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Blowup for semilinear wave equation with space-dependent damping and combined nonlinearities

Abstract

This paper is concerned with the Cauchy problem for semilinear wave equation with space-dependent scattering damping and combined nonlinearities. The blowup results of solution are established by introducing proper test functions. Moreover, upper bound lifespan estimates of a solution to the Cauchy problem with small initial values are derived. To the best of our knowledge, the results in Theorems 1.11.2 are new.

Introduction and main results

In this work, we consider the following Cauchy problem of wave equation with space-dependent damping and combined nonlinearities:

$$\begin{aligned} \textstyle\begin{cases} u_{tt}- \Delta u + \frac{ \mu }{ (1+ \vert x \vert )^{\beta} } u_{ t} = \vert u_{t} \vert ^{p}+ \vert u \vert ^{q},\quad (t,x)\in [0,T)\times \mathbb{R}^{n} , \\ u(0,x)=\varepsilon f(x), \qquad u_{t}(0,x)=\varepsilon g(x), \quad x\in \mathbb{R}^{n}, \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(\mu >0\), \(\beta >1\), \(p>1\), \(q>1\), \(n\geq 2\). The compactly supported nonnegative initial values satisfy \((f,g)\in H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\) (\(n\geq 2\)) and

$$ f(x)\geq 0, g(x)\geq 0 \text{ a.e.},\quad f(x)=g(x)=0\quad \text{for } \vert x \vert >1. $$
(1.2)

In addition, \(f(x)\), \(g(x)\not \equiv 0\).

The study of formation of singularity for semilinear wave equation has a long history (see detailed illustrations in [3, 5, 9, 11, 2225, 2730, 33, 34, 3942] and the references therein). In fact, problem (1.1) originates from the following three problems:

$$\begin{aligned}& \textstyle\begin{cases} u_{tt}- \Delta u = \vert u \vert ^{p},\quad (t,x)\in [0,T)\times \mathbb{R}^{n}, n\geq 1, \\ u(0,x)=\varepsilon f(x), \qquad u_{t}(0,x)=\varepsilon g(x), \quad x\in \mathbb{R}^{n}, \end{cases}\displaystyle \end{aligned}$$
(1.3)
$$\begin{aligned}& \textstyle\begin{cases} u_{tt}- \Delta u = \vert u_{t} \vert ^{p},\quad (t,x)\in [0,T)\times \mathbb{R}^{n}, n\geq 1, \\ u(0,x)=\varepsilon f(x), \qquad u_{t}(0,x)=\varepsilon g(x), \quad x\in \mathbb{R}^{n}, \end{cases}\displaystyle \end{aligned}$$
(1.4)

and

$$\begin{aligned} \textstyle\begin{cases} u_{t}- \Delta u = \vert u \vert ^{p},\quad (t,x)\in [0,T)\times \mathbb{R}^{n}, n\geq 1, \\ u(0,x)=\varepsilon f(x), \quad x\in \mathbb{R}^{n}. \end{cases}\displaystyle \end{aligned}$$
(1.5)

Problem (1.3) is known as the Strauss conjecture (see [35]), which shows that the solution blows up in finite time when \(1< p\leq p_{S}(n)\) (\(n\geq 2\)) and \(p_{S}(1)=+\infty \) for \(n=1\), whereas the solution exists globally in time when \(p>p_{S}(n)\). Here \(p_{S}(n)\) is the Strauss critical exponent, which is the positive root of the quadratic equation

$$ \begin{aligned} \gamma (p,n)=-(n-1)p^{2}+(n+1)p+2=0.\end{aligned} $$

Problem (1.4) is known as the Glassey conjecture (see [6]), where the Glassey critical exponent is \(p_{G}(n)=\frac{n+1}{n-1}\). It is shown in [4] that the Cauchy problem of heat equation (1.5) possesses the Fujita critical exponent \(p_{F}(n) =1+\frac{2}{n}\).

Scholars investigated the blowup dynamics of a semilinear wave equation with damping term

$$ \begin{aligned} u_{tt}-\Delta u+h(u_{t})=f(u,u_{t}), \end{aligned} $$
(1.6)

where \(h(u_{t})=\frac{\mu}{(1+t)^{\beta}}u_{t}\), \(\frac{\mu}{(1+|x|)^{\beta}}u_{t}\) (\(\mu >0\), \(\beta \in \mathbb{R}\)) and \(f(u,u_{t})=|u|^{p},|u_{t}|^{p}\), \(|u_{t}|^{p}+|u|^{q}\) (\(p>1\), \(q>1\)). When the critical exponent of the damped wave equation (1.6) is related to the Srauss exponent \(p_{S}(n)\) or the Glassey exponent \(p_{G}(n)\), we say that the equation behaves like that of the wave equation. This means that the damping term in the equation makes no effect. When the critical exponent is related to the Fujita exponent \(p_{F}(n)\), we say that the damping term makes an effect. According to the range of β, we use the following table to show the effect of damping terms (we can see it in [18, 21]).

Blowup and global existence results in connection with the semilinear wave equation with time-dependent damping \(\frac{\mu}{(1+t)^{\beta}}u_{t} \) are established in [1, 13, 16, 20, 31, 37, 38]. Energy estimates of solution to semilinear wave equation with space-dependent damping are derived in [14, 15, 36]. Nishihara et al. [32] investigated the blowup and global existence for a semilinear wave equation with space- and time-dependent damping. In the present paper, we mainly concentrate on the problem with space-dependent scattering damping case \(\frac{\mu}{(1+|x|)^{\beta}}u_{t}\) (\(\beta >1\)). Namely, the behavior of a solution is similar to that of the wave equation in this case. Lai and Tu [17] considered upper bound lifespan estimates of a solution to the wave equation with space-dependent damping \(\frac{\mu}{(1+|x|)^{\beta}}u_{t}\) (\(\beta >2\), \(n\geq 2\)) and \(f(u,u_{t})=|u|^{p}\), \(|u_{t}|^{p}\) for both subcritical and critical exponents. Especially, for the power nonlinearity \(|u|^{p}\) (\(\frac{n}{n-1}< p\leq p_{S}(n)\)) and derivative-type nonlinearity \(|u_{t}|^{p}\) (\(1< p\leq p_{G}(n)\)), they obtained the same critical exponents and upper bound lifespan estimates of solutions as in the situation without damping by using the test function method. Lai et al. [17] obtained upper bound lifespan estimate of solution when \(f(u,u_{t})=|u|^{p}\) and \(\beta >1\). Meanwhile, the lifespan estimate for the case \(1< p<\frac{n}{n-1}\) was also improved.

Table 1 The effect of the damping terms

We are in the position to present some known results related to the semilinear wave equation (1.6) with combined nonlinearities \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\). Han and Zhou [10] obtained an upper bound lifespan estimate of solution to the Cauchy problem without damping term by constructing a proper test function and solving ordinary differential inequalities. Hidano et al. [12] established the sharp lower bound lifespan estimate of a solution to the problem. Dai et al. [2] derived the sharp lifespan estimate of a solution to the nonlinear wave equation when \(p\geq q_{S}(n)\) and \(q=q_{S}(n)\) (\(n=2, 3\)), where \(q_{S}(n)\) is the Strauss critical exponent of the semilinear wave equation with power nonlinearity \(|u|^{q}\). Lai and Takamura [19] illustrated blowup results and upper bound lifespan estimates of a solution to the problem with time-dependent damping term \(\frac{\mu}{(1+t)^{\beta}}u_{t}\) (\(\beta >1\)) by using a multiplier and iteration argument. Blowup of a solution to the problem with scale-invariant damping \(\frac{\mu}{ 1+t }u_{t} \) was investigated by applying test function approach (see [7, 8]). Liu and Wang [26] consider problem (1.1) for the more general nonlinearity \(f(u,u_{t})=c_{1}|u_{t}|^{p}+c_{2}|u|^{q}\) on asymptotically Euclidean manifolds. Upper bound lifespan estimates of solution with different values of \(c_{1}\) and \(c_{2}\) are obtained. In addition, the existence of a solution is established.

Inspired by the works [10, 17, 19, 21], we consider blowup and upper bound lifespan estimates of a solution to problem (1.1). To our best knowledge, the blowup for the space-dependent damped wave equation with combined nonlinearities has not been discussed yet. The purpose of this paper is to fill this gap. We establish upper bound lifespan estimates of a solution. It is worth mentioning that in this paper, we employ the test function method different from the technique in [10, 19]. We bear in mind that lifespan estimates of solutions to the problems with space-dependent damping \(\frac{\mu}{(1+|x|)^{\beta}}u_{t}\) (\(\beta >2\)) and \(f(u,u_{t})=|u|^{p}, |u_{t}|^{p}\) are investigated in [21]. Thanks to the work [17], we obtain upper bound lifespan estimates of a solution to problem (1.1) with \(\frac{\mu}{(1+|x|)^{\beta}}u_{t}\) (\(\beta >1\)) and combined nonlinearities \(|u_{t}|^{p}+|u |^{q}\) (see the new results in Theorems 1.11.2 in this paper).

The main results in this paper are described as follows.

Theorem 1.1

Let \(n\geq 2\), \(\mu >0\), and \(\beta >1\), and let f and g satisfy (1.2). Suppose that problem (1.1) has an energy solution u such that

$$ \operatorname{supp}(u, u_{t})\subset \bigl\{ (t,x)\in [0,T)\times \mathbb{R}^{n} | \vert x \vert \leq t+1 \bigr\} .$$

Then we have the following lifespan estimates of solution:

$$ T(\varepsilon )\leq \textstyle\begin{cases} C\varepsilon ^{-\frac{2p(q-1)}{-(n-1)pq+(n-1)p+2q+2}}, & \max \{1, \frac{2}{n-1}\}< p< \frac{4n-2}{n-1}, \\ &\frac{n}{n-1}< q< 1+\frac{4}{(n-1)p-2}, \\ C\varepsilon ^{-\frac{2(p-1)}{n+1-(n-1)p}}, & 1< p< \frac{n+1}{n-1}, 1< q, \end{cases} $$
(1.7)

where C is a positive constant.

Theorem 1.2

Let \(n\geq 2\), \(\mu >0\), and \(\beta >1\), and let f and g satisfy (1.2). Suppose that problem (1.1) has an energy solution u such that

$$ \operatorname{supp}(u, u_{t})\subset \bigl\{ (t,x)\in [0,T)\times \mathbb{R}^{n} | \vert x \vert \leq t+1 \bigr\} .$$

Then the lifespan estimates of solution satisfy

$$ T(\varepsilon )\leq \textstyle\begin{cases} C\varepsilon ^{-\frac{2q(p-1)}{-(n-1)qp+(n-1)q+2p}}, & 1< p, n=2,3 \textit{ or } 1< p< \frac{n-1}{n-3}, n>3, \\ & 1< q< \frac{2p}{(n-1)(p-1)}, \\ C\varepsilon ^{-\frac{2(q-1)}{(n+1)-(n-1)q}}, & 1< p, 1< q< \frac{n+1}{n-1}. \end{cases} $$
(1.8)

Remark 1.1

In Theorem 1.1, for \(\max \{1+\frac{1}{2(n-1)},\frac{2}{n-1}\}< p<\frac{n+1}{n-1}\) and \(\frac{n}{n-1}< q<2p-1\), we have

$$ \frac{2p(q-1)}{-(n-1)pq+(n-1)p+2q+2}< \frac{2(p-1)}{n+1-(n-1)p},$$

where we have used the fact \(2p-1<1+\frac{4}{(n-1)p-2}\) for \(p<\frac{n+1}{n-1}\). When \(\max \{1,\frac{2}{n-1}\}< p<\frac{n+1}{n-1}\) and \(\max \{2p-1,\frac{n}{n-1}\}< q<1+\frac{4}{(n-1)p-2}\), we obtain

$$ \frac{2p(q-1)}{-(n-1)pq+(n-1)p+2q+2}>\frac{2(p-1)}{n+1-(n-1)p}.$$

We use Fig. 1 to make a simple description for \(n=2\).

Figure 1
figure 1

The case \(n=2\) in Theorem 1.1

For \(p,q\in B\cup C \cup E\), we have the first lifespan estimate in (1.7). For \(p,q\in A\cup B\cup C\cup D\), we obtain the second lifespan estimate in (1.7), whereas for \(p,q\in B\), the second lifespan estimate in (1.7) is better than the first one. For \(p,q\in C\), the first lifespan estimate in (1.7) is better than the second one.

Remark 1.2

In Theorem 1.2, for \(1< q< p<\frac{n+1}{n-1}\) or \(\frac{n+1}{n-1}< p\) (\(n=2,3\)), \(\frac{n+1}{n-1}< p<\frac{n-1}{n-3}\) (\(n>3\)), \(1< q<\frac{2p}{(n-1)(p-1)}\), we have

$$ \frac{2(q-1)}{(n+1)-(n-1)q}< \frac{2q(p-1)}{-(n-1)qp+(n-1)q+2p}.$$

When \(1< p< q<\frac{n+1}{n-1}\), we have

$$ \frac{2(q-1)}{(n+1)-(n-1)q}>\frac{2q(p-1)}{-(n-1)qp+(n-1)q+2p}.$$

Similarly, we use Fig. 2 to illustrate the specific comparison for \(n=2\).

Figure 2
figure 2

The case \(n=2\) in Theorem 1.2

For \(p,q\in F\cup G\cup H\), we obtain the first lifespan estimate in (1.8). For \(p,q\in G\cup H\cup I\), we have the second lifespan estimate in (1.8). For \(p,q\in G\), the first lifespan estimate in (1.8) is better than the second one, and for \(p,q\in H\), the second lifespan estimate in (1.8) is better than the first one.

Remark 1.3

Let \(n\geq 2\), \(\mu >0\), and \(\beta >1\). The assumptions in Theorems 1.1 and 1.2 hold. Combining the results in [17, 21] with (1.7) and (1.8), we derive

$$ T(\varepsilon )\leq \textstyle\begin{cases} \exp (C\varepsilon ^{-(p-1)}), & p= p_{G}(n), q> \frac{n+3}{n-1}, \\ C\varepsilon ^{-\Gamma _{G}(n,p)^{-1}}, & q>2p-1, 1< p< \frac{n+1}{n-1}, \\ C\varepsilon ^{-{\Gamma _{1}(p,q,n)}^{-1}},& \frac{n}{n-1}< q, p\leq q< 2p-1, \Gamma _{1}(p,q,n)>0, \\ C\varepsilon ^{-{\Gamma _{S}(n,q)}^{-1}}, & p>q, \frac{n}{n-1}< q< p_{S}(n), \\ C\varepsilon ^{-{\Gamma _{2}(p,q,n)}^{-1}}, & p< q< \frac{n}{n-1}, \\ C\varepsilon ^{-\frac{2(q-1)}{(n+1)-(n-1)q}},& p>q, q< \frac{n}{n-1}, \\ \exp (C\varepsilon ^{-q(q-1)}),& p\geq q=p_{S}(n), \end{cases} $$
(1.9)

where

$$\begin{aligned}& \Gamma _{1}(n,p,q)=\frac{-(n-1)pq+(n-1)p+2q+2}{2p(q-1)}, \\& \Gamma _{2}(n,p,q)=\frac{-(n-1)pq+(n-1)q+2p}{2q(p-1)}, \\& \Gamma _{G}(n,p)=\frac{n+1-(n-1)p}{2(p-1)}, \\& \Gamma _{S}(n,q)= \frac{\gamma (q,n)}{2q(q-1)}, \end{aligned}$$

\(p_{S}(n)\) denotes the Strauss critical exponent, and \(p_{G}(n)\) represents the Glassey critical exponent.

Throughout this paper, C denotes a positive constant independent of ε, which may vary from line to line.

Preliminaries

In this section, we present several basic definitions and lemmas.

Definition 2.1

A function u is called an energy solution of problem (1.1) on \([0,T)\) if

$$ u\in \bigcap_{i=0}^{1}C^{i} \bigl([0,T\bigr);H^{1-i}\bigl(\mathbb{R}^{n}\bigr))\cap C^{1}\bigl([0,T\bigr);L^{p}\bigl( \mathbb{R}^{n} \bigr))\cap L_{\mathrm{loc}}^{q}\bigl([0,T\bigr)\times \mathbb{R}^{n})$$

satisfies \(u(0,x)=\varepsilon f(x)\) and \(u_{t}(0,x)=\varepsilon g(x)\). Moreover, we have

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\varphi (0,x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\varphi (t,x)\,dx\,dt \\ & \qquad {} + \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}\varepsilon f(x)\varphi (0,x)\,dx \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl(-\partial _{t}u(t,x) \partial _{t}\varphi (t,x)+\nabla u(t,x)\nabla \varphi (t,x)\bigr)\,dx\,dt \\ & \qquad {} - \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u(t,x)\partial _{t} \varphi (t,x)\,dx\,dt, \end{aligned} $$
(2.1)

where \(\varphi (t,x)\in C_{0}^{\infty}([0,T)\times \mathbb{R}^{n})\) and \(T\in (1,T(\varepsilon ))\). Here \(T(\varepsilon )\) represents the upper bound lifespan estimate of a solution to problem (1.1), which satisfies

$$ T(\varepsilon )=\sup \{T>0, \text{there exists an energy solution to problem (1.1)}\}.$$

Definition 2.2

The cutoff function \(\eta (t)\in C^{\infty}([0,\infty ))\) is defined by

$$ \eta (t)=\textstyle\begin{cases} 1, & t\leq \frac{1}{2}, \\ \text{decreasing}, & t\in (\frac{1}{2}, 1), \\ 0, & t\geq 1,\end{cases} $$

which satisfies \(|\eta '(t)|,|\eta ''(t)|< C\). Let \(\eta _{T}(t)=\eta (t/T)\) and \(\gamma >1\). We have that

$$\begin{aligned}& \partial _{t}\eta _{T}^{2\gamma}=\frac{2\gamma}{T} \eta _{T}^{2\gamma -1} \eta ', \\& \partial ^{2}_{t}\eta _{T}^{2\gamma}= \frac{2\gamma (2\gamma -1)}{T^{2}}\eta _{T}^{2\gamma -2} \bigl\vert \eta ' \bigr\vert ^{2}+ \frac{2\gamma}{T^{2}}\eta _{T}^{2\gamma -1}\eta ''. \end{aligned}$$

Lemma 2.3

(Lemma 3.1 in [21])

If \(\beta >0\), then for all \(\alpha \in \mathbb{R}\) and a fixed constant R, there exists a positive constant C such that

$$ \begin{aligned} \int _{0}^{t+R}(1+r)^{\alpha}e^{-\beta (t-r)} \,dr\leq C(t+R)^{\alpha}. \end{aligned} $$
(2.2)

Lemma 2.4

(Lemma 2.5 in [17])

Let \(n\geq 2\), \(\beta >1\), and \(\mu \geq 0\). Then the equation

$$ \begin{aligned} \Delta \phi (x)-\frac{\mu}{(1+ \vert x \vert )^{\beta}}\phi (x)=\phi (x) \end{aligned} $$
(2.3)

admits a solution \(\phi (x)\). Moreover, there exists a constant \(C_{1}\in (0,1)\) such that

$$ \begin{aligned} C_{1}\bigl(1+ \vert x \vert \bigr)^{-\frac{n-1}{2}}e^{ \vert x \vert }< \phi (x)< C_{1}^{-1} \bigl(1+ \vert x \vert \bigr)^{- \frac{n-1}{2}}e^{ \vert x \vert }. \end{aligned} $$
(2.4)

Let \(\psi (t,x)=e^{-t}\phi (x)\). Then we have

$$ \partial _{t}^{2}\psi (t,x)-\Delta \psi (t,x)- \frac{\mu}{(1+ \vert x \vert )^{\beta}}\partial _{t}\psi (t,x)=0.$$

Proof of Theorem 1.1

In this section, we illustrate the proof of Theorem 1.1.

Case \(p\geq q\)

First, we choose \(\varphi (t,x)=\eta _{T}^{2q'}\) as the test function, where \(q'\) satisfies \(\frac{1}{q}+\frac{1}{q'}=1\). From (2.1) we obtain

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\eta _{T}^{2q'} \,dx\,dt + \int _{ \mathbb{R}^{n}}\frac{\mu}{(1+ \vert x \vert )^{\beta}}\varepsilon f(x)\,dx \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u\partial _{t}^{2} \eta _{T}^{2q'}\,dx\,dt - \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial _{t}\eta _{T}^{2q'}\,dx\,dt \\ &\quad =I_{1}+I_{2}, \end{aligned} $$
(3.1)

where we have used the fact that \(\partial _{t}\eta _{T}(0)=0\) and \(\eta _{T}(T)=0\).

Using the Hölder and Young inequalities, we have that for \(q>\frac{n}{n-1}\),

$$\begin{aligned}& \begin{aligned} I_{1}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}u\partial _{t}^{2} \eta _{T}^{2q'}\,dx\,dt \\ & \leq \frac{C}{T^{2}} \int _{0}^{T} \int _{ \mathbb{R}^{n}} \bigl\vert u\eta _{T}^{2q'-2} \bigr\vert \,dx\,dt \\ & \leq \frac{C}{T^{2}}\biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2q'}\,dx\,dt\biggr)^{\frac{1}{q}} \biggl( \int _{0}^{T} \int _{\{r\leq t+1\}}1\,dx\,dt\biggr)^{\frac{1}{q'}} \\ &\leq CT^{n+1-2q'}+\frac{1}{3} \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2q'}\,dx\,dt, \end{aligned} \end{aligned}$$
(3.2)
$$\begin{aligned}& \begin{aligned} I_{2}&=- \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial _{t}\eta _{T}^{2q'}\,dx\,dt \\ & \leq \frac {C}{T} \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}} \bigl\vert u\eta _{T}^{2q'-2} \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2q'}\,dx\,dt \biggr)^{\frac{1}{q}} \biggl( \int _{0}^{T} \int _{0}^{t+1}\frac{r^{n-1-q'}}{(1+r)^{q'(\beta -1)}}\,dr\,dt \biggr)^{\frac{1}{q'}} \\ & \leq CT^{n+1-2q'}+\frac{1}{3} \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2q'}\,dx\,dt. \end{aligned} \end{aligned}$$
(3.3)

Combining (3.1)–(3.3), we deduce

$$ \begin{aligned} C_{1}(f,g)\varepsilon + \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr) \eta _{T}^{2q'} \,dx\,dt \leq CT^{n+1-2q'}, \end{aligned} $$
(3.4)

where \(C_{1}(f,g)=C(\int _{\mathbb{R}^{n}}g(x)\,dx+\int _{\mathbb{R}^{n}} \frac{\mu}{(1+|x|)^{\beta}}f(x)\,dx)\).

Let \(\varphi (t,x)=\partial _{t}\Phi _{1}(t,x)\), where \(\Phi _{1}(t,x)=-\eta _{T}^{2q'}\psi (t,x)=-\eta _{T}^{2q'}e^{-t} \phi (x)\), and \(\psi (t,x)\) is defined in Lemma 2.4. Applying (2.1), we have

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\partial _{t} \Phi _{1}\,dx\,dt \\ & \qquad {} + \int _{\mathbb{R}^{n}}\frac{\mu}{(1+ \vert x \vert )^{\beta}} \varepsilon f(x)\phi (x)\,dx \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}-\partial _{t}u \partial _{t}^{2}\Phi _{1}\,dx\,dt+ \int _{0}^{T} \int _{\mathbb{R}^{n}} \nabla u\nabla \partial _{t}\Phi _{1}\,dx\,dt \\ &\qquad {} - \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial ^{2}_{t} \Phi _{1}\,dx\,dt, \end{aligned} $$

where we have employed the fact \(\partial _{t}\Phi _{1}(0,x)=\phi (x)\). Since

$$\begin{aligned}& \int _{0}^{T} \int _{\mathbb{R}^{n}}-\partial _{t}u \partial _{t}^{2}\Phi _{1}\,dx\,dt = \int _{0}^{T} \int _{ \mathbb{R}^{n}}\partial _{t}u\bigl(\partial _{t}^{2}\eta _{T}^{2q'}\psi + \eta _{T}^{2q'}\psi -2\partial _{t}\eta _{T}^{2q'}\psi \bigr)\,dx\,dt, \\& \int _{0}^{T} \int _{\mathbb{R}^{n}}\nabla u \nabla \partial _{t}\Phi _{1}\,dx\,dt \\& \quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}\partial _{t}( \nabla u\nabla \Phi _{1})\,dx\,dt- \int _{0}^{T} \int _{\mathbb{R}^{n}} \nabla u_{t}\nabla \Phi _{1} \,dx\,dt \\& \quad = \int _{\mathbb{R}^{n}}\varepsilon \nabla f(x)\nabla \phi (x)\,dx- \int _{0}^{T} \int _{\mathbb{R}^{n}}\nabla u_{t}\nabla \Phi _{1}\,dx \,dt \\& \quad = -\varepsilon \int _{\mathbb{R}^{n}}\Delta \phi f(x)\,dx- \int _{0}^{T} \int _{\mathbb{R}^{n}}\nabla u_{t}\nabla \Phi _{1}\,dx \,dt \\& \quad = -\varepsilon \int _{\mathbb{R}^{n}}\Delta \phi f(x)\,dx- \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t}\eta _{T}^{2q'} \Delta \psi \,dx\,dt, \end{aligned}$$

and

$$ \begin{aligned} & \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial ^{2}_{t} \Phi _{1}\,dx\,dt \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}\partial _{t}\biggl( \frac{\mu}{(1+ \vert x \vert )^{\beta}}u \partial _{t}\Phi _{1}\biggr)- \frac{\mu}{(1+ \vert x \vert )^{\beta}} u_{t}\partial _{t}\Phi _{1}\,dx\,dt \\ & \quad = - \int _{\mathbb{R}^{n}}\frac{\mu}{(1+ \vert x \vert )^{\beta}} \varepsilon f(x)\phi (x)\,dx- \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}} u_{t}\bigl(\eta _{T}^{2q'}\psi -\partial _{t} \eta _{T}^{2q'}\psi \bigr)\,dx\,dt, \end{aligned} $$

we have

$$\begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\partial _{t} \Phi _{1}\,dx\,dt \\ & \qquad {} +\varepsilon \int _{\mathbb{R}^{n}}\frac{\mu}{(1+ \vert x \vert )^{\beta}}f(x) \phi (x)\,dx \\ & \quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t}\bigl(\partial _{t}^{2} \eta _{T}^{2q'}\psi +\eta _{T}^{2q'}\psi -2\partial _{t}\eta _{T}^{2q'} \psi \bigr)\,dx \,dt- \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \eta _{T}^{2q'} \Delta \psi \,dx\,dt \\ & \qquad {} - \varepsilon \int _{\mathbb{R}^{n}}\biggl(1+ \frac{\mu}{(1+ \vert x \vert )^{\beta}}\biggr)f(x)\phi (x)\,dx + \int _{ \mathbb{R}^{n}}\frac{\mu}{(1+ \vert x \vert )^{\beta}}\varepsilon f(x)\phi (x)\,dx \\ & \qquad {} + \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u_{t}\bigl[\eta _{T}^{2q'}\psi -\partial _{t} \eta _{T}^{2q'}\psi \bigr]\,dx\,dt. \end{aligned}$$

It follows that

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\partial _{t} \Phi _{1}\,dx\,dt \\ & \qquad {} +\varepsilon \int _{\mathbb{R}^{n}}\biggl(1+ \frac{\mu}{(1+ \vert x \vert )^{\beta}}\biggr) f(x)\phi (x)\,dx \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t}\biggl(\partial _{t}^{2} \eta _{T}^{2q'}\psi +2\partial _{t}\eta _{T}^{2q'}\partial _{t}\psi - \frac{\mu}{(1+ \vert x \vert )^{\beta}} \partial _{t}\eta _{T}^{2q'}\psi \biggr)\,dx\,dt \\ & \qquad {} + \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t}\eta _{T}^{2q'} \biggl[- \Delta \psi +\psi -\frac{\mu}{(1+ \vert x \vert )^{\beta}}\partial _{t}\psi \biggr] \,dx\,dt \\ & \quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t}\biggl(\partial _{t}^{2} \eta _{T}^{2q'}\psi +2\partial _{t}\eta _{T}^{2q'}\partial _{t}\psi - \frac{\mu}{(1+ \vert x \vert )^{\beta}} \partial _{t}\eta _{T}^{2q'}\psi \biggr)\,dx\,dt \\ & \quad =I_{3}+I_{4}+I_{5}, \end{aligned} $$
(3.5)

where we have applied Lemma 2.4.

We are in the position to derive the estimates for \(I_{3}\), \(I_{4}\), and \(I_{5}\).

Employing Lemma 2.3 leads to

$$\begin{aligned}& I_{3}= \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \partial _{t}^{2} \eta _{T}^{2q'}\psi \,dx\,dt \\& \hphantom{I_{3}} \leq \frac {C}{T^{2}} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2q'-2}\psi \bigr\vert \,dx\,dt \\& \hphantom{I_{3}} \leq \frac {C}{T^{2}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{(2q'-2)p}\,dx\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert \psi \vert ^{p'}\,dx\,dt \biggr)^{ \frac{1}{p'}} \end{aligned}$$
(3.6)
$$\begin{aligned}& \hphantom{I_{3}} \leq \frac {C}{T^{2}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\,dx\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert \psi \vert ^{p'}\,dx\,dt \biggr)^{ \frac{1}{p'}} \\& \hphantom{I_{3}} \leq CT^{-2+(n-\frac{n-1}{2}p')\frac{1}{p'}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\,dx\,dt \biggr)^{\frac{1}{p}}, \\& \begin{aligned} I_{4}&=2 \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \partial _{t}\eta _{T}^{2q'}\partial _{t}\psi \,dx\,dt \\ & \leq \frac {C}{T} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2q'-1}\psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p} \eta _{T}^{2q'}\,dx\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert \psi \vert ^{p'}\,dx\,dt \biggr)^{\frac{1}{p'}} \\ & \leq CT^{-1+(n-\frac{n-1}{2}p')\frac{1}{p'}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\,dx\,dt \biggr)^{\frac{1}{p}}, \end{aligned} \end{aligned}$$
(3.7)
$$\begin{aligned}& \vert I_{5} \vert \leq C I_{4}\leq CT^{-1+(n-\frac{n-1}{2}p') \frac{1}{p'}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\,dx\,dt \biggr)^{\frac{1}{p}}. \end{aligned}$$
(3.8)

A direct calculation gives rise to

$$ \begin{aligned} \partial _{t}\Phi _{1}=\eta _{T}^{2q'}\psi -2q'\eta _{T}^{2q'-1} \partial _{t}\eta _{T} \psi \geq \eta _{T}^{2q'} \psi >0. \end{aligned} $$

Using (3.5)–(3.8), we have

$$ \varepsilon C_{2}(f,g)\leq CT^{-1+(n-\frac{n-1}{2}p') \frac{1}{p'}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\,dx\,dt \biggr)^{\frac{1}{p}}, $$

which implies

$$ C\varepsilon ^{p}T^{n-\frac{n-1}{2}p}\leq \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\,dx\,dt, $$
(3.9)

where \(C_{2}(f,g)=C(\int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+\int _{\mathbb{R}^{n}}(1+ \frac{\mu}{(1+|x|)^{\beta}}) f(x)\phi (x)\,dx)\).

Combining (3.4) and (3.9) and using the assumptions \(\max \{1, \frac{2}{n-1}\}< p<\frac{4n-2}{n-1}\), \(\frac{n}{n-1}< q<1+\frac{4}{(n-1)p-2}\), and \(q\leq p\), we obtain

$$ \begin{aligned} T(\varepsilon )\leq C\varepsilon ^{- \frac{2p(q-1)}{-(n-1)pq+(n-1)p+2q+2}}. \end{aligned} $$

On the other hand, according to (3.5), we derive

$$\begin{aligned}& \begin{aligned} I_{3}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \partial _{t}^{2} \eta _{T}^{2q'}\psi \,dx\,dt \\ & \leq \frac {C}{T^{2}} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2q'-2}\psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T^{2}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{(2q'-2)p}\psi \,dx\,dt \biggr)^{ \frac{1}{p}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}}\psi \,dx\,dt \biggr)^{ \frac{1}{p'}} \\ & \leq CT^{-2+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\psi \,dx\,dt \biggr)^{\frac{1}{p}}, \end{aligned} \end{aligned}$$
(3.10)
$$\begin{aligned}& \begin{aligned} I_{4}&=2 \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \partial _{t}\eta _{T}^{2q'}\partial _{t}\psi \,dx\,dt \\ & \leq \frac {C}{T} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2q'-1}\psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p} \eta _{T}^{2q'}\psi \,dx\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}}\psi \,dx\,dt \biggr)^{\frac{1}{p'}} \\ & \leq CT^{-1+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\psi \,dx\,dt \biggr)^{\frac{1}{p}}, \end{aligned} \end{aligned}$$
(3.11)
$$\begin{aligned}& \vert I_{5} \vert \leq C I_{4}\leq CT^{-1+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2q'}\psi \,dx\,dt \biggr)^{\frac{1}{p}}. \end{aligned}$$
(3.12)

Taking into account (3.5) and (3.10)–(3.12) and using the Young inequality, we have

$$ \begin{aligned} \varepsilon C_{2}(f,g)\leq C T^{-p'+\frac{n+1}{2}}. \end{aligned} $$

Therefore, for \(1< p<\frac{n+1}{n-1}\) and \(1< q\leq p\), we have that

$$ \begin{aligned} T(\varepsilon )\leq C \varepsilon ^{-\frac{2(p-1)}{n+1-(n-1)p}}. \end{aligned} $$
(3.13)

Case \(p< q\)

Taking \(\varphi (t,x)=\eta _{T}^{2p'}\) in (2.1) yields

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\eta _{T}^{2p'} \,dx\,dt + \int _{ \mathbb{R}^{n}}\frac{\mu}{(1+ \vert x \vert )^{\beta}}\varepsilon f(x)\,dx \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u\partial _{t}^{2} \eta _{T}^{2p'}\,dx\,dt - \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial _{t}\eta _{T}^{2p'}\,dx\,dt \\ & \quad =I_{6}+I_{7}. \end{aligned} $$
(3.14)

Applying the fact that \(p< q\) and \(q>\frac{n}{n-1}\), we deduce

$$\begin{aligned}& \begin{aligned} I_{6}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}u\partial _{t}^{2} \eta _{T}^{2p'}\,dx\,dt \\ & \leq \frac {C}{T^{2}} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u \eta _{T}^{2p'-2} \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T^{2}}\biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q} \eta _{T}^{2p'}\,dx\,dt\biggr)^{\frac{1}{q}} \biggl( \int _{0}^{T} \int _{\{r\leq t+1\}}1\,dx\,dt\biggr)^{ \frac{1}{q'}} \\ & \leq CT^{n+1-2q'}+\frac{1}{3} \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2p'}\,dx\,dt, \end{aligned} \end{aligned}$$
(3.15)
$$\begin{aligned}& I_{7}=- \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial _{t}\eta _{T}^{2p'}\,dx\,dt \\& \begin{aligned}&\hphantom{I_{7}} \leq \frac {C}{T} \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}} \bigl\vert u\eta _{T}^{2p'-2} \bigr\vert \,dx\,dt \\ &\hphantom{I_{7}} \leq \frac {C}{T} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q} \eta _{T}^{2p'}\,dx\,dt \biggr)^{\frac{1}{q}} \biggl( \int _{0}^{T} \int _{0}^{t+1} \frac{r^{n-1-q'}}{(1+r)^{q'(\beta -1)}}\,dr\,dt \biggr)^{\frac{1}{q'}} \end{aligned} \\& \hphantom{I_{7}} \leq CT^{n+1-2q'}+\frac{1}{3} \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2p'}\,dx\,dt. \end{aligned}$$
(3.16)

Combining (3.14)–(3.16), we get

$$ \begin{aligned} C_{1}(f,g)\varepsilon + \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr) \eta _{T}^{2p'} \,dx\,dt \leq CT^{n+1-2q'}. \end{aligned} $$
(3.17)

We set \(\varphi (t,x)=\partial _{t}\Phi _{2}(x,t)\) in (2.1), where \(\Phi _{2}(x,t)=-\eta _{T}^{2p'}\psi (x,t)=-\eta _{T}^{2p'}e^{-t} \phi (x)\). Therefore we have

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\partial _{t} \Phi _{2}\,dx\,dt \\ &\qquad {} +\varepsilon \int _{\mathbb{R}^{n}}\biggl(1+ \frac{\mu}{(1+ \vert x \vert )^{\beta}}\biggr) f(x)\phi (x)\,dx \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t}\biggl(\partial _{t}^{2} \eta _{T}^{2p'}\psi +2\partial _{t}\eta _{T}^{2p'}\partial _{t}\psi - \frac{\mu}{(1+ \vert x \vert )^{\beta}} \partial _{t}\eta _{T}^{2p'}\psi \biggr)\,dx\,dt \\ &\quad =I_{8}+I_{9}+I_{10}. \end{aligned} $$
(3.18)

Similarly to the deduction in (3.9), we obtain

$$ \begin{aligned} C\varepsilon ^{p}T^{n-\frac{n-1}{2}p} \leq \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2p'}\,dx\,dt. \end{aligned} $$
(3.19)

From (3.17) and (3.19) the conditions \(\max \{1, \frac{2}{n-1}\}< p<\frac{4n-2}{n-1}\) and \(\max \{p,\frac{n}{n-1}\}< q<1+\frac{4}{(n-1)p+2}\) lead to

$$ \begin{aligned} T(\varepsilon )\leq C\varepsilon ^{- \frac{2p(q-1)}{-(n-1)pq+(n-1)p+2q+2}}.\end{aligned} $$

By (3.18) we have

$$\begin{aligned}& \begin{aligned} I_{8}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \partial _{t}^{2} \eta _{T}^{2p'}\psi \,dx\,dt \\ & \leq \frac {C}{T^{2}} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2p'-2}\psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T^{2}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{(2p'-2)p}\psi \,dx\,dt \biggr)^{ \frac{1}{p}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}}\psi \,dx\,dt \biggr)^{ \frac{1}{p'}} \\ & \leq CT^{-2+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2p'}\psi \,dx\,dt \biggr)^{\frac{1}{p}}, \end{aligned} \end{aligned}$$
(3.20)
$$\begin{aligned}& I_{9}=2 \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \partial _{t}\eta _{T}^{2p'}\partial _{t}\psi \,dx\,dt \\& \begin{aligned}\hphantom{I_{9}}& \leq \frac {C}{T} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2p'-1}\psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p} \eta _{T}^{2p'}\psi \,dx\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}}\psi \,dx\,dt \biggr)^{\frac{1}{p'}}\end{aligned} \end{aligned}$$
(3.21)
$$\begin{aligned}& \hphantom{I_{9}}\leq CT^{-1+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2p'}\psi \,dx\,dt \biggr)^{\frac{1}{p}}, \\& \vert I_{10} \vert \leq C I_{9}\leq CT^{-1+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2p'}\psi \,dx\,dt \biggr)^{\frac{1}{p}}. \end{aligned}$$
(3.22)

Combining (3.18) and (3.20)–(3.22), for \(1< p<\min \{q,\frac{n+1}{n-1}\}\) and \(q>1\), we have

$$ \begin{aligned} T(\varepsilon )\leq C \varepsilon ^{-\frac{2(p-1)}{n+1-(n-1)p}}. \end{aligned} $$
(3.23)

This completes the proof of Theorem 1.1.

Proof of Theorem 1.2

Taking \(\varphi (t,x)=\eta _{T}^{2k}\psi (t,x)\) in (2.1), where \(k=\max \{p',q'\}\) and \(\psi (t,x)=e^{-t}\phi (x)\), we obtain

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\eta _{T}^{2k} \psi \,dx\,dt \\ &\qquad {} + \int _{\mathbb{R}^{n}}\frac{\mu}{(1+ \vert x \vert )^{\beta}}\varepsilon f(x) \phi (x)\,dx \\ & \quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}-\partial _{t}u \partial _{t} \bigl[\eta _{T}^{2k}\psi \bigr]\,dx\,dt+ \int _{0}^{T} \int _{ \mathbb{R}^{n}}\nabla u\nabla \bigl[\eta _{T}^{2k} \psi \bigr]\,dx\,dt \\ & \qquad {} - \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial _{t}\bigl[ \eta _{T}^{2k}\psi \bigr]\,dx\,dt. \end{aligned} $$

A direct calculation shows that

$$\begin{aligned}& \int _{0}^{T} \int _{\mathbb{R}^{n}}-\partial _{t}u \partial _{t} \bigl[\eta _{T}^{2k}\psi \bigr]\,dx\,dt \\& \quad =- \int _{0}^{T} \int _{\mathbb{R}^{n}}\partial _{t} \bigl(u\partial _{t}\bigl[\eta _{T}^{2k}\psi \bigr] \bigr)-u \partial _{t}^{2}\bigl[ \eta _{T}^{2k} \psi \bigr]\,dx\,dt \\& \quad =- \int _{\mathbb{R}^{n}}\varepsilon f(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}u\bigl[\partial _{t}^{2}\eta _{T}^{2k} \psi +2\partial _{t}\eta _{T}^{2k}\partial _{t}\psi +\eta _{T}^{2k} \partial _{t}^{2}\psi \bigr] \,dx\,dt, \\& \int _{0}^{T} \int _{\mathbb{R}^{n}}\nabla u\nabla \bigl( \eta _{T}^{2k} \psi \bigr)\,dx\,dt = - \int _{0}^{T} \int _{ \mathbb{R}^{n}}u\eta _{T}^{2k}\Delta \psi \,dx \,dt, \end{aligned}$$

and

$$ \begin{aligned} & - \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\partial _{t}\bigl( \eta _{T}^{2k}\psi \bigr)\,dx\,dt \\ &\quad =- \int _{0}^{T} \int _{\mathbb{R}^{n}} \frac{\mu}{(1+ \vert x \vert )^{\beta}}u\bigl[\partial _{t} \eta _{T}^{2k}\psi + \eta _{T}^{2k} \partial _{t}\psi \bigr]\,dx\,dt. \end{aligned} $$

It follows that

$$\begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\eta _{T}^{2k} \psi \,dx\,dt \\ & \qquad {} + \int _{\mathbb{R}^{n}}\biggl(1+\frac{\mu}{(1+ \vert x \vert )^{\beta}}\biggr) \varepsilon f(x) \phi (x)\,dx \\ \begin{aligned}&\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u\biggl[\partial _{t}^{2} \eta _{T}^{2k}\psi +2\partial _{t}\eta _{T}^{2k}\partial _{t}\psi - \frac{\mu}{(1+ \vert x \vert )^{\beta}} \partial _{t}\eta _{T}^{2k}\psi \biggr]\,dx\,dt \\ & \qquad {} + \int _{0}^{T} \int _{\mathbb{R}^{n}}u\eta _{T}^{2k}\biggl[ \partial _{t}^{2}\psi -\Delta \psi -\frac {\mu}{(1+ \vert x \vert )^{\beta}} \partial _{t}\psi \biggr]\,dx\,dt \end{aligned} \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u\biggl[\partial _{t}^{2} \eta _{T}^{2k}\psi +2\partial _{t}\eta _{T}^{2k}\partial _{t}\psi - \frac{\mu}{(1+ \vert x \vert )^{\beta}} \partial _{t}\eta _{T}^{2k}\psi \biggr]\,dx\,dt \\ &\quad =I_{11}+I_{12}+I_{13}. \end{aligned}$$
(4.1)

Employing (2.2), we have

$$\begin{aligned}& \begin{aligned} I_{11}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}u\partial _{t}^{2} \eta _{T}^{2k}\psi \,dx\,dt \\ & \leq \frac {C}{T^{2}} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u \eta _{T}^{2k-2} \psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T^{2}}\biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q} \eta _{T}^{2k}\psi \,dx\,dt\biggr)^{\frac{1}{q}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}}\psi \,dx\,dt\biggr)^{\frac{1}{q'}} \\ & \leq CT^{-2+\frac{n+1}{2q'}}\biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt\biggr)^{\frac{1}{q}}, \end{aligned} \end{aligned}$$
(4.2)
$$\begin{aligned}& \begin{aligned} I_{12}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}2\partial _{t} \eta _{T}^{2k} \partial _{t}\psi \,dx\,dt \\ & \leq \frac {C}{T} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u \eta _{T}^{2k-1} \psi \bigr\vert \,dx\,dt \\ & \leq CT^{-1+\frac{n+1}{2q'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt \biggr)^{\frac{1}{q}}, \end{aligned} \end{aligned}$$
(4.3)
$$\begin{aligned}& \begin{aligned} \vert I_{13} \vert &\leq CI_{12} \\ & \leq CT^{-1+\frac{n+1}{2q'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt \biggr)^{\frac{1}{q}}. \end{aligned} \end{aligned}$$
(4.4)

Combining (4.1)–(4.4), we deduce

$$ \begin{aligned} C\varepsilon ^{q}T^{q-\frac{n+1}{2}(q-1)} \leq \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt. \end{aligned} $$
(4.5)

On the other hand, we take \(\varphi (t,x)=\partial _{t}\Phi (t,x)\) in (2.1), where \(\Phi (t,x)=-\eta _{T}^{2k}\psi (t,x)=-\eta _{T}^{2k}e^{-t}\phi (x)\). Similarly to the derivation in (3.5) and (3.18), we acquire

$$ \begin{aligned} & \varepsilon \int _{\mathbb{R}^{n}}g(x)\phi (x)\,dx+ \int _{0}^{T} \int _{\mathbb{R}^{n}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\partial _{t} \Phi \,dx\,dt \\ &\qquad {} +\varepsilon \int _{\mathbb{R}^{n}}\biggl(1+ \frac{\mu}{(1+ \vert x \vert )^{\beta}}\biggr) f(x)\phi (x)\,dx \\ &\quad = \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t}\biggl(\partial _{t}^{2} \eta _{T}^{2k}\psi +2\partial _{t}\eta _{T}^{2k}\partial _{t}\psi - \frac{\mu}{(1+ \vert x \vert )^{\beta}} \partial _{t}\eta _{T}^{2k}\psi \biggr)\,dx\,dt \\ & \quad =I_{14}+I_{15}+I_{16}. \end{aligned} $$
(4.6)

It follows that

$$\begin{aligned} I_{14}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}u_{t} \partial _{t}^{2} \eta _{T}^{2k}\psi \,dx\,dt \\ & \leq \frac {C}{T^{2}} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2k-2}\psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T^{2}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2k}\psi \,dx\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}}\psi \,dx\,dt \biggr)^{\frac{1}{p'}} \\ & \leq CT^{-2+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2k}\psi \,dx\,dt \biggr)^{\frac{1}{p}} \\ & \leq CT^{-2p'+\frac{n+1}{2}}+\frac{1}{3} \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2k}\psi \,dx\,dt. \end{aligned}$$
(4.7)

Similarly, we conclude that

$$\begin{aligned}& \begin{aligned} I_{15}&= \int _{0}^{T} \int _{\mathbb{R}^{n}}2u_{t} \partial _{t}\eta _{T}^{2k}\partial _{t}\psi \,dx\,dt \\ & \leq \frac {C}{T} \int _{0}^{T} \int _{\mathbb{R}^{n}} \bigl\vert u_{t} \eta _{T}^{2k-2}\psi \bigr\vert \,dx\,dt \\ & \leq \frac {C}{T} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p} \eta _{T}^{2k}\psi \,dx\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}}\psi \,dx\,dt \biggr)^{\frac{1}{p'}} \\ & \leq CT^{-1+\frac{n+1}{2p'}} \biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2k}\psi \,dx\,dt \biggr)^{\frac{1}{p}} \\ & \leq CT^{-p'+\frac{n+1}{2}}+\frac{1}{3} \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2k}\psi \,dx\,dt, \end{aligned} \end{aligned}$$
(4.8)
$$\begin{aligned}& \begin{aligned} \vert I_{16} \vert & \leq CI_{15} \\ & \leq CT^{-p'+\frac{n+1}{2}}+\frac{1}{3} \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u_{t} \vert ^{p}\eta _{T}^{2k}\psi \,dx\,dt. \end{aligned} \end{aligned}$$
(4.9)

Employing the fact \(\partial _{t}\Phi =\eta _{T}^{2k}\psi -2k\eta _{T}^{2k-1}\partial _{t} \eta _{T}\psi \geq \eta _{T}^{2k}\psi >0\) and (4.5)–(4.9), we have

$$ \begin{aligned} C{\varepsilon}^{q}T^{q-\frac{(n+1)(q-1)}{2}}\leq \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt\leq CT^{-p'+ \frac{n+1}{2}},\end{aligned} $$

which implies

$$ \begin{aligned} T(\varepsilon )\leq C\varepsilon ^{- \frac{2q(p-1)}{-(n-1)qp+(n-1)q+2p}} \end{aligned} $$
(4.10)

for \(p>1\) (\(n=2,3\)), \(1< p<\frac{n-1}{n-3}\) (\(n>3\)), and \(1< q<\frac{2p}{(n-1)(p-1)}\).

On the other hand, (4.2)–(4.4) yield

$$\begin{aligned}& \begin{aligned} I_{11}& \leq CT^{-2+\frac{n+1}{2q'}}\biggl( \int _{0}^{T} \int _{ \mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt\biggr)^{\frac{1}{q}} \\ & \leq CT^{-2q'+\frac{n+1}{2}}+\frac {1}{3} \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt, \end{aligned} \end{aligned}$$
(4.11)
$$\begin{aligned}& \begin{aligned} I_{12}&\leq CT^{-1+\frac{n+1}{2q'}} \biggl( \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt \biggr)^{ \frac{1}{q}} \\ & \leq CT^{-q'+\frac{n+1}{2}}+\frac {1}{3} \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt, \end{aligned} \end{aligned}$$
(4.12)
$$\begin{aligned}& \begin{aligned} \vert I_{13} \vert &\leq CI_{12} \\ & \leq CT^{-q'+\frac{n+1}{2}}+\frac {1}{3} \int _{0}^{T} \int _{\mathbb{R}^{n}} \vert u \vert ^{q}\eta _{T}^{2k}\psi \,dx\,dt. \end{aligned} \end{aligned}$$
(4.13)

From (4.1) and (4.11)–(4.13) we obtain

$$ \begin{aligned} \varepsilon C_{2}(f,g)\leq CT^{-q'+\frac{n+1}{2}},\end{aligned} $$

which implies

$$ \begin{aligned} T(\varepsilon )\leq C\varepsilon ^{-\frac{2(q-1)}{(n+1)-(n-1)q}}\end{aligned} $$

for \(p>1\) and \(1< q<\frac{n+1}{n-1}\). The proof of Theorem 1.2 is finished.

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Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. The author Sen Ming would like to express his sincere thank to Professor Yi Zhou for his guidance and encouragements during the postdoctoral study in Fudan University. The authors would like to express their gratitude to Professor Ning-An Lai for his useful suggestions.

Funding

The project is supported by Natural Science Foundation of Shanxi Province of China (No. 201901D211276), Fundamental Research Program of Shanxi Province (Nos. 20210302123021 and 20210302123045), Innovative Research Team of North University of China (No. TD201901), Science and Technology Innovation Project of Higher Education Institutions in Shanxi (No. 2020L0277), and Science Foundation of North University of China (No. XJJ201922).

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Yao, J., Ming, S. & Fan, X. Blowup for semilinear wave equation with space-dependent damping and combined nonlinearities. Adv Cont Discr Mod 2022, 47 (2022). https://doi.org/10.1186/s13662-022-03719-3

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MSC

  • 35L70
  • 58J45

Keywords

  • Space-dependent damping
  • Semilinear wave equation
  • Combined nonlinearities
  • Blowup
  • Lifespan estimates