A New Method for Blow-Up to Scale-Invariant Damped Wave Equations with Derivatives and Combined Nonlinear Terms

Abstract

The Cauchy problems of scale-invariant damped wave equations with derivative nonlinear terms and with combined nonlinear terms are studied. A new method is provided to show that the solutions will blow up in a finite time, if the nonlinear powers satisfy some conditions. The method is based on constructing appropriate test functions, by using the solution of an ordinary differential equation. It may be useful to prove the nonexistence for global solutions for other nonlinear evolution equations.

1. Introduction

Many researchers have studied the damped wave equation, such as Usamah [1], who performed symmetry analysis and exhibited exact solutions for various forms of diffusivity and viscosity, but in the present, work we study the model:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_{tt}-\Delta u+\frac{\mu }{1+t}{u}_t={\left|u_t\right|}^p,\mathrm{in}[0,T)\times {\mathbb{R}}^n,\hfill \\ u(x,0)=\epsilon f\left(x\right),u_t(x,0)=\epsilon g\left(x\right),x\in {\mathbb{R}}^n,\hfill \end{array}\right.\end{array}$$

(1)

and:

$$\left\{\begin{array}{c}{u}_{tt}-\Delta u+\frac{\mu }{1+t}{u}_t=|u_t{|}^p+{\left|u\right|}^q,\mathrm{in}[0,T)\times {\mathbb{R}}^n,\hfill \\ u(x,0)=\epsilon f\left(x\right),u_t(x,0)=\epsilon g\left(x\right),\hfill \end{array}\right.$$

(2)

where $\mu >0$ is a constant and $f\left(x\right),g\left(x\right)$ are the initial data with compact support, which satisfy:

$$f\left(x\right)\in H^1\left({\mathbb{R}}^n\right),g\left(x\right)\in L^2\left({\mathbb{R}}^n\right),$$

and

$$\begin{array}{c}\hfill suppf\left(x\right),g\left(x\right)\in \{x:|x|\le 1\}.\end{array}$$

(3)

The semilinear wave equation with scale-invariant damping has attracted more and more attention recently: on one the hand, it is the border of the “wave-like” and “heat-like” phenomena of the damped wave equation; on the other hand, it has a close relation to the Tricomi equation, which is used to describe gas dynamics. There are many literature works that have studied the semilinear wave equations with scale-invariant damping; see [2,3,4,5,6,7,8,9,10,11,12,13,14] and the references therein. For Problem (1), Lai and Takamura [15] showed the blow-up for $1<p\le p_G(n+2\mu )$, which seems not to be the sharp blow-up power, since Palmieri and Tu [16] proved a blow-up result in the range $1<p\le p_G(n+\sigma )$ for:

$$\sigma =\left\{\begin{array}{c}2\mu ,for\mu \in [0,1),\hfill \\ 2,for\mu \in [1,2),\hfill \\ \mu ,for\mu \in [2,\infty ).\hfill \end{array}\right.$$

Obviously, when $\mu \in [0,1]$, the former result coincides with that in [15], and there is some improvement for $\mu \in (1,2)$, but still some gap for $p_G(n+\mu )$, while for $[2,\infty )$, they improved the blow-up power to the expected $p_G(n+\mu )$. Recently, Hamouda and Hamza [17] showed blow-up results for (1) when $1<p\le p_G(n+\mu )$ and for (2) when:

$$\gamma (p,q,n+\mu )<4,$$

with:

$$\gamma (p,q,n)=(q-1)\left((n-1)p-2\right),$$

(4)

which improved the results in [15,16,18], by using a similar method in [11]. We should mention that there are many blow-up results for other nonlinear evolution equations; see [19,20,21] and the references therein.

In this work, we aim to show the blow-up results and lifespan estimate in [17] by using a new method, based on the works [22,23]. We constructed a special test function, the key ingredient of which is the solution of a ordinary differential equation. Inspired by [24], we can obtain the explicit solution of the ODE and furthermore obtain the asymptotic behavior.

2. Main Result

Definition 1.

We define the upper bound of the lifespan for (1) and (2) as:

$$T_{\epsilon }=sup\{T>0;thereexistsanenergysolutionto\left(1\right)and\left(2\right)in[0,T)\}.$$

Then, our main results read as follows:

Theorem 1.

Let $1<p\le p_G(n+\mu )$. Assume that the initial data $f,g$ are non-negative and do not vanish identically. Furthermore, the compact support assumption (3) holds. If we further assume the energy solution satisfies:

$$suppu(t,x)\subset \left\{x\right|:\left|x\right|\le t+1\},$$

(5)

then the solution of (1) will blow up in a finite time, and the upper bound of the lifespan will satisfy:

$$T\le \left\{\begin{array}{cc}C{\epsilon }^{-(p-1)/\{1-(n+\mu -1\left)\right(p-1)/2\}}\hfill & for1<p<p_G(n+\mu ),\hfill \\ exp\left(C{\epsilon }^{-(p-1)}\right)\hfill & forp=p_G(n+\mu ),\hfill \end{array}\right.$$

(6)

where C denotes a positive constant, which may have a different value from line to line and is independent of ε.

Theorem 2.

Let $\gamma (p,q,n+\mu )<4$. Assume that the initial data $f,g$ are non-negative and do not vanish identically. Furthermore, the compact support assumption (3) holds. If we further assume the energy solution satisfies:

$$suppu(t,x)\subset \left\{x\right|:\left|x\right|\le t+1\},$$

(7)

then the solution of (2) will blow up in a finite time, and the upper bound of the lifespan will satisfy:

$$T\le C{\epsilon }^{-\frac{2p(q-1)}{4-\gamma (p,q,n+\mu )}},$$

(8)

where C denotes a positive constant, which may have a different value from line to line and is independent of ε.

3. Test Function

As mentioned above, the key ingredient of the test function is one of the solutions of the following ODEs:

$${\lambda }^{″}\left(t\right)-\frac{\mu }{1+t}{\lambda }^{\prime }\left(t\right)-\lambda \left(t\right)=0.$$

(9)

Lemma 1.

The ODE (9) admits one solution:

$$\lambda \left(t\right)={(1+t)}^{\frac{\mu +1}{2}}{K}_{\frac{\mu +1}{2}}(1+t),$$

where $K_{\nu }\left(z\right)$ is the modified Bessel functions of the second kind. In particular, λ is a real and positive function satisfying:

$$\lambda \left(0\right)=K_{\frac{\mu +1}{2}}\left(1\right)>0,{\lambda }^{\prime }\left(0\right)=-K_{\frac{\mu -1}{2}}\left(1\right)<0,{\lambda }^{\prime }\left(t\right)<0,$$

(10)

and, for large t,

$$\lambda \left(t\right)=\frac{1}{e}\sqrt{\frac{\pi }{2}}{(1+t)}^{\frac{\mu }{2}}{e}^{-t}\times \left(1+O\left(\frac{1}{1+t}\right)\right)=-{\lambda }^{\prime }\left(t\right).$$

(11)

Proof. 

We first collect some useful relations, which can be found in [25].

$$K_{\nu }\left(z\right)=K_{-\nu }\left(z\right)=\frac{\pi }{2}\frac{I_{-\nu }\left(z\right)-I_{\nu }\left(z\right)}{sin\left(\nu \pi \right)},$$

(12)

where $I_{\nu }\left(z\right)$ is the modified Bessel functions of the first kind, and when $\nu$ is an integer, the right hand-side of this equation is replaced by its limiting value.

$$K_{\nu }^{\prime }\left(z\right)=-K_{\nu -1}\left(z\right)-\frac{\nu }{z}{K}_{\nu }\left(z\right),K_{\nu }^{\prime }\left(z\right)=-K_{\nu +1}\left(z\right)+\frac{\nu }{z}{K}_{\nu }\left(z\right),$$

(13)

$$K_{\nu }\left(z\right)=\sqrt{\frac{\pi }{2}}{z}^{-1/2}{e}^{-z}\times (1+O\left(z^{-1}\right)),for\left|z\right|largeand|argz|<\frac{3}{2}\pi .$$

(14)

Then, it is easy to check from (13):

$$\begin{array}{cc}\hfill {\lambda }^{\prime }\left(t\right)& =\frac{\mu +1}{2}{(1+t)}^{\frac{\mu -1}{2}}{K}_{\frac{\mu +1}{2}}(1+t)+{(1+t)}^{\frac{\mu +1}{2}}{K}_{\frac{\mu +1}{2}}^{\prime }(1+t)\hfill \\ \hfill & =\frac{\mu +1}{2}{(1+t)}^{\frac{\mu -1}{2}}{K}_{\frac{\mu +1}{2}}(1+t)+{(1+t)}^{\frac{\mu +1}{2}}[-K_{\frac{\mu -1}{2}}(1+t)\hfill \\ \hfill & -\frac{\mu +1}{2(1+t)}{K}_{\frac{\mu +1}{2}}(1+t)]\hfill \\ \hfill & =-{(1+t)}^{\frac{\mu +1}{2}}{K}_{\frac{\mu -1}{2}}(1+t),\hfill \\ \hfill {\lambda }^{″}\left(t\right)& =-\frac{\mu +1}{2}{(1+t)}^{\frac{\mu -1}{2}}{K}_{\frac{\mu -1}{2}}(1+t)-{(1+t)}^{\frac{\mu +1}{2}}{K}_{\frac{\mu -1}{2}}^{\prime }(1+t)\hfill \\ \hfill & =-\frac{\mu +1}{2}{(1+t)}^{\frac{\mu -1}{2}}{K}_{\frac{\mu -1}{2}}(1+t)+{(1+t)}^{\frac{\mu +1}{2}}{K}_{\frac{\mu +1}{2}}(1+t)\hfill \\ \hfill & -\frac{\mu -1}{2}{(1+t)}^{\frac{\mu -1}{2}}{K}_{\frac{\mu -1}{2}}(1+t)\hfill \\ \hfill & =-\mu {(1+t)}^{\frac{\mu -1}{2}}{K}_{\frac{\mu -1}{2}}(1+t)+{(1+t)}^{\frac{\mu +1}{2}}{K}_{\frac{\mu +1}{2}}(1+t)\hfill \\ \hfill & =\frac{\mu }{1+t}{\lambda }^{\prime }\left(t\right)+\lambda \left(t\right).\hfill \end{array}$$

Hence, $\lambda$ solves (9). Thanks to the identity (12), we know that $K_{\nu }\left(z\right)=K_{\left|\nu \right|}\left(z\right)$ for every $\nu \in \mathbb{R}$. Since $K_{\nu }\left(z\right)$ is real and positive for $z>0$, also $\lambda$ is real and positive, whereas ${\lambda }^{\prime }$ is negative. Then, we achieve (10), while exploiting Formula (14):

$$\begin{array}{c}\hfill K_{\frac{\mu +1}{2}}(1+t)=K_{\frac{\mu -1}{2}}(1+t)=\sqrt{\frac{\pi }{2}}{(1+t)}^{-1/2}{e}^{-(1+t)}\times \left(1+O\left(\frac{1}{1+t}\right)\right),\end{array}$$

$$\begin{array}{cc}\hfill \lambda \left(t\right)& ={(1+t)}^{\frac{\mu +1}{2}}\sqrt{\frac{\pi }{2}}{(1+t)}^{-1/2}{e}^{-(1+t)}\times \left(1+O\left(\frac{1}{1+t}\right)\right)\hfill \\ \hfill & =\sqrt{\frac{\pi }{2}}{(1+t)}^{\frac{\mu }{2}}{e}^{-(1+t)}\times \left(1+O\left(\frac{1}{1+t}\right)\right)\hfill \\ \hfill & =-{\lambda }^{\prime }\left(t\right).\hfill \end{array}$$

We obtain the relations (11). □

4. Proof of the Theorem 1

As in [22], we introduce two cut-off functions:

$$\eta \left(t\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{for}t\le \frac{1}{2},\hfill \\ decreasing\hfill & \mathrm{for}\frac{1}{2}<t<1,\hfill \\ 0\hfill & \mathrm{for}t\ge 1,\hfill \end{array}\right.\theta \left(t\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{for}t<\frac{1}{2},\hfill \\ \eta \left(t\right)\hfill & \mathrm{for}t\ge \frac{1}{2},\hfill \end{array}\right.$$

with:

$$|{\eta }^{\prime }\left(t\right)|\le C,|{\eta }^{″}\left(t\right)|\le C,$$

and

$${\eta }_M\left(t\right)=\eta \left(\frac{t}{M}\right).$$

Then, we construct our test function as:

$$\Phi (t,x)=-{\partial }_t\left({\eta }_M^{2p^{\prime }}\left(t\right)\lambda \left(t\right)\varphi \left(x\right)\right),$$

where $M\in (1,T]$ for any $T\in [1,T(\epsilon \left)\right]$ and:

$$\varphi \left(x\right)={\int }_{S^{n-1}}{e}^{x·\omega }d\sigma ,$$

which satisfies:

$$\begin{array}{c}\hfill 0<\varphi \left(x\right)\le C{(1+|x\left|\right)}^{-\frac{n-1}{2}}{e}^{\left|x\right|}.\end{array}$$

(15)

Then, we have:

$$\begin{array}{cc}\hfill \Phi (t,x)& =-\left({\partial }_t{\eta }_M^{2p^{\prime }}\left(t\right)\lambda \left(t\right)\varphi \left(x\right)+{\eta }_M^{2p^{\prime }}\left(t\right){\lambda }^{\prime }\left(t\right)\varphi \left(x\right)\right)\hfill \\ \hfill & \ge {\eta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)\hfill \\ \hfill & \ge 0,\hfill \end{array}$$

(16)

where we used the fact that both ${\eta }_M\left(t\right)$ and $\lambda \left(t\right)$ are non-increasing functions.

Remark 1.

Note that the test function $\varphi \left(x\right)$ admits some good properties. First, it is non-negative, and it satisfies:

$$\Delta \varphi =\varphi .$$

Finally, it has the asymptotic behavior (15).

Multiplying the equation in (1) with $\Phi (t,x)$ and integrating over $[0,T]\times {\mathbb{R}}^n$, then by integration by parts, we obtain:

$$\begin{array}{cc}\hfill & -\epsilon {\int }_{{\mathbb{R}}^n}{\lambda }^{\prime }\left(0\right)g\left(x\right)\varphi \left(x\right)dx+\epsilon {\int }_{{\mathbb{R}}^n}\lambda \left(0\right)f\left(x\right)\varphi \left(x\right)dx\hfill \\ \hfill & +{\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\hfill \\ \hfill \le & {\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\eta }_M^{2p^{\prime }}\left(t\right)\left({\lambda }^{″}\left(t\right)-\frac{\mu }{1+t}{\lambda }^{\prime }\left(t\right)-\lambda \left(t\right)\right)dxdt\hfill \\ \hfill & +{\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\partial }_t^2{\eta }_M^{2p^{\prime }}\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt\hfill \\ \hfill & +2{\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\partial }_t{\eta }_M^{2p^{\prime }}\left(t\right){\lambda }^{\prime }\left(t\right)\varphi \left(x\right)dxdt\hfill \\ \hfill & -{\int }_0^T{\int }_{{\mathbb{R}}^n}\frac{\mu }{1+t}{u}_t{\partial }_t{\eta }_M^{2p^{\prime }}\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt,\hfill \end{array}$$

(17)

which yields for some positive constant $C_1=C(f,g,\mu )$ by combining (9) and (10):

$$\begin{array}{cc}\hfill & C_1\epsilon +{\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\hfill \\ \hfill \le & {\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\partial }_t^2{\eta }_M^{2p^{\prime }}\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt\hfill \\ \hfill & +2{\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\partial }_t{\eta }_M^{2p^{\prime }}\left(t\right){\lambda }^{\prime }\left(t\right)\varphi \left(x\right)dxdt\hfill \\ \hfill & -{\int }_0^T{\int }_{{\mathbb{R}}^n}\frac{\mu }{1+t}{u}_t{\partial }_t{\eta }_M^{2p^{\prime }}\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt,\hfill \\ \hfill \triangleq & I+II+III.\hfill \end{array}$$

(18)

We estimate the three terms $I,II,III$ by the nonlinear term by using the Hölder inequality. For I, it follows from (11) and (15) that:

$$\begin{array}{cc}\hfill I\le & C{M}^{-2}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\theta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\right)}^{\frac{1}{p}}\hfill \\ \hfill & \times {\left({\int }_{\frac{M}{2}}^M{\int }_{\left|x\right|\le 1+t}|{\lambda }^{\prime }{\left(t\right)|}^{-\frac{1}{p-1}}{\left|\lambda \left(t\right)\right|}^{\frac{p}{p-1}}\varphi \left(x\right)dxdt\right)}^{\frac{1}{p^{\prime }}}\hfill \\ \hfill \le & C{M}^{-2+\frac{n+\mu +1}{2}\frac{1}{p^{\prime }}}\times {\left({\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\right)}^{\frac{1}{p}}.\hfill \end{array}$$

(19)

In the same way for $II$ and $III$, we have:

$$\begin{array}{cc}\hfill II:\le & C{M}^{-1+\frac{n+\mu +1}{2}\frac{1}{p^{\prime }}}\times {\left({\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\right)}^{\frac{1}{p}},\hfill \end{array}$$

(20)

and:

$$\begin{array}{cc}\hfill III:\le & C{M}^{-2+\frac{n+\mu +1}{2}\frac{1}{p^{\prime }}}\times {\left({\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\right)}^{\frac{1}{p}}.\hfill \end{array}$$

(21)

By combining (18)–(21), we obtain:

$$\begin{array}{cc}\hfill & C_1\epsilon +{\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\hfill \\ \hfill \le & C{M}^{-1+\frac{n+\mu +1}{2}\frac{1}{p^{\prime }}}\times {\left({\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\theta }_M^{2p^{\prime }}\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\right)}^{\frac{1}{p}}.\hfill \end{array}$$

(22)

If for a function $w(t,x)$, we set:

$$Y\left[w\right]\left(M\right)={\int }_1^M\left({\int }_0^T{\int }_{{\mathbb{R}}^n}w(t,x){\theta }_{\sigma }^{2p^{\prime }}\left(t\right)dxdt\right){\sigma }^{-1}d\sigma ,$$

then as in [24], we have:

$$\begin{array}{cc}\hfill & Y\left[|u_t{|}^p\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)\right]\left(M\right)\hfill \\ \hfill =& {\int }_1^M\left({\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right){\theta }_{\sigma }^{2p^{\prime }}\left(t\right)dxdt\right){\sigma }^{-1}d\sigma \hfill \\ \hfill \le & Clog2{\int }_0^T{\int }_{{\mathbb{R}}^n}{\eta }_M^{2p^{\prime }}|u_t{|}^p\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt.\hfill \end{array}$$

(23)

For simplicity, we denote $Y\left(M\right)$ for $Y\left[|u_t{|}^p\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)\right]\left(M\right)$, then by (23), we have:

$$\begin{array}{c}\hfill Y\left(M\right)\le Clog2{\int }_0^T{\int }_{{\mathbb{R}}^n}{\eta }_M^{2p^{\prime }}|u_t{|}^p\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt,\end{array}$$

(24)

$$\begin{array}{c}\hfill \frac{d}{dM}Y\left(M\right)=M^{-1}{\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right){\theta }_M^{2p^{\prime }}\left(t\right)dxdt.\end{array}$$

(25)

Hence, by combining (22), (24), and (25), we know there exist positive constants $C_2,C_3$ such that:

$$\begin{array}{c}\hfill M{Y}^{\prime }\left(M\right)\ge C{M}^{p-\frac{(n+\mu +1)(p-1)}{2}}{\left(C_2\epsilon +C_{3}Y\left(M\right)\right)}^p,\end{array}$$

(26)

which leads to the lifespan estimate (6).

5. Proof for Theorem 2

For the problem with combined nonlinearity, we have to introduce another cut-off function:

$$\zeta \left(t\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{for}t\le \frac{1}{4},\hfill \\ increasing\hfill & \mathrm{for}\frac{1}{4}<t<\frac{1}{2},\hfill \\ \theta \left(t\right)\hfill & \mathrm{for}t\ge \frac{1}{2}.\hfill \end{array}\right.$$

Let:

$${\zeta }_M\left(t\right)=\zeta \left(\frac{t}{M}\right),{\psi }_M\left(t\right)={\zeta }_M^k\left(t\right),$$

with $k>0$, which will be determined later, and $M\in (1,T)$. It is easy to obtain:

$$\begin{array}{c}|{\partial }_t^2{\psi }_M\left(t\right)|\le C{M}^{-2}{\psi }_M^{1-\frac{2}{k}}\left(t\right),\hfill \\ |{\partial }_t{\psi }_M\left(t\right)|=C{M}^{-1}{\psi }_M^{1-\frac{1}{k}}\left(t\right).\hfill \end{array}$$

(27)

Multiplying the equation in (2) with ${\psi }_M(t,x)$ and integrating over $[0,T]\times {\mathbb{R}}^n$, then by integration by parts, we obtain:

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{{\mathbb{R}}^n}\left(|u_t{|}^p+{\left|u\right|}^q\right){\psi }_M\left(t\right)dxdt\hfill \\ \hfill =& {\int }_0^T{\int }_{{\mathbb{R}}^n}u{\partial }_t^2{\psi }_{M}dxdt+{\int }_0^T{\int }_{{\mathbb{R}}^n}\frac{\mu }{{(1+t)}^2}u{\psi }_{M}dxdt-{\int }_0^T{\int }_{{\mathbb{R}}^n}\frac{\mu }{1+t}u{\partial }_t{\psi }_{M}dxdt\hfill \\ \hfill \triangleq & IV+V+VI.\hfill \end{array}$$

(28)

We estimate $IV$ as:

$$\begin{array}{cc}\hfill IV=& {\int }_0^T{\int }_{{\mathbb{R}}^n}u{\partial }_t^2{\psi }_{M}dxdt\hfill \\ \hfill \le & C{M}^{-2}{\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u\right|{\psi }_M^{1-\frac{2}{k}-\frac{1}{2q^{\prime }}}{\psi }_M^{\frac{1}{2q^{\prime }}}dxdt\hfill \\ \hfill \le & C{M}^{-2}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^q{\psi }_M^{q(1-\frac{2}{k}-\frac{1}{2q^{\prime }})}dxdt\right)}^{\frac{1}{q}}{\left({\int }_0^T{\int }_{\left|x\right|\le 1+t}{\psi }_M^{\frac{1}{2}}dxdt\right)}^{\frac{1}{q^{\prime }}}\hfill \\ \hfill \le & C{R}^{-2+\frac{n+1}{q^{\prime }}}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^q{\psi }_{M}dxdt\right)}^{\frac{1}{q}},\hfill \end{array}$$

(29)

if we choose k large enough such that:

$$q\left(1-\frac{2}{k}-\frac{1}{2q^{\prime }}\right)\ge 1.$$

(30)

For V, although there is no derivative on the cut-off function ${\psi }_M$, note that:

$$supp{\psi }_M\subset \left[\frac{R}{4},R\right],$$

We can obtain in a similar way as for $IV$:

$$\begin{array}{c}\hfill V,VI\le C{R}^{-2+\frac{n+1}{q^{\prime }}}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^q{\psi }_{M}dxdt\right)}^{\frac{1}{q}},\end{array}$$

(31)

where we need to choose $k>0$ satisfying for $VI$:

$$q\left(1-\frac{1}{k}-\frac{1}{2q^{\prime }}\right)\ge 1.$$

(32)

By combining (28), (29) and (31), we have:

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{{\mathbb{R}}^n}\left(|u_t{|}^p+{\left|u\right|}^q\right){\psi }_M\left(t\right)dxdt\hfill \\ \hfill \le & C{R}^{-2+\frac{n+1}{q^{\prime }}}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^q{\psi }_{M}dxdt\right)}^{\frac{1}{q}}\hfill \\ \hfill \le & C{M}^{n-\frac{q+1}{q-1}}+\frac{1}{2}{\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u\right|}^q{\psi }_{M}dxdt,\hfill \end{array}$$

(33)

which yields:

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u_t\right|}^p{\psi }_M\left(t\right)dxdt\le C{M}^{n-\frac{q+1}{q-1}}.\hfill \end{array}$$

(34)

The next step is to use the test function:

$$\Psi (t,x)=-{\partial }_t\left({\eta }_M^k\left(t\right)\lambda \left(t\right)\varphi \left(x\right)\right)\ge 0$$

to obtain the lower bound of the nonlinear term. Multiplying the equation in (2) with $\Psi (t,x)$ and integrating over $[0,T]\times {\mathbb{R}}^n$, then by integration by parts, we obtain:

$$\begin{array}{cc}\hfill & -\epsilon {\int }_{{\mathbb{R}}^n}{\lambda }^{\prime }\left(0\right)g\left(x\right)\varphi \left(x\right)dx+\epsilon {\int }_{{\mathbb{R}}^n}\lambda \left(0\right)f\left(x\right)\varphi \left(x\right)dx\hfill \\ \hfill & +{\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^k\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\hfill \\ \hfill =& {\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\eta }_M^k\left(t\right)\left({\lambda }^{″}\left(t\right)-\frac{\mu }{1+t}{\lambda }^{\prime }\left(t\right)-\lambda \left(t\right)\right)dxdt\hfill \\ \hfill & +{\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\partial }_t^2{\eta }_M^k\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt\hfill \\ \hfill & +2{\int }_0^T{\int }_{{\mathbb{R}}^n}{u}_t{\partial }_t{\eta }_M^k\left(t\right){\lambda }^{\prime }\left(t\right)\varphi \left(x\right)dxdt\hfill \\ \hfill & -{\int }_0^T{\int }_{{\mathbb{R}}^n}\frac{\mu }{1+t}{u}_t{\partial }_t{\eta }_M^k\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt,\hfill \end{array}$$

(35)

which yields for some positive constant $C_2=C(f,g,\mu )$ by combining (9) and (10):

$$\begin{array}{cc}\hfill & C_2\epsilon +{\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t{|}^p{\eta }_M^k\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\hfill \\ \hfill \le & {\int }_0^T{\int }_{{\mathbb{R}}^n}\left|u_t\right|{\partial }_t^2{\eta }_M^k\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt\hfill \\ \hfill & +2{\int }_0^T{\int }_{{\mathbb{R}}^n}|u_t|{\partial }_t{\eta }_M^k\left(t\right)\left|{\lambda }^{\prime }\left(t\right)\right|\varphi \left(x\right)dxdt\hfill \\ \hfill & +{\int }_0^T{\int }_{{\mathbb{R}}^n}\frac{\mu }{1+t}\left|u_t\right|{\partial }_t{\eta }_M^k\left(t\right)\lambda \left(t\right)\varphi \left(x\right)dxdt,\hfill \\ \hfill \triangleq & I_c+I{I}_c+II{I}_c.\hfill \end{array}$$

(36)

Furthermore, it is easy to obtain:

$$\begin{array}{cc}\hfill & |{\partial }_t^2{\eta }_M^k\left(t\right)|\le C{M}^{-2}{\theta }_M^{k-2}\left(t\right),\hfill \\ \hfill & |{\partial }_t{\eta }_M\left(t\right)|=C{M}^{-1}{\theta }_M^{k-1}\left(t\right).\hfill \end{array}$$

(37)

As above, the term $I_c$ can be estimated as:

$$\begin{array}{cc}\hfill I_c\le & C{M}^{-2}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u_t\right|}^p{\theta }_M^k\left(t\right)dxdt\right)}^{\frac{1}{p}}\hfill \\ \hfill & \times {\left({\int }_{\frac{M}{2}}^M{\int }_{\left|x\right|\le 1+t}{\theta }_M^{k-2p^{\prime }}{\left|\lambda \left(t\right)\varphi \left(x\right)\right|}^{p^{\prime }}dxdt\right)}^{\frac{1}{p^{\prime }}}\hfill \\ \hfill \le & C{M}^{-2}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u_t\right|}^p{\theta }_M^k\left(t\right)dxdt\right)}^{\frac{1}{p}}\hfill \\ \hfill & \times {\left({\int }_{\frac{M}{2}}^M{\int }_{\left|x\right|\le 1+t}{(1+t)}^{\frac{\mu p^{\prime }}{2}}{e}^{-t{p}^{\prime }}{(1+r)}^{n-1-\frac{n-1}{2}{p}^{\prime }}{e}^{p^{\prime }r}dxdt\right)}^{\frac{1}{p^{\prime }}}\hfill \\ \hfill \le & C{M}^{-2+\frac{\mu }{2}+\frac{n}{p^{\prime }}-\frac{n-1}{2}}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u_t\right|}^p{\theta }_M^k\left(t\right)dxdt\right)}^{\frac{1}{p}}.\hfill \end{array}$$

(38)

In the same way, we have:

$$\begin{array}{c}\hfill I{I}_c,II{I}_c\le C{M}^{-1+\frac{\mu }{2}+\frac{n}{p^{\prime }}-\frac{n-1}{2}}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u_t\right|}^p{\theta }_M^k\left(t\right)dxdt\right)}^{\frac{1}{p}}.\end{array}$$

(39)

Then, (36), (38) and (39), yield:

$$\begin{array}{c}\hfill C\epsilon \le C{M}^{-1+\frac{\mu }{2}+\frac{n}{p^{\prime }}-\frac{n-1}{2}}{\left({\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u_t\right|}^p{\theta }_M^k\left(t\right)dxdt\right)}^{\frac{1}{p}},\end{array}$$

(40)

which in turn yields:

$$\begin{array}{c}\hfill C{\epsilon }^p{M}^{n-\frac{(n+\mu -1)p}{2}}\le {\int }_0^T{\int }_{{\mathbb{R}}^n}{\left|u_t\right|}^p{\theta }_M^k\left(t\right)dxdt.\end{array}$$

(41)

Since:

$${\theta }_M^k\left(t\right)\le {\psi }_M\left(t\right)={\zeta }_M^k\left(t\right),$$

we then conclude the lifespan (8) by combining (34) and (41).