passing to the limit as $T\to \infty$ in (40), we deduce ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0$, which contradicts the assumption (6).

(2) When $1<q<1+{\displaystyle \frac{1}{n-1}}(n\ge 2)$, taking $T\to \infty$ in (39) gives rise to

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0,\end{array}$$

which contradicts the assumption (6). This completes the proof of Theorem 1. □

4. Proof of Theorem 2

Proof. 

When $-1<\sigma <0$, we come to

$$\begin{array}{c}\hfill {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt={\int }_0^T{t}^{\sigma }{T}^{-\lambda }{(T-t)}^{\lambda }dt{\int }_{{\mathbb{R}}^n}w\left(x\right){\eta }^k{\left(\right|x|}^2{T}^{-2\rho })dx.\end{array}$$

Taking variable transformation $s={\displaystyle \frac{t}{T}}$, we acquire

$$\begin{array}{ccc}& & {\int }_0^T{t}^{\sigma }{T}^{-\lambda }{(T-t)}^{\lambda }dt={\int }_0^1{\left(Ts\right)}^{\sigma }{T}^{-\lambda }{(T-Ts)}^{\lambda }Tds\hfill \\ & & =T^{\sigma +1}{\int }_0^1{s}^{\sigma }{(1-s)}^{\lambda }ds\hfill \\ & & =B(\sigma +1,\lambda +1)T^{\sigma +1},\hfill \end{array}$$

where $B(\sigma +1,\lambda +1)$ denotes the Beta function. We see that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right){\eta }^k{\left(\right|x|}^2{T}^{-2\rho })dx\ge {\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\end{array}$$

Therefore, we conclude

$$\begin{array}{c}\hfill {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\ge C{T}^{\sigma +1}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\end{array}$$

(41)

Employing (26) and (41), we get

$$\begin{array}{c}\hfill C{T}^{\sigma +1}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{{\displaystyle \frac{n}{2}}-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

It follows that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{{\displaystyle \frac{n}{2}}-{\displaystyle \frac{\alpha +1}{p-1}}-\sigma -1}.\end{array}$$

(42)

(1) When $n=1$, we derive

$${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}-\sigma -{\displaystyle \frac{1}{2}}}.$$

(43)

We discuss the following two situations. If $-1<\sigma <-{\displaystyle \frac{1}{2}}$ and $1<p<1-{\displaystyle \frac{2(\alpha +1)}{2\sigma +1}}$, sending $T\to \infty$ in (43), we obtain ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0$. If $-{\displaystyle \frac{1}{2}}\le \sigma <0$ and $p>1$, passing to the limit as $T\to \infty$ in (43), we deduce ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0$. These contradict the assumption ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0$.

(2) When $n=2$, $1<p<1-{\displaystyle \frac{\alpha +1}{\sigma }}$, we rewrite (42) as

$${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}-\sigma }.$$

(44)

Passing to the limit as $T\to \infty$ in (44), we come to ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0$. This contradicts the assumption ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0$.

(3) When $n\ge 3$, $1<p<1+{\displaystyle \frac{2(\alpha +1)}{n-2\sigma -2}}$, taking $T\to \infty$ in (42), we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0,\end{array}$$

which contradicts the assumption ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0$.

From (34) and (41), we achieve

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{T}^{{\displaystyle \frac{n(p+\alpha )(q-1)}{(p-1)q}}-{\displaystyle \frac{\alpha +1}{p-1}}-\sigma -1}.\end{array}$$

(45)

When $n\ge 1$ and $(p+\alpha )\left(\right(n-1)q-n)-\sigma q(p-1)<0$, taking $T\to \infty$ in (45), we arrive at

$${\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0,$$

which contradicts the assumption ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0$. The proof of Theorem 2 is finished. □

5. Proof of Theorem 3

Proof. 

We are in the position to consider the case of $\sigma =0$, $n\ge 3$, $0<\xi <n$, $p>1+{\displaystyle \frac{2(\alpha +1)}{n-2}}$, $q>1+{\displaystyle \frac{1}{n-1}}$, $w\left(x\right)\in I_{\xi }^{+}$.

Assume that there exist global solutions to problem (1) in this situation. We choose $\phi (t,x)=a_T\left(t\right)b_T\left(x\right)=T^{-\lambda }{(T-t)}^{\lambda }{\eta }^k{\left(\right|x|}^2{T}^{-2\rho })$ as the test function. Elementary computations show

$$\begin{array}{ccc}\hfill & & {\int }_0^T{\int }_{{\mathbb{R}}^n}w\left(x\right)\phi (t,x)dxdt={\displaystyle \frac{T}{\lambda +1}}{\int }_{\left\{x\right|\left|x\right|<\sqrt{2}{T}^{\rho }\}}w\left(x\right){\eta }^k{\left(\right|x|}^2{T}^{-2\rho })dx\hfill \\ & & \ge CT{\int }_{\left\{x\right|\left|x\right|\le T^{\rho }\}}w\left(x\right)dx\hfill \\ & & \ge CT{\int }_{\{x|{\displaystyle \frac{T^{\rho }}{2}}\le |x|\le T^{\rho }\}}{\left|x\right|}^{-\xi }dx\hfill \\ & & \ge C{T}^{(n-\xi )\rho +1}.\hfill \end{array}$$

(46)

Taking $\rho ={\displaystyle \frac{1}{2}}$ and employing (26) and (46), we have

$$\begin{array}{c}\hfill C{T}^{{\displaystyle \frac{1}{2}}(n-\xi )+1}\le C{T}^{{\displaystyle \frac{n}{2}}-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

It turns out that

$$\begin{array}{c}\hfill 0<C\le T^{{\displaystyle \frac{\xi }{2}}-1-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(47)

If $0<\xi <{\displaystyle \frac{2(p+\alpha )}{p-1}}$, passing to the limit as $T\to \infty$ in (47), we derive the contradiction.

When $\rho ={\displaystyle \frac{(p+\alpha )(q-1)}{(p-1)q}}$, from (34) and (46), we deduce

$$\begin{array}{c}\hfill C{T}^{{\displaystyle \frac{(n-\xi )(p+\alpha )(q-1)}{(p-1)q}}+1}\le C{T}^{{\displaystyle \frac{n(p+\alpha )(q-1)}{(p-1)q}}-{\displaystyle \frac{\alpha +1}{p-1}}},\end{array}$$

which leads to

$$\begin{array}{c}\hfill 0<C\le T^{{\displaystyle \frac{\xi (p+\alpha )(q-1)}{(p-1)q}}-1-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(48)

If $0<\xi <{\displaystyle \frac{q}{q-1}}$, we achieve the contradiction by taking $T\to \infty$ in (48). The proof of Theorem 3 is completed. □

6. Proof of Theorem 4

It is supposed that u is the global weak solution to problem (1).

Proof. 

When $\sigma >0$, we set $\phi (t,x)=a_T\left(t\right)b_R\left(x\right)=T^{-\lambda }{(T-t)}^{\lambda }{\eta }^k{\left(\right|x|}^2{R}^{-2\rho })$. Direct calculation shows

$$\begin{array}{c}\hfill \underset{R\to \infty }{lim}{\int }_{{\mathbb{R}}^n}w\left(x\right){\eta }^k{\left(\right|x|}^2{R}^{-2\rho })dx={\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\end{array}$$

For sufficient large R, we have

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right){\eta }^k{\left(\right|x|}^2{R}^{-2\rho })dx\ge {\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\end{array}$$

Applying the similar procedure in the proof of Theorem 2, we come to

$$\begin{array}{ccc}\hfill & & {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\ge C{T}^{\sigma +1}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx.\hfill \end{array}$$

(49)

Utilizing the change of variable $x=R^{\rho }\zeta$, we calculate

$$\begin{array}{ccc}\hfill & & {\int }_{{\mathbb{R}}^n}{b}_R\left(x\right)dx={\int }_{\left\{x\right|\left|x\right|<\sqrt{2}{R}^{\rho }\}}{\eta }^k{\left(\right|x|}^2{R}^{-2\rho })dx\hfill \\ & & ={\int }_{\left\{\zeta \right|\left|\zeta \right|<\sqrt{2}\}}{\eta }^k{\left(\right|\zeta |}^2)R^{n\rho }d\zeta \hfill \\ & & \le C{R}^{n\rho }.\hfill \end{array}$$

(50)

From (16) and (50), we deduce

$$\begin{array}{c}\hfill I_1={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_{tt}\right|}^{p^{\prime }}dxdt\le C{R}^{n\rho }{T}^{-1-{\displaystyle \frac{2+\alpha }{p-1}}}.\end{array}$$

(51)

Making use of (19) and (50), we achieve

$$\begin{array}{c}\hfill I_4={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{\phi }_t\right|}^{p^{\prime }}dxdt\le C{R}^{n\rho }{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(52)

When $k>2p^{\prime }$, it follows from some computations that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\left(b_R\left(x\right)\right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta b_R\left(x\right)|}^{p^{\prime }}dx\le C{R}^{n\rho -2\rho p^{\prime }}.\end{array}$$

(53)

Applying (22) and (53), we derive

$$\begin{array}{c}\hfill I_2={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{|\Delta \phi |}^{p^{\prime }}dxdt\le C{R}^{n\rho -2\rho p^{\prime }}{T}^{1-{\displaystyle \frac{\alpha }{p-1}}}.\end{array}$$

(54)

Taking advantage of (19) and (53), we have

$$\begin{array}{c}\hfill I_3={\int }_0^T{\int }_{{\mathbb{R}}^n}{\left(I_{t|T}^{\alpha }\phi \right)}^{-{\displaystyle \frac{1}{p-1}}}{\left|{(\Delta \phi )}_t\right|}^{p^{\prime }}dxdt\le C{R}^{n\rho -2\rho p^{\prime }}{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}.\end{array}$$

(55)

Employing (14), (49), (51), (52), (54) and (55), we acquire

$$\begin{array}{c}\hfill C{T}^{\sigma +1}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{R}^{n\rho }{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}+C{R}^{n\rho -2\rho p^{\prime }}{T}^{1-{\displaystyle \frac{\alpha }{p-1}}},\end{array}$$

which results in

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{R}^{n\rho }{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}-\sigma -1}+C{R}^{n\rho -2\rho p^{\prime }}{T}^{-{\displaystyle \frac{\alpha }{p-1}}-\sigma }.\end{array}$$

Fixing R and taking $T\to \infty$, we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0.\end{array}$$

This contradicts the assumption ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0$.

On the other hand, similar to the derivation of (29), we arrive at

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\left(b_R\left(x\right)\right)}^{-{\displaystyle \frac{1}{q-1}}}{|\nabla b_R\left(x\right)|}^{q^{\prime }}dx\le C{R}^{n\rho -\rho q^{\prime }}.\end{array}$$

(56)

Utilizing (28) and (56), one has the estimate

$$\begin{array}{c}\hfill I_5={\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{|\nabla \phi |}^{q^{\prime }}dxdt\le C{R}^{n\rho -\rho q^{\prime }}T.\end{array}$$

(57)

From (32) and (56), we achieve

$$\begin{array}{c}\hfill I_6={\int }_0^T{\int }_{{\mathbb{R}}^n}{\phi }^{-{\displaystyle \frac{1}{q-1}}}{\left|{(\nabla \phi )}_t\right|}^{q^{\prime }}dxdt\le C{R}^{n\rho -\rho q^{\prime }}{T}^{1-q^{\prime }}.\end{array}$$

(58)

Taking advantage of (15), (51), (52), (57) and (58) yields

$$\begin{array}{ccc}\hfill & & {\int }_0^T{\int }_{{\mathbb{R}}^n}{t}^{\sigma }w\left(x\right)\phi (t,x)dxdt\hfill \\ & & \le C{R}^{n\rho }{T}^{-1-{\displaystyle \frac{2+\alpha }{p-1}}}+C{R}^{n\rho }{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}+C{R}^{n\rho -\rho q^{\prime }}T+C{R}^{n\rho -\rho q^{\prime }}{T}^{1-q^{\prime }}\hfill \\ & & \le C{R}^{n\rho }{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}+C{R}^{n\rho -\rho q^{\prime }}T,\hfill \end{array}$$

(59)

which together with (49) implies

$$\begin{array}{c}\hfill C{T}^{\sigma +1}{\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{R}^{n\rho }{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}}+C{R}^{n\rho -\rho q^{\prime }}T.\end{array}$$

It turns out that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le C{R}^{n\rho }{T}^{-{\displaystyle \frac{\alpha +1}{p-1}}-\sigma -1}+C{R}^{n\rho -\rho q^{\prime }}{T}^{-\sigma }.\end{array}$$

Fixing R and letting $T\to \infty$, we conclude

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}w\left(x\right)dx\le 0,\end{array}$$

which contradicts the assumption ${\int }_{{\mathbb{R}}^n}w\left(x\right)dx>0$. This finishes the proof of Theorem 4. □