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Blow-up of solution to semilinear wave equations with strong damping and scattering damping

Abstract

This paper is devoted to investigating the initial boundary value problem for a semilinear wave equation with strong damping and scattering damping on an exterior domain. By introducing suitable multipliers and applying the test-function technique together with an iteration method, we derive the blow-up dynamics and an upper-bound lifespan estimate of the solution to the problem with power-type nonlinearity \(|u|^{p}\), derivative-type nonlinearity \(|u_{t}|^{p}\), and combined type nonlinearities \(|u_{t}|^{p}+|u|^{q}\) in the scattering case, respectively. The novelty of the present paper is that we establish the upper-bound lifespan estimate of the solution to the problem with strong damping and scattering damping, which are associated with the well-known Strauss exponent and Glassey exponent.

1 Introduction

In the present paper, we mainly consider the following initial boundary value problem

$$ \textstyle\begin{cases} u_{tt}-\Delta u-\Delta u_{t}+\frac{\mu}{(1+t)^{\beta}}u_{t}=f(u,u_{t}), & x\in \Omega ^{c},t>0, \\ u(x,0)=\varepsilon u_{0}(x),\qquad u_{t}(x,0)=\varepsilon u_{1}(x), &x\in \Omega ^{c}, \\ u|_{\partial \Omega ^{c}}=0,&t>0, \end{cases} $$
(1.1)

where \(\Delta =\sum_{i=1}^{n} \frac{\partial ^{2}}{\partial x_{i}^{2}}\), \(\mu >0\), \(\beta \geq 1\), \(f(u,u_{t})=|u|^{p}, |u_{t}|^{p}, |u_{t}|^{p}+|u|^{q}\) for \(1< p, q<\infty \), \(\Omega =B_{1}(0)\) is the unit ball in R n , Ω c = R n B 1 (0) (\(n \geq 1\)). The initial data \(u_{0}(x)\in H^{1}(\Omega ^{c})\), \(u_{1}(x) \in L^{2}(\Omega ^{c})\) are compactly supported functions and \(\operatorname{supp} (u_{0}(x), u_{1}(x))\subset B_{R}(0) \cap \Omega ^{c}\), where \(B_{R}(0)=\{x | |x|\leq R\}\), \(R>2\). \(\varepsilon \in (0,1)\) is a small parameter.

In recent years, many researchers are interested in exploring blow-up results and lifespan estimates of solutions to the semilinear wave equation. The following classical Cauchy problems without damping

{ u t Δ u = | u | p , x R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , x R n
(1.2)

and

{ u t t Δ u = f ( u , u t ) , x R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x R n
(1.3)

have been studied extensively (see [117]). It is worth noting that problem (1.2) admits the Fujita exponent \(p_{F}(n)=1+\frac{2}{n}\). Fujita [3] shows that the solution to problem (1.2) blows up in finite time when \(1< p< p_{F}(n)\), while there exists a global solution when \(p>p_{F}(n)\). Xu and Su [14] discuss the asymptotic behavior of the solution to problem (1.2) with strong damping on the exterior domain. Global existence and the blow-up dynamics of solution are verified under certain assumptions on the initial value. The problem (1.3) with \(f(u,u_{t})=|u|^{p}\) asserts the Strauss exponent \(p_{S}(n)\). If \(n=1\), \(p_{S}(n)=\infty \). If \(n\geq 2\), \(p_{S}(n)\) is the positive root of the quadratic equation

$$ r(p,n)=-(n-1)p^{2}+(n+1)p+2=0.$$

John [7] establishes the blow-up result of solution in the case \(1< p< p_{S}(3)\) and the existence of a global solution in the case \(p>p_{S}(3)\). Zhou and Han [17] consider the blow-up dynamics and the upper-bound lifespan estimate of the solution to the variable-coefficient wave equation with \(f(u,u_{t})=|u|^{p}\) by applying the test-function method and the Kato lemma when \(n \geq 3\). Nonexistence of a global solution to problem (1.3) with \(f(u,u_{t})=|u|^{p}\) in the critical case \(p=p_{S}(n) (n\geq 4)\) is demonstrated by taking advantage of the Kato lemma (see [15]). Han [4, 5] investigates the nonexistence of a global solution and the upper-bound lifespan estimate of the solution to the variable-coefficient wave equation with \(f(u,u_{t})=|u|^{p}\) by using the Kato lemma when \(n=1, 2\). Wakasa and Yordanov [13] verify the sharp lifespan estimate of the solution in the critical case \(p=p_{S}(n) (n \geq 2)\). The strategy of proof is based on the test-function method and an iteration technique. Lai et al. [8] acquire local well-posedness and an upper-bound lifespan estimate of the solution to problem (1.3) with \(f(u,u_{t})=|u|^{p} (1< p\leq p_{S}(2))\) on an asymptotically Euclidean exterior domain. The Cauchy problem (1.3) with \(f(u,u_{t})=|u_{t}|^{p}\) affirms the Glassey exponent \(p_{G}(n)\). When \(n=1\), \(p_{G}(n)=\infty \). When \(n\geq 2\), \(p_{G}(n)=\frac{n+1}{n-1}\). Zhou [16] illustrates the blow-up phenomenon and the upper-bound lifespan estimate of the solution to problem (1.3) with \(f(u,u_{t})=|u_{t}|^{p}\). Zhou and Han [17] establish the upper-bound lifespan estimate of the solution to the variable-coefficient wave equation with \(f(u,u_{t})=|u_{t}|^{p}\) by deriving an ordinary differential inequality. Han and Zhou [6] employ the Kato lemma to present the blow-up dynamics for problem (1.3) with \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\).

Let us turn our attention to the following problem with a damping term

{ u t t Δ u + μ ( 1 + t ) β u t = 0 , x R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x R n .
(1.4)

The behaviors of the solution are divided into four cases according to the decay rate β. When \(\beta \in (-\infty ,-1)\), the damping term is overdamping, which means that the solution dose not decay to zero when \(t\rightarrow \infty \). When \(\beta \in [-1,1)\), the damping term is effective and the solution behaves like that of the heat equation. When \(\beta =1\), the damping term is scale invariant by imposing the scaling \(u(x,t)=u(\lambda x,\lambda (1+t)-1) (\lambda >0)\). When \(\beta \in (1,\infty )\), the solution scatters to the free-wave equation when \(t\rightarrow \infty \). This is the scattering case. The corresponding nonlinear damped wave equation

{ u t t Δ u + μ ( 1 + t ) β u t = f ( u , u t ) , x R n , t > 0 , u ( x , 0 ) = ε u 0 ( x ) , u t ( x , 0 ) = ε u 1 ( x ) , x R n
(1.5)

causes extensive attention (see detailed illustrations in [1830]), where \(f(u,u_{t})=|u|^{p}, |u_{t}|^{p}, |u_{t}|^{p}+|u|^{q}\), respectively. In the scattering case \(\beta >1\), Lai et al. [24] derive the blow-up result and the upper-bound lifespan estimate of the solution to problem (1.5) with \(f(u,u_{t})=|u|^{p}\) by employing an appropriate multiplier and an iteration method in the subcritical case \(1< p< p_{S}(n)\). Wakasa and Yordanov [30] demonstrate the upper-bound lifespan estimate of the solution to the variable-coefficient wave equation with \(f(u,u_{t})=|u|^{p}\) in the critical case \(p=p_{S}(n) (n \geq 2)\). The existence of the global solution in the super-critical case \(p>p_{S}(n) (n=3,4)\) is investigated (see [27]). Lai et al. [25] deduce the upper-bound lifespan estimate of solution to problem (1.5) with \(f(u,u_{t})=|u_{t}|^{p}\) by introducing some multipliers and constructing the Riccati equation in the scattering and scale-invariant cases, respectively. The upper-bound lifespan estimate of the solution to problem (1.5) with \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\) is discussed by utilizing an iteration approach (see [26]). Ming et al. [28] investigate the upper-bound lifespan estimates of the solutions to the coupled system of semilinear wave equations with \(|v_{t}|^{p_{1}}+|v|^{q_{1}}\), \(|u_{t}|^{p_{2}}+|u|^{q_{2}}\) in the subcritical and critical cases. The methods applied in the proofs are the functional method and an iteration technique. In the scale-invariant case \(\beta =1\), Imai et al. [20] establish the lifespan estimate of the solution to problem (1.5) with \(f(u,u_{t})=|u|^{p}\) when \(\mu =2\) and \(1< p\leq p_{F}(2)\). Wakasa [29] and Kato et al. [22] verify the blow-up and lifespan estimate of the solution in the subcritical and critical cases \(1< p\leq p_{F}(1)\) when \(\mu =2\). Kato et al. [21] discuss the blow-up dynamics and lifespan estimate of the solution to problem (1.5) with \(\mu =2\), \(\beta =1\), \(f(u,u_{t})=|u|^{p} (1< p\leq p_{S}(5))\) in three space dimensions. Moreover, it is shown that the existence of the global solution without a spherically symmetric assumption is obtained when \(p> p_{S}(5)\). Lai et al. [23] consider the Cauchy problem with scale-invariant damping \(\frac{\mu _{1}}{1+t}u_{t}\) and mass term \(\frac{\mu _{2}}{(1+t)^{2}}u\). Formation of a singularity of the solution is derived by imposing certain assumptions on the initial values. Chen [31] illustrates blow-up phenomena and the upper-bound lifespan estimate of the solution to problem (1.5) with \(\beta =1\) and \(f(u,u_{t})=|u_{t}|^{p}, |u_{t}|^{p}+|u|^{q}\), respectively. The key tool used in the proofs is the test-function technique. We refer the readers to the works in [3236] for more details.

Many scholars focus on studying the Cauchy problem for the wave equation with a strong damping term

{ u t t Δ u Δ u t = f ( u , u t ) , x R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x R n .
(1.6)

D’Abbicco and Reissig [37] prove the blow-up of the solution to problem (1.6) with \(f(u,u_{t})=|u|^{p}\) when \(1< p<1+\frac{2}{n}\). In addition, the existence of a global solution is obtained when \(p>1+\frac{3}{n-1}(n\geq 2)\). Ikehata and Inoue [38] demonstrate the global existence of the solution to problem (1.6) with \(f(u,u_{t})=|u|^{p}\) on the exterior domain in two space dimensions when \(p>6\). Fino [39] considers the initial boundary value problem for wave equation with \(f(u,u_{t})=|u|^{p}\) and small initial values. The blow-up dynamics of the solution to the problem is established by applying the Kato lemma. Fino [40] employs the test-function technique to present the blow-up result of the solution to problem (1.6) with \(f(u,u_{t})=|u|^{p}\) on an exterior domain. Lian and Xu [41] investigate the existence of a global solution to the initial boundary value problem of the wave equation with weak and strong dampings and \(f(u,u_{t})=u\ln |u|\) by utilizing the contraction mapping principle. Yang and Zhou [42] study the existence of the global solution to the fractional Kirchhoff wave equation with structural damping or strong damping on an exterior domain. Chen and Fino [43] discuss the formation of the singularity of the solution to the initial boundary value problem (1.6) with \(f(u,u_{t})=|u_{t}|^{p}\) and \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\) by taking advantage of the test-function method. However, the upper bound of the lifespan estimate of solution has not been derived in [39, 40, 43].

Motivated by the previous works in [2426, 40, 43], we are concerned with the blow-up dynamics and the upper-bound lifespan estimate of the solution to problem (1.1) on an exterior domain. The nonlinear terms are presented in the form of \(f(u,u_{t})=|u|^{p}, |u_{t}|^{p}, |u_{t}|^{p}+|u|^{q}\), respectively. The method utilized in this paper is different from the Kato lemma employed in [46, 17], where the semilinear wave equations without damping are discussed. We observe that Lai et al. [2426] consider the nonexistence of a global solution and the upper-bound lifespan estimate of the solution to the Cauchy problem with scattering damping in n space dimensions by applying an iteration approach. The results derived in [2426] are extended to the case on an exterior domain. Fino et al. [40, 43] illustrate the blow-up of the solution to the initial boundary value problem with strong damping by making use of the test-function technique. Concerning that the upper-bound lifespan estimate of the solution to the semilinear wave equation with strong damping has not been established yet in [40, 43], we fill this gap by introducing appropriate multipliers and utilizing the test-function technique together with an iteration method. It is worth mentioning that the interaction between the strong damping term and the scattering damping term on the blow-up of the solution is analyzed. The main new contribution is that we provide an upper-bound lifespan estimate of the solution to the problem with two types of damping terms if the exponents in nonlinear terms and initial values satisfy some conditions. To the best of our knowledge, the results in Theorems 1.1–1.7 are new for problem (1.1).

The main results in this paper are stated by the following theorems.

Theorem 1.1

Let \(n\geq 3\), \(1< p< p_{S}(n)\), \(\mu >0\) and \(\beta >1\). Assume that \((u_{0},u_{1})\in H^{1}(\Omega ^{c})\times L^{2}(\Omega ^{c})\) are nonnegative functions and \(u_{0}\) does not vanish identically. If a solution u to problem (1.1) with \(f(u,u_{t})=|u|^{p}\) satisfies \(\operatorname{supp} u \subset \{(x,t)\in \Omega ^{c}\times [0,T) | |x| \leq t+R\}\), then u blows up in finite time. Moreover, there exists a constant \(\varepsilon _{0}=\varepsilon _{0}(u_{0}, u_{1}, n, p, \mu , \beta , R)>0\) such that the lifespan estimate of solution \(T(\varepsilon )\) satisfies

$$ T(\varepsilon ) \leq C\varepsilon ^{-\frac{2p(p-1)}{r(p,n)}}, $$
(1.7)

where \(0<\varepsilon \leq \varepsilon _{0}\), \(C>0\) is a constant independent of ε.

Theorem 1.2

Let \(n=2\), \(1< p< p_{S}(2)\), \(\mu >0\), and \(\beta >1\). Assume that the initial values satisfy the same conditions in Theorem 1.1. If

$$ \begin{aligned} \int _{\Omega ^{c}}u_{1}(x)\phi _{0}(x)\,dx \neq 0,\end{aligned} $$

then (1.7) is replaced by

$$ T(\varepsilon )\leq \textstyle\begin{cases} C\varepsilon ^{-\frac{p-1}{3-p}}, & 1< p< 2, \\ C\varepsilon ^{-\frac{2p(p-1)}{r(p,2)}}, & 2\leq p< p_{S}(2). \end{cases} $$

Theorem 1.3

Let \(n=1\), \(\mu >0\) and \(\beta >1\). Assume that the initial values satisfy the same conditions in Theorem 1.1. If

$$ \begin{aligned} \int _{\Omega ^{c}}u_{1}(x)\phi _{0}(x)\,dx \neq 0,\end{aligned} $$

then (1.7) is replaced by

$$ T(\varepsilon )\leq \textstyle\begin{cases} C\varepsilon ^{-\frac{p-1}{3-p}}, & 1< p< 2, \\ C\varepsilon ^{-\frac{p(p-1)}{2}}, & 2\leq p< +\infty . \end{cases} $$

Theorem 1.4

Let \(n\geq 1\), \(\mu >0\) and \(\beta >1\). Assume that \((u_{0},u_{1})\in H^{1}(\Omega ^{c})\times L^{2}(\Omega ^{c})\) are nonnegative functions and \(u_{1}\) does not vanish identically. If a solution u to problem (1.1) with \(f(u,u_{t})=|u_{t}|^{p}\) satisfies \(\operatorname{supp} u \subset \{(x,t)\in \Omega ^{c}\times [0,T) | |x| \leq t+R\}\), then u blows up in finite time. Moreover, the lifespan estimate of solution \(T(\varepsilon )\) satisfies

$$ T(\varepsilon )\leq \textstyle\begin{cases} C\varepsilon ^{-\frac{2(p-1)}{2-(n-1)(p-1)}},& 1< p< p_{G}(n), \\ \exp (C\varepsilon ^{-(p-1)}),& p=p_{G}(n), \end{cases} $$
(1.8)

where \(C>0\) is a constant independent of ε.

Theorem 1.5

Let \(n\geq 1\), \(\mu >0\), and \(\beta =1\). Assume that the initial values satisfy the same conditions in Theorem 1.4. The lifespan estimate of solution \(T(\varepsilon )\) satisfies

$$ T(\varepsilon )\leq \textstyle\begin{cases} C\varepsilon ^{-\frac{2(p-1)}{2-(n+2\mu -1)(p-1)}},& 1< p< p_{G}(n+2 \mu ), \\ \exp (C\varepsilon ^{-(p-1)}),& p=p_{G}(n+2\mu ), \end{cases} $$
(1.8)

where \(C>0\) is a constant independent of ε.

Theorem 1.6

Let \(n\geq 1\), \(\mu >0\) and \(\beta >1\). Assume that \((u_{0},u_{1})\in H^{1}(\Omega ^{c})\times L^{2}(\Omega ^{c})\) are nonnegative functions and \(u_{1}\) does not vanish identically. Suppose that a solution u to problem (1.1) with \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\) satisfies \(\operatorname{supp} u \subset \{(x,t)\in \Omega ^{c}\times [0,T) | |x| \leq t+R\}\). If \(p>1\) and

$$ \textstyle\begin{cases} 1< q< \min \{1+\frac{4}{(n-1)p-2},\frac{2n}{n-2}\},& n\geq 2, \\ 1< q,& n=1, \end{cases} $$
(1.9)

then u blows up in finite time. Moreover, the lifespan estimate of solution \(T(\varepsilon )\) satisfies

$$ T(\varepsilon )\leq \textstyle\begin{cases} C\varepsilon ^{-\frac{2p(q-1)}{2q+2-(n-1)p(q-1)}}, & n\geq 2, \\ C\varepsilon ^{-\frac{p(q-1)}{2}},& n=1, \end{cases} $$

where \(C>0\) is a constant independent of ε.

Theorem 1.7

Let \(n\geq 2\), \(\mu >0\), and \(\beta >1\). Assume that the initial values satisfy the same conditions in Theorem 1.6. If \(p>\frac{2n}{n-1}\) and \(1< q<\frac{n+1}{n-1}\), then a solution u to problem (1.1) with \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\) blows up in finite time. Moreover, the lifespan estimate of solution \(T(\varepsilon )\) satisfies

$$ T(\varepsilon )\leq C\varepsilon ^{-\frac{q-1}{q+1-n(q-1)}},$$

where \(C>0\) is a constant independent of ε.

Remark 1.1

The restriction \(q<\frac{2n}{n-2}\) is necessary to guarantee the integrability of the nonlinear term \(|u|^{q}\) in Theorem 1.6.

2 Preliminaries

In order to prove the main results, we present several related lemmas and the definition of a weak solution.

Lemma 2.1

([4, 5, 17])

There exists a function \(\phi _{0}(x) \in C^{2}(\Omega ^{c})\) that satisfies

$$ \textstyle\begin{cases} \Delta \phi _{0}(x)=0,\quad x \in \Omega ^{c}, n\geq 1, \\ \phi _{0}|_{\partial \Omega ^{c}}=0. \end{cases} $$

In the case \(n\geq 3\), \(\phi _{0}(x)\rightarrow 1\) as \(|x|\rightarrow \infty \), \(0<\phi _{0}(x)<1\) for all \(x \in \Omega ^{c}\). In the case \(n=2\), \(\phi _{0}(x)\rightarrow \infty \) as \(|x|\rightarrow \infty \), \(0<\phi _{0}(x)\leq C\ln |x|\) for all \(x \in \Omega ^{c}\). In the case \(n=1\), \(\phi _{0}(x)\rightarrow \infty \) as \(|x|\rightarrow \infty \), \(C_{1}x\leq \phi _{0}(x)\leq C_{2}x\) for all \(x \in \Omega ^{c}\). Here, C, \(C_{1}\), \(C_{2}\) are positive constants.

Lemma 2.2

([39])

There exists a function \(\varphi _{1}(x) \in C^{2}(\Omega ^{c})\) satisfying the following boundary value problem

{ Δ φ 1 ( x ) = 1 2 φ 1 ( x ) , x Ω c , n 1 , φ 1 | Ω c = 0 , | x | , φ 1 ( x ) S n 1 e x ω d ω .

Moreover, \(0<\varphi _{1}(x)<C(1+|x|)^{-(n-1)/2}e^{|x|}\) for all \(x \in \Omega ^{c}\), where \(C>0\) is a constant.

We define \(\psi _{1}(x,t)=e^{-t}\varphi _{1}(x)\). Direct calculation shows

$$\begin{aligned} (\psi _{1})_{t}=-\psi _{1},\qquad (\psi _{1})_{tt}=\psi _{1},\qquad \Delta \psi _{1}=\frac{1}{2}\psi _{1}. \end{aligned}$$
(2.1)

Lemma 2.3

([17])

Let \(n\geq 1\), \(p>1\). It holds that

$$ \int _{\Omega ^{c}\cap \{|x|\leq t+R\}}\bigl(\phi _{0}(x) \bigr)^{-\frac{1}{p-1}}\bigl( \psi _{1}(x,t)\bigr)^{p'}\,dx \leq C(t+R)^{n-1-\frac{(n-1)p'}{2}},$$

where \(p'=\frac{p}{p-1}\), C is a positive constant.

Lemma 2.4

([17])

Let \(n\geq 1\), \(p>1\). For all \(t\geq 0\), it holds that

$$\begin{aligned} \int _{\Omega ^{c}\cap \{|x|\leq t+R\}}\psi _{1}(x,t)\,dx\leq C (t+R)^{(n-1)/2}, \end{aligned}$$

where C is a positive constant.

Definition 2.1

Assume that \(u\in C([0,T), H^{1}(\Omega ^{c}))\cap C^{1}([0,T), L^{2}(\Omega ^{c}))\). \(u \in L_{\mathrm{loc}}^{p}([0,T)\times \Omega ^{c})\) when the nonlinear term is \(f(u,u_{t})=|u|^{p}\). \(u_{t} \in L_{\mathrm{loc}}^{p}([0,T)\times \Omega ^{c})\) when the nonlinear term is \(f(u,u_{t})=|u_{t}|^{p}\). \(u_{t} \in L_{\mathrm{loc}}^{p}([0,T)\times \Omega ^{c})\) and \(u \in L_{\mathrm{loc}}^{q}([0,T)\times \Omega ^{c})\) when the nonlinear term is \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\). It holds that

$$\begin{aligned}& \int _{\Omega ^{c}}u_{t}(x,t)\phi (x,t)\,dx- \int _{\Omega ^{c}} \varepsilon u_{1}(x)\phi (x,0)\,dx \\& \qquad {}- \int _{0}^{t} \int _{\Omega ^{c}}\bigl\{ u_{s}(x,s)\phi _{s}(x,s)+u(x,s) \Delta \phi (x,s)+u_{s}(x,s)\Delta \phi (x,s)\bigr\} \,dx\,ds \\& \qquad {}+ \int _{0}^{t} \int _{\Omega ^{c}}\frac{\mu}{(1+s)^{\beta}}u_{s}(x,s) \phi (x,s)\,dx\,ds \\& \quad = \int _{0}^{t} \int _{\Omega ^{c}}f(u,u_{t})\phi (x,s)\,dx\,ds, \end{aligned}$$
(2.2)

where \(u(x,0)=\varepsilon u_{0}(x)\), \(\phi \in C^{\infty}_{0}([0,T)\times \Omega ^{c})\).

Employing the integration by parts in (2.2) and letting \(t\rightarrow T\), we obtain

$$\begin{aligned}& \int _{0}^{T} \int _{\Omega ^{c}}u(x,s)\biggl\{ \phi _{ss}(x,s)-\Delta \phi (x,s) +\Delta \phi _{s}(x,s)-\biggl( \frac{\mu \phi (x,s)}{(1+s)^{\beta}} \biggr)_{s}\biggr\} \,dx\,ds \\& \quad = \int _{\Omega ^{c}}\varepsilon u_{1}(x) \phi (x,0)\,dx - \int _{ \Omega ^{c}}\varepsilon u_{0}(x) \phi _{t}(x,0)\,dx - \int _{\Omega ^{c}} \varepsilon u_{0}(x) \Delta \phi (x,0)\,dx \\& \qquad {} + \int _{\Omega ^{c}}\varepsilon u_{0}(x) \mu \phi (x,0)\,dx + \int _{0}^{T} \int _{\Omega ^{c}}f(u,u_{t})\phi (x,s)\,dx\,ds. \end{aligned}$$

We present a multiplier

$$ m(t)=\exp \biggl(\mu \frac{(1+t)^{1-\beta}}{1-\beta}\biggr). $$
(2.3)

In the case \(\beta >1\), we have

$$ m'(t)=\frac{\mu}{(1+t)^{\beta}}m(t), \quad 0< m(0) \leq m(t)\leq 1, t\geq 0. $$
(2.4)

3 Proof of Theorem 1.1

We set

$$\textstyle\begin{cases} F_{0}(t)=\int _{\Omega ^{c}}u(x,t)\phi _{0}(x)\,dx, \\ F_{1}(t)=\int _{\Omega ^{c}}u(x,t)\psi _{1}(x,t)\,dx, \\ F_{2}(t)=\int _{\Omega ^{c}}u_{t}(x,t)\psi _{1}(x,t)\,dx. \end{cases} $$

Lemma 3.1

Let \(n\geq 1\). Under the same assumptions in Theorem 1.1, for all \(t\geq 0\), it holds that

$$F_{1}(t)\geq \frac{1}{3}m(0)\varepsilon \int _{\Omega ^{c}} u_{0}(x) \varphi _{1}(x)\,dx > 0.$$

Proof of Lemma 3.1

Employing (2.2), we deduce

$$\begin{aligned}& \frac{d}{dt} \int _{\Omega ^{c}}u_{t}(x,t)\phi (x,t)\,dx+ \int _{ \Omega ^{c}}\frac{\mu}{(1+t)^{\beta}}u_{t}(x,t)\phi (x,t)\,dx \\& \qquad {}- \int _{\Omega ^{c}}\bigl\{ u_{t}(x,t)\phi _{t}(x,t)+u(x,t)\Delta \phi (x,t)+u_{t}(x,t) \Delta \phi (x,t)\bigr\} \,dx \\& \quad = \int _{\Omega ^{c}}f(u,u_{t})\phi (x,t)\,dx. \end{aligned}$$
(3.1)

Multiplying (3.1) by \(m(t)\) and integrating over \((0,t)\) yields

$$\begin{aligned}& m(t) \int _{\Omega ^{c}}u_{t}(x,t)\phi (x,t)\,dx-m(0) \int _{\Omega ^{c}} \varepsilon u_{1}(x) \phi (x,0)\,dx \\& \qquad {}- \int _{0}^{t} \int _{\Omega ^{c}}m(s)\bigl\{ u_{s}(x,s)\phi _{s}(x,s)+u(x,s) \Delta \phi (x,s)+u_{s}(x,s)\Delta \phi (x,s)\bigr\} \,dx\,ds \\& \quad = \int _{0}^{t} \int _{\Omega ^{c}}m(s)f(u,u_{t})\phi (x,s)\,dx\,ds. \end{aligned}$$

Replacing \(\phi (x,t)\) by \(\psi _{1}(x,t)\), we come to

$$\begin{aligned} m(t) \biggl(F'_{1}(t)+\frac{3}{2}F_{1}(t) \biggr) =&m(0) \int _{\Omega ^{c}} \varepsilon \biggl(u_{1}(x)+ \frac{1}{2}u_{0}(x)\biggr)\varphi _{1}(x)\,dx \\ &{}+\frac{1}{2} \int _{0}^{t} \int _{ \Omega ^{c}}\frac{\mu}{(1+s)^{\beta}}m(s)u(x,s)\psi _{1}(x,s)\,dx\,ds \\ & {}+ \int _{0}^{t} \int _{\Omega ^{c}}m(s)f(u,u_{t}) \psi _{1}(x,s)\,dx\,ds. \end{aligned}$$

It follows that

$$\begin{aligned} F'_{1}(t)+\frac{3}{2}F_{1}(t) \geq& m(0) \int _{\Omega ^{c}} \varepsilon \biggl(u_{1}(x)+ \frac{1}{2}u_{0}(x)\biggr)\varphi _{1}(x)\,dx \\ & {} +\frac{1}{2m(t)} \int _{0}^{t} \frac{\mu}{(1+s)^{\beta}}m(s)F_{1}(s)\,ds. \end{aligned}$$
(3.2)

From straightforward computations, we observe

$$\begin{aligned} e^{\frac{3}{2}t}F_{1}(t) \geq& m(0) \int _{\Omega ^{c}}\varepsilon u_{0}(x) \varphi _{1}(x)\,dx \\ &{} +m(0)\frac{2}{3}\bigl(e^{\frac{3}{2}t}-1\bigr) \int _{ \Omega ^{c}}\varepsilon \biggl(u_{1}(x)+ \frac{1}{2}u_{0}(x)\biggr)\varphi _{1}(x)\,dx \\ \geq& \frac{1}{3}m(0)e^{\frac{3}{2}t} \int _{ \Omega ^{c}}\varepsilon u_{0}(x)\varphi _{1}(x)\,dx, \end{aligned}$$

which results in

$$\begin{aligned} F_{1}(t)\geq \frac{1}{3}m(0)\varepsilon \int _{\Omega ^{c}} u_{0}(x) \varphi _{1}(x)\,dx > 0. \end{aligned}$$

This completes the proof of Lemma 3.1. □

Proof of Theorem 1.1

Letting \(\phi (x,t)= \phi _{0}(x)\) in (2.2) with \(f(u,u_{t})=|u|^{p}\) and applying Lemma 2.1, we derive

$$ F_{0}''(t)+ \frac{\mu}{(1+t)^{\beta}}F'_{0}(t)= \int _{\Omega ^{c}} \vert u \vert ^{p} \phi _{0}(x)\,dx. $$
(3.3)

Multiplying (3.3) by \(m(t)\), we acquire

$$ \bigl[m(t)F'_{0}(t) \bigr]'=m(t) \int _{\Omega ^{c}} \vert u \vert ^{p}\phi _{0}(x)\,dx. $$
(3.4)

Integrating (3.4) over the interval \((0,t)\) and employing (2.4) yields

$$ F'_{0}(t)\geq m(t)F'_{0}(t) \geq m(0) \int _{0}^{t} \int _{\Omega ^{c}} \vert u \vert ^{p} \phi _{0}(x)\,dx\,dt. $$
(3.5)

Applying the Holder inequality and Lemma 2.1 gives rise to

$$ \begin{aligned}[b] \int _{\Omega ^{c}} \vert u \vert ^{p}\phi _{0}(x)\,dx& \geq \frac{ \vert \int _{\Omega ^{c}}u \phi _{0}(x) \,dx \vert ^{p}}{(\int _{\Omega ^{c}\cap \{ \vert x \vert \leq t+R\}} \phi _{0}(x) \,dx)^{p-1}} \\ & =C_{3}(t+R)^{-n(p-1)} \bigl\vert F_{0}(t) \bigr\vert ^{p}. \end{aligned} $$
(3.6)

Inserting (3.6) into (3.5), we deduce

$$ F_{0}(t)\geq C_{4} \int _{0}^{t} \int _{0}^{s}(r+R)^{-n(p-1)} \bigl\vert F_{0}(r) \bigr\vert ^{p}\,dr\,ds. $$
(3.7)

Taking advantage of the Holder inequality and Lemma 2.3, we come to

$$\begin{aligned} \int _{\Omega ^{c}} \vert u \vert ^{p}\phi _{0}(x)\,dx \geq& \frac{ \vert \int _{\Omega ^{c}}u \psi _{1}\,dx \vert ^{p}}{(\int _{\Omega ^{c}\cap \{ \vert x \vert \leq t+R\}}(\phi _{0})^{-\frac{1}{p-1}}(\psi _{1})^{\frac{p}{p-1}})^{p-1}} \\ \geq& C_{5}(t+R)^{(n-1)(1-p/2)} \bigl\vert F_{1}(t) \bigr\vert ^{p}. \end{aligned}$$
(3.8)

Combining (3.5) and (3.8) with Lemma 3.1, we conclude

$$\begin{aligned} F_{0}(t) \geq& C_{6}\varepsilon ^{p} \int _{0}^{t} \int _{0}^{s}(r+R)^{(n-1)(1-p/2)}\,dr\,ds \\ \geq& \frac{C_{6}}{n(n+1)}\varepsilon ^{p}(t+R)^{-(n-1)p/2} t^{n+1}, \end{aligned}$$
(3.9)

where \(C_{6}=C_{5}m(0)(\frac{1}{3}m(0)\int _{\Omega ^{c}}u_{0}(x)\varphi _{1}(x)\,dx)^{p}>0\).

Suppose that

F 0 (t) D j ( t + R ) a j t b j ,t0,j N
(3.10)

with \(D_{1}=\frac{C_{6}}{n(n+1)}\varepsilon ^{p}\), \(a_{1}=\frac{(n-1)p}{2}\), \(b_{1}=n+1\). Inserting (3.10) into (3.7) leads to

$$F_{0}(t)\geq D_{j+1}(t+R)^{-a_{j+1}}t^{b_{j+1}},$$

where

$$D_{j+1}\geq \frac{C_{4} D_{j}^{p}}{(b_{j}p+2)^{2}},\quad a_{j+1}=a_{j}p+n(p-1), b_{j+1}=b_{j}p+2.$$

Direct calculation indicates

$$\begin{aligned}& a_{j}=p^{j-1}\bigl((n-1)p/2+n\bigr)-n, \\& b_{j}=p^{j-1}\bigl(n+1+2/(p-1)\bigr)-2/(p-1), \\& D_{j}\geq \frac{C_{7}D_{j-1}^{p}}{p^{2(j-1)}}, \end{aligned}$$

where \(C_{7}=\frac{C_{4}}{(n+1+2/(p-1))^{2}}\). A straightforward calculation shows

$$D_{j}\geq \exp \bigl(p^{j-1}\bigl(\log D_{1}-S_{p}(\infty )\bigr)\bigr),$$

where \(S_{p}(\infty )=\lim_{j \rightarrow \infty}\sum^{j-1}_{k=1} \frac{2k\log p-\log C_{7}}{p^{k}}\). From (3.10), we acquire

$$ F_{0}(t)\geq (t+R)^{n}t^{-\frac{2}{p-1}} \exp \bigl(p^{j-1}J(t)\bigr), $$
(3.11)

where \(J(t)=-((n-1)p/2+n)\log (t+R)+(n+1+2/(p-1))\log t+\log D_{1}-S_{p}( \infty )\). When \(t\geq R\), we come to

$$J(t)\geq \log \bigl(t^{\frac{\gamma (p,n)}{2(p-1)}}D_{1}\bigr)-C_{8},$$

where \(C_{8}=((n-1)p/2+n)\log 2+S_{p}(\infty )>0\). It turns out that \(J(t)>1\) when \(t> C_{9}\varepsilon ^{-\frac{2p(p-1)}{r(p,n)}}\). It is deduced from (3.11) that \(F_{0}(t)\rightarrow \infty \) as \(j\rightarrow \infty \). Consequently, we arrive at the lifespan estimate

$$ T(\varepsilon ) \leq C\varepsilon ^{-\frac{2p(p-1)}{r(p,n)}}.$$

The proof of Theorem 1.1 is finished. □

4 Proofs of Theorems 1.2 and 1.3

4.1 Proof of Theorem 1.2

Taking advantage of Lemma 2.1, we obtain

$$\begin{aligned} \int _{\Omega ^{c}\cap \{ \vert x \vert \leq t+R\}}\phi _{0}(x)\,dx &\leq \int _{\{ \vert x \vert \leq t+R\}\backslash B_{1}(0)}\ln \vert x \vert \,dx \\ & \leq C_{10}(t+R)^{2}\ln (t+R). \end{aligned} $$

Thus, we have

$$ \int _{\Omega ^{c}} \vert u \vert ^{p}\phi _{0}(x)\,dx \geq C_{11}(t+R)^{-2(p-1)} \bigl\vert F_{0}(t) \bigr\vert ^{p}, $$
(4.1)

where \(C_{11}=C_{10}^{-(p-1)}(\ln (T+R))^{-(p-1)}>0\). Inserting (4.1) into (3.5), we deduce

$$ F_{0}(t)\geq C_{11}m(0) \int _{0}^{t} \int _{0}^{s}(r+R)^{-2(p-1)} \bigl\vert F_{0}(r) \bigr\vert ^{p}\,dr\,ds. $$
(4.2)

Similar to the proof of Theorem 1.1, we achieve

$$ F_{0}(t) \geq \frac{C_{6}}{6}\varepsilon ^{p}(t+R)^{-\frac{p}{2}} t^{3}. $$
(4.3)

Assume that

F 0 (t)> D ¯ j ( t + R ) a ¯ j t b ¯ j ,t0,j N
(4.4)

with \(\bar{D}_{1}=\frac{C_{6}}{6}\varepsilon ^{p}\), \(\bar{a}_{1}=\frac{p}{2}\), \(\bar{b}_{1}=3\). Inserting (4.4) into (4.2) yields

$$F_{0}(t)\geq \frac{C_{11}m(0) \bar{D}_{j}^{p}}{(\bar{b}_{j}p+2)^{2}}(t+R)^{- \bar{a}_{j}p-2(p-1)}t^{\bar{b}_{j}p+2}.$$

Direct computation gives rise to

$$F_{0}(t)\geq (t+R)^{2}t^{-\frac{2}{p-1}}\exp \bigl(p^{j-1}\bar{J}(t)\bigr),$$

where \(\bar{J}(t)\geq \log (\bar{D}_{1}t^{\frac{-p^{2}+3p+2}{2(p-1)}})-C_{12}\). Consequently, we come to the lifespan estimate

$$ T(\varepsilon ) \leq C\varepsilon ^{-\frac{2p(p-1)}{r(p,2)}}. $$
(4.5)

On the other hand, thanks to the conditions \(u_{1}(x)\geq 0\), \(\int _{\Omega ^{c}}u_{1}(x)\phi _{0}(x)\,dx \neq 0\), we derive \(\int _{\Omega ^{c}}u_{1}(x)\phi _{0}(x)\,dx > 0\). Applying (3.4) and (2.4) leads to

$$ F'_{0}(t)\geq m(0)F'_{0}(0)=C_{13} \varepsilon ,$$

where \(C_{13}=m(0)\int _{\Omega ^{c}}u_{1}(x)\phi _{0}(x)\,dx>0\). It follows that

$$ F_{0}(t)\geq C_{13} \varepsilon + \int _{\Omega ^{c}}\varepsilon u_{0}(x) \phi _{0}(x)\,dx\geq C_{14}\varepsilon (1+t), $$
(4.6)

where \(C_{14}=\min \{C_{13}, \int _{\Omega ^{c}} u_{0}(x)\phi _{0}(x)\,dx\}>0\). Inserting (4.6) into (4.2), we arrive at

$$\begin{aligned} F_{0}(t) \geq& C_{15}\varepsilon ^{p} \int _{0}^{t} \int _{0}^{s}(r+R)^{-(p-2)}\,dr\,ds \\ \geq &\frac{C_{15}\varepsilon ^{p}}{6}(t+R)^{-(p-1)}t^{3}, \end{aligned}$$
(4.7)

where \(C_{15}=C_{11}m(0)C_{14}^{p}>0\). Assume that

F 0 (t) D ¯ j ( t + R ) a ¯ j t b ¯ j ,t0,j N
(4.8)

with \(\bar{D}_{1}=\frac{C_{15}\varepsilon ^{p}}{6}\), \(\bar{a}_{1}=p-1\), \(\bar{b}_{1}=3\). Combining (4.8) with (4.2), we acquire

$$F_{0}(t)\geq \bar{D}_{j+1}(t+R)^{-\bar{a}_{j+1}}t^{\bar{b}_{j+1}},$$

where

$$\bar{D}_{j+1}> \frac{C_{11}m(0)\bar{D}_{j}^{p}}{(\bar{b}_{j}p+2)^{2}}, \qquad \bar{a}_{j+1}=p\bar{a}_{j}+2(p-1), \bar{b}_{j+1}=p\bar{b}_{j}+2.$$

Straightforward calculation shows

$$\begin{aligned}& \bar{a}_{j}=p^{j-1}(p+1)-2,\qquad \bar{b}_{j}=p^{j-1} \biggl(3+\frac{2}{p-1}\biggr)- \frac{2}{p-1}, \\& \bar{D}_{j}\geq \frac{C_{16}\bar{D}_{j-1}^{p}}{p^{2(j-1)}}\geq \exp \bigl(p^{j-1}\bigl(\log \bar{D}_{1}-\bar{S}_{p}( \infty )\bigr)\bigr), \end{aligned}$$

where \(\bar{S}_{p}(\infty )=\lim_{j\rightarrow \infty}\sum^{j-1}_{k=1}\frac{2k\log p-\log C_{16}}{p^{k}}\). Thus, taking advantage of (4.8) generates

$$F_{0}(t)\geq (t+R)^{2}t^{-\frac{2}{p-1}} \exp \bigl(p^{j-1}\bar{J}(t)\bigr),$$

where \(\bar{J}(t)\geq \log (\bar{D}_{1}t^{\frac{p(3-p)}{p-1}})-C_{17}\) with \(t\geq R\). When \(1< p<2\) and \(t\geq C\varepsilon ^{-\frac{p-1}{3-p}}\), we have \(J(t)>1\). Therefore, we derive the lifespan estimate

$$ T(\varepsilon )\leq C\varepsilon ^{-\frac{p-1}{3-p}}. $$
(4.9)

It is easy to check that (4.9) is stronger than (4.5) when \(1< p<2\), which is equivalent to

$$\frac{p-1}{3-p}< \frac{2p(p-1)}{r(p,2)}.$$

This completes the proof of Theorem 1.2.

4.2 Proof of Theorem 1.3

Utilizing Lemma 2.1, we obtain

$$ \int _{\Omega ^{c}\cap \{|x|\leq t+R\}}\phi _{0}(x)\,dx \leq \int _{1}^{t+R}C_{2}x\,dx \leq \frac{C_{2}}{2}(t+R)^{2}.$$

Thus, we arrive at

$$ \int _{\Omega ^{c}} \vert u \vert ^{p}\phi _{0}(x)\,dx \geq C_{18}(t+R)^{-2(p-1)} \bigl\vert F_{0}(t) \bigr\vert ^{p}. $$
(4.10)

According to (3.5) and (4.10), we conclude

$$ F_{0}(t)\geq C_{18}m(0) \int _{0}^{t} \int _{0}^{s}(r+R)^{-2(p-1)} \bigl\vert F_{0}(r) \bigr\vert ^{p}\,dr\,ds. $$
(4.11)

Similar to the discussion in Theorem 1.1, we derive

$$F_{0}(t) \geq \frac{C_{6}}{2}\varepsilon ^{p} t^{2}.$$

Assume that

F 0 (t)> D ¯ j ( t + R ) a ¯ j t b ¯ j ,t0,j N

with \(\bar{D}_{1}=\frac{C_{6}}{2}\varepsilon ^{p}\), \(\bar{a}_{1}=0\), \(\bar{b}_{1}=2\). Direct calculation gives rise to

$$\bar{J}(t)\geq \log \bigl(\bar{D}_{1}t^{\frac{2}{p-1}} \bigr)-C_{19}.$$

As a result, we come to the lifespan estimate

$$ T(\varepsilon ) \leq C\varepsilon ^{-\frac{p(p-1)}{2}}. $$
(4.12)

On the other hand, similar to the proof of Theorem 1.2, we have

$$ F_{0}(t)\geq C_{14}\varepsilon (t+1). $$
(4.13)

Inserting (4.13) into (4.11), we deduce

$$\begin{aligned} F_{0}(t)\geq C_{20}\varepsilon ^{p}(t+R)^{-(p-4)}. \end{aligned}$$

Assume that

F 0 (t)> D ¯ j ( t + R ) a ¯ j t b ¯ j ,t0,j N
(4.14)

with \(\bar{D}_{1}=C_{20}\varepsilon ^{p}\), \(\bar{a}_{1}=p-4\), \(\bar{b}_{1}=0\). A series of calculations show \(J(t)\geq \log (D_{1}t^{\frac{p(3-p)}{p-1}})-C_{21}\). Consequently, we obtain the lifespan estimate

$$ T(\varepsilon )\leq C\varepsilon ^{-\frac{p-1}{3-p}}. $$
(4.15)

We deduce that (4.15) is stronger than (4.12) when \(1< p<2\), which is equivalent to

$$\frac{p-1}{3-p}< \frac{p(p-1)}{2}.$$

This finishes the proof of Theorem 1.3.

5 Proof of Theorem 1.4

Replacing \(\phi (x,t)\) by \(\psi _{1}(x,t)\) in (2.2) with \(f(u,u_{t})=|u_{t}|^{p}\) and utilizing (2.1), we deduce

$$ \frac{d}{dt} \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx+ \int _{\Omega ^{c}} \frac{1}{2}(u_{t}-u)\psi _{1}\,dx + \int _{\Omega ^{c}} \frac{\mu}{(1+t)^{\beta}}u_{t}\psi _{1}\,dx= \int _{\Omega ^{c}} \vert u_{t} \vert ^{p} \psi _{1}\,dx, $$
(5.1)

which leads to

$$\frac{d}{dt} \int _{\Omega ^{c}}\biggl(u_{t}+\frac{1}{2}u \biggr)\psi _{1}\,dx= \int _{ \Omega ^{c}} \vert u_{t} \vert ^{p}\psi _{1}\,dx - \int _{\Omega ^{c}} \frac{\mu}{(1+t)^{\beta}}u_{t}\psi _{1}\,dx.$$

It follows that

$$\frac{d}{dt}\biggl(m(t) \int _{\Omega ^{c}}\biggl(u_{t}+\frac{1}{2}u \biggr)\psi _{1}\,dx\biggr)=m(t) \int _{\Omega ^{c}} \vert u_{t} \vert ^{p}\psi _{1}\,dx +\frac{1}{2} \frac{\mu}{(1+t)^{\beta}}m(t)F_{1}(t).$$

Making use of Lemma 3.1 yields

$$\begin{aligned}& m(t) \int _{\Omega ^{c}}\biggl(u_{t}+\frac{1}{2}u \biggr)\psi _{1}\,dx -m(0) \int _{ \Omega ^{c}}\varepsilon \biggl(u_{1}(x)+ \frac{1}{2}u_{0}(x)\biggr)\varphi _{1}(x)\,dx \\& \quad \geq \int _{0}^{t} \int _{\Omega ^{c}}m(t) \vert u_{t} \vert ^{p}\psi _{1}\,dx\,dt. \end{aligned}$$
(5.2)

On the other hand, multiplying (5.1) by \(m(t)\) leads to

$$\begin{aligned}& \frac{d}{dt}\biggl[m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx\biggr] +m(t) \int _{ \Omega ^{c}}\frac{1}{2}(u_{t}-u)\psi _{1}\,dx=m(t) \int _{\Omega ^{c}} \vert u_{t} \vert ^{p} \psi _{1}\,dx. \end{aligned}$$
(5.3)

From (5.2) and (5.3), we arrive at

$$\begin{aligned}& \frac{d}{dt}\biggl[m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx\biggr] + \frac{3}{2}m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx \\& \quad \geq m(0) \int _{\Omega ^{c}}\varepsilon \biggl(u_{1}(x)+ \frac{1}{2}u_{0}(x)\biggr) \varphi _{1}(x)\,dx +m(t) \int _{\Omega ^{c}} \vert u_{t} \vert ^{p}\psi _{1}\,dx \\& \qquad {} + \int _{0}^{t} \int _{\Omega ^{c}}m(t) \vert u_{t} \vert ^{p}\psi _{1}\,dx\,dt. \end{aligned}$$
(5.4)

Setting

$$\begin{aligned} G_{1}(t) =&m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx - \frac{2}{3}m(0) \int _{\Omega ^{c}}\varepsilon u_{1}(x)\varphi _{1}(x)\,dx -\frac{2}{3} \int _{0}^{t} \int _{\Omega ^{c}}m(t) \vert u_{t} \vert ^{p} \psi _{1}\,dx\,dt, \end{aligned}$$

we have \(G_{1}(0)=\frac{1}{3}m(0)\int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx>0\). It is deduced from (5.4) that

$$\begin{aligned} \frac{d}{dt}G_{1}(t)+\frac{3}{2}G_{1}(t) \geq& \frac{1}{3}m(t) \int _{ \Omega ^{c}} \vert u_{t} \vert ^{p}\psi _{1}\,dx \\ & {} +\frac{1}{2}m(0)\varepsilon \int _{\Omega ^{c}}u_{0}(x)\varphi _{1}(x)\,dx \\ \geq &0. \end{aligned}$$

It follows that \(G_{1}(t)\geq e^{-\frac{3}{2}t}G_{1}(0)> 0\). Thus, we achieve

$$\begin{aligned} m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx \geq& \frac{2}{3}m(0) \int _{ \Omega ^{c}}\varepsilon u_{1}(x)\varphi _{1}(x)\,dx \\ & {} +\frac{2}{3} \int _{0}^{t} \int _{ \Omega ^{c}}m(t) \vert u_{t} \vert ^{p}\psi _{1}\,dx\,dt. \end{aligned}$$

Taking \(H_{1}(t)=\frac{2}{3}m(0)\int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx +\frac{2}{3}\int _{0}^{t}\int _{\Omega ^{c}}m(t)|u_{t}|^{p} \psi _{1}\,dx\,dt\), we have

$$\begin{aligned}& m(t)F_{2}(t)\geq H_{1}(t) \end{aligned}$$
(5.5)

and \(H_{1}(0)=\frac{2}{3}m(0)\int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx\). Utilizing the Holder inequality and Lemma 2.4 indicates

$$\begin{aligned} \int _{\Omega ^{c}} \vert u_{t} \vert ^{p}\psi _{1}\,dx \geq& \frac{ \vert \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx \vert ^{p}}{(\int _{\Omega ^{c}\cap \{ \vert x \vert \leq t+R\}}\psi _{1}\,dx)^{p-1}} \\ = & C^{-(p-1)}(t+R)^{-\frac{(n-1)(p-1)}{2}} \bigl\vert F_{2}(t) \bigr\vert ^{p}. \end{aligned}$$
(5.6)

Taking into account (2.4), (5.5), and (5.6), we achieve

$$\begin{aligned} H_{1}'(t) =&\frac{2}{3}m(t) \int _{\Omega ^{c}} \vert u_{t} \vert ^{p}\psi _{1}\,dx \\ \geq &\frac{2}{3}C^{-(p-1)}(t+R)^{-\frac{(n-1)(p-1)}{2}}H_{1}^{p}(t). \end{aligned}$$
(5.7)

By the property of the Riccati equation, we derive that the solution blows up for \(\frac{(n-1)(p-1)}{2}\leq 1\).

When \(\frac{(n-1)(p-1)}{2}<1\) (namely, \(1< p< p_{G}(n)\)), solving the ordinary differential inequality (5.7), we come to

$$H(t)\geq \biggl(\biggl(\frac{2}{3}m(0) \int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx\biggr)^{-(p-1)}+C(t+R)^{1-(n-1)(p-1)/2}-1 \biggr)^{- \frac{1}{p-1}}.$$

Therefore, we arrive at the lifespan estimate

$$T(\varepsilon )\leq C\varepsilon ^{-\frac{2(p-1)}{2-(n-1)(p-1)}}.$$

When \(\frac{(n-1)(p-1)}{2}=1\) (namely, \(p=p_{G}(n)\)), direct computation gives rise to

$$H(t)\geq \biggl(\biggl(\frac{2}{3}m(0) \int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx\biggr)^{-(p-1)}+C\ln (t+R)\biggr)^{-\frac{1}{p-1}}.$$

As a consequence, we obtain the lifespan estimate

$$ T(\varepsilon )\leq \exp \bigl(C\varepsilon ^{-(p-1)}\bigr).$$

The proof of Theorem 1.4 is completed.

6 Proof of Theorem 1.5

We define a multiplier

$$\begin{aligned}& m_{1}(t)=(1+t)^{\mu}. \end{aligned}$$
(6.1)

It follows that \(1=m_{1}(0)\leq m_{1}(t)\), \(\frac{m'_{1}(t)}{m_{1}(t)}=\frac{\mu}{1+t}\).

Lemma 6.1

Let \(n\geq 1\). Under the same conditions in Theorem 1.5, for all \(t\geq 0\), it holds that

$$F_{1}(t)\geq \frac{1}{3m_{1}(t)}\varepsilon \int _{\Omega ^{c}} u_{0}(x) \varphi _{1}(x)\,dx \geq 0.$$

Proof of Lemma 6.1

The proof of Lemma 6.1 is similar to the proof of Lemma 3.1. By employing the multiplier \(m_{1}(t)\), we acquire the desired result when \(f(u,u_{t})=|u_{t}|^{p}\). We omit the detailed proof. □

Proof of Theorem 1.5

Similar to the proof of Theorem 1.4, we have

$$\begin{aligned}& \frac{d}{dt}\biggl[m_{1}(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx\biggr] + \frac{3}{2}m_{1}(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx \\& \quad \geq \int _{\Omega ^{c}}\varepsilon \biggl(u_{1}(x)+ \frac{1}{2}u_{0}(x)\biggr) \varphi _{1}(x)\,dx +m_{1}(t) \int _{\Omega ^{c}} \vert u_{t} \vert ^{p}\psi _{1}\,dx \\& \qquad {} + \int _{0}^{t} \int _{\Omega ^{c}}m_{1}(t) \vert u_{t} \vert ^{p}\psi _{1}\,dx\,dt. \end{aligned}$$

We set

$$\begin{aligned} G_{2}(t) =&m_{1}(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx - \frac{2}{3} \int _{\Omega ^{c}}\varepsilon u_{1}(x)\varphi _{1}(x)\,dx -\frac{2}{3} \int _{0}^{t} \int _{\Omega ^{c}}m_{1}(t) \vert u_{t} \vert ^{p} \psi _{1}\,dx\,dt. \end{aligned}$$

It turns out that

$$m_{1}(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx\geq \frac{2}{3} \int _{ \Omega ^{c}}\varepsilon u_{1}(x)\varphi _{1}(x)\,dx+\frac{2}{3} \int _{0}^{t} \int _{\Omega ^{c}}m_{1}(t) \vert u_{t} \vert ^{p}\psi _{1}\,dx\,dt.$$

Let

$$H_{2}(t)=\frac{2}{3}\varepsilon \int _{\Omega ^{c}}u_{1}(x)\varphi _{1}(x)\,dx +\frac{2}{3} \int ^{t}_{0} \int _{\Omega ^{c}}m_{1}(t) \vert u_{t} \vert ^{p} \psi _{1}\,dx\,dt.$$

It follows that \(m_{1}(t)F_{2}(t)\geq H_{2}(t)\). We bear in mind

$$H_{2}(0)=\frac{2}{3} \int _{\Omega ^{c}}\varepsilon u_{1}(x)\varphi _{1}(x)\,dx>0.$$

Applying the Holder inequality, (6.1) and Lemma 2.4, we obtain

$$\begin{aligned} H_{2}'(t) =&\frac{2}{3} \int _{\Omega ^{c}}m_{1}(t) \vert u_{t} \vert ^{p}\psi _{1}\,dx \\ \geq &\frac{2C^{-(p-1)}}{3(t+R)^{(p-1)(n+2\mu -1)/2}}H_{2}^{p}(t). \end{aligned}$$
(6.2)

By solving the ordinary differential inequality (6.2), we derive the lifespan estimate (1.8) in Theorem 1.5. The proof of Theorem 1.5 is finished. □

7 Proofs of Theorems 1.6 and 1.7

7.1 Proof of Theorem 1.6

First, we present a lemma that will be utilized in the proof.

Lemma 7.1

Let \(n\geq 1\). Under the same conditions in Theorem 1.6, for all \(t\geq 0\), it holds that

$$F_{2}(t)\geq \frac{2}{3}m(0) \int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx.$$

Proof of Lemma 7.1

When \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\), similar to the proof of Theorem 1.4, we deduce

$$\begin{aligned}& \frac{d}{dt}\biggl[m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx\biggr] + \frac{3}{2}m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx \\& \quad \geq m(0) \int _{\Omega ^{c}}\varepsilon \biggl(u_{1}(x)+ \frac{1}{2}u_{0}(x)\biggr) \varphi _{1}(x)\,dx +m(t) \int _{\Omega ^{c}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\psi _{1}\,dx \\& \qquad {} + \int _{0}^{t} \int _{\Omega ^{c}}m(s) \bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr) \psi _{1}\,dx\,ds. \end{aligned}$$

We denote

$$\begin{aligned} G_{3}(t)={}&m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx - \frac{2}{3}m(0) \int _{\Omega ^{c}}\varepsilon u_{1}(x)\varphi _{1}(x)\,dx \\ & {} -\frac{2}{3} \int _{0}^{t} \int _{\Omega ^{c}}m(s) \vert u_{t} \vert ^{p} \psi _{1}\,dx\,ds. \end{aligned} $$

It is worth noting that \(G_{3}(0)=\frac{1}{3}m(0)\int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx>0\). An elementary calculation generates

$$\begin{aligned}& \begin{aligned} \frac{d}{dt}G_{3}(t)+ \frac{3}{2}G_{3}(t)\geq{}& \frac{1}{2}m(0) \int _{ \Omega ^{c}}\varepsilon u_{0}(x)\varphi _{1}(x)\,dx \\ &{} +m(t) \int _{\Omega ^{c}}\biggl( \frac{1}{3} \vert u_{t} \vert ^{p}+ \vert u \vert ^{q} \biggr)\psi _{1}\,dx + \int _{0}^{t} \int _{ \Omega ^{c}}m(s) \vert u \vert ^{q}\psi _{1}\,dx\,ds \\ \geq {}&0. \end{aligned} \end{aligned}$$

It follows that \(G_{3}(t)\geq e^{-\frac{3}{2}t}G_{3}(0)>0\), which leads to

$$ \begin{aligned} m(t) \int _{\Omega ^{c}}u_{t}\psi _{1}\geq {}& \frac{2}{3}m(0) \int _{ \Omega ^{c}}\varepsilon u_{1}(x)\varphi _{1}(x)\,dx \,dx \\ & {} +\frac{2}{3} \int _{0}^{t} \int _{\Omega ^{c}}m(s) \vert u_{t} \vert ^{p} \psi _{1}\,dx\,ds. \end{aligned} $$
(7.1)

Thus, we have

$$F_{2}(t)\geq \frac{2}{3}m(0) \int _{\Omega ^{c}}\varepsilon u_{1}(x) \varphi _{1}(x)\,dx.$$

This completes the proof of Lemma 7.1. □

Proof of Theorem 1.6

Taking \(\phi (x,t)=\phi _{0}(x)\) in (2.2) with \(f(u,u_{t})=|u_{t}|^{p}+|u|^{q}\), we come to

$$ F''_{0}(t)+ \frac{\mu}{(1+t)^{\beta}}F'_{0}(t)= \int _{\Omega ^{c}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr) \phi _{0}(x)\,dx. $$
(7.1)

Multiplying (7.1) by \(m(t)\) leads to

$$ \bigl[m(t)F'_{0}(t) \bigr]'=m(t) \int _{\Omega ^{c}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr)\phi _{0}(x)\,dx. $$
(7.2)

Employing (2.4) gives rise to

$$ F'_{0}(t)\geq m(0) \int _{0}^{t} \int _{\Omega ^{c}}\bigl( \vert u_{t} \vert ^{p}+ \vert u \vert ^{q}\bigr) \phi _{0}(x)\,dx\,dt. $$
(7.3)

Utilizing the Holder inequality and Lemma 2.3, we obtain

$$\begin{aligned} \int _{\Omega ^{c}} \vert u_{t} \vert ^{p}\phi _{0}(x)\,dx \geq& \frac{ \vert \int _{\Omega ^{c}}u_{t}\psi _{1}\,dx \vert ^{p}}{(\int _{\Omega ^{c}\cap \{ \vert x \vert \leq t+R\}}(\phi _{0})^{-\frac{1}{p-1}}(\psi _{1})^{\frac{p}{p-1}})^{p-1}} \\ \geq& C^{-(p-1)}(t+R)^{-(n-1)(\frac{p}{2}-1)} \bigl\vert F_{2}(t) \bigr\vert ^{p}. \end{aligned}$$
(7.4)

It is deduced from (7.3), (7.4), and Lemma 7.1 that

$$ \begin{aligned}[b] F'_{0}(t)& \geq m(0) \int _{0}^{t} \int _{\Omega ^{c}} \vert u_{t} \vert ^{p} \phi _{0}(x)\,dx\,dt. \\ & \geq C_{22}m(0)\varepsilon ^{p} \int _{0}^{t}(t+R)^{-(n-1)( \frac{p}{2}-1)}\,dt, \end{aligned} $$
(7.5)

where \(C_{22}=C^{-(p-1)}(\frac{2}{3}m(0)\int _{\Omega ^{c}}u_{1}(x)\varphi _{1}(x)\,dx)^{p}>0\). It follows that

$$ F_{0}(t)>C_{23}\varepsilon ^{p}(t+R)^{-(n-1)\frac{p}{2}} t^{n+1}, $$
(7.6)

where \(C_{23}=\frac{C_{22}m(0)}{n(n+1)}\).

In the case of \(n\geq 3\), applying the Holder inequality and Lemma 2.1 yields

$$ \begin{aligned}[b] \int _{\Omega ^{c}} \vert u \vert ^{q}\phi _{0}(x)\,dx &\geq \frac{ \vert \int _{\Omega ^{c}}u \phi _{0}(x)\,dx \vert ^{q}}{(\int _{\Omega ^{c}\cap \{ \vert x \vert \leq t+R\}}1\,dx)^{q-1}} \\ & \geq C_{24}(t+R)^{-n(q-1)} \bigl\vert F_{0}(t) \bigr\vert ^{q}. \end{aligned} $$
(7.7)

From (7.3) and (7.7), we achieve

$$ \begin{aligned}[b] F'_{0}(t)&\geq m(0) \int _{0}^{t} \int _{\Omega ^{c}} \vert u \vert ^{q}\phi _{0}(x)\,dx\,dt \\ & \geq C_{24}m(0) \int _{0}^{t}(t+R)^{-n(q-1)} \bigl\vert F_{0}(t) \bigr\vert ^{q}\,dt, \end{aligned} $$
(7.8)

which implies

$$ F_{0}(t)\geq C_{24}m(0) \int _{0}^{t} \int _{0}^{s}(r+R)^{-n(q-1)} \bigl\vert F_{0}(r) \bigr\vert ^{q}\,dr\,ds. $$
(7.8)

In the case of \(n=2\), we acquire

$$ \begin{aligned} \int _{\Omega ^{c}} \vert u \vert ^{q}\phi _{0}(x)\,dx &\geq \frac{ \vert \int _{\Omega ^{c}}u \phi _{0}(x)\,dx \vert ^{q}}{(\int _{\{ \vert x \vert \leq t+R\}\backslash B_{1}(0)}\ln \vert x \vert \,dx)^{q-1}} \\ & \geq C_{25}(t+R)^{-2(q-1)} \bigl\vert F_{0}(t) \bigr\vert ^{q}, \end{aligned} $$

where \(C_{25}=C^{-(q-1)}(\ln (T+R))^{-(q-1)}>0\). In the case of \(n=1\), we derive

$$\begin{aligned} \int _{\Omega ^{c}} \vert u \vert ^{q}\phi _{0}(x)\,dx &\geq \frac{ \vert \int _{\Omega ^{c}}u \phi _{0}(x)\,dx \vert ^{q}}{(\int _{1}^{t+R}C_{2}x \,dx)^{q-1}} \\ & \geq C_{26}(t+R)^{-2(q-1)} \bigl\vert F_{0}(t) \bigr\vert ^{q}. \end{aligned} $$

It turns out that

$$ F_{0}(t)\geq C_{27}m(0) \int _{0}^{t} \int _{0}^{s}(r+R)^{-2(q-1)} \bigl\vert F_{0}(r) \bigr\vert ^{q}\,dr\,ds. $$
(7.9)

Suppose that

F 0 (t) D j ( t + R ) a j t b j ,t0,j N
(7.10)

with \(D_{1}=C_{23}\varepsilon ^{p}\), \(a_{1}=(n-1)p\), \(b_{1}=n+1\).

When \(n\geq 3\), inserting (7.10) into (7.8), we come to

$$F_{0}(t)\geq D_{j+1}(t+R)^{-a_{j+1}}t^{b_{j+1}},$$

where \(D_{j+1}\geq \frac{C_{24} m(0) D_{j}^{q}}{(b_{j}p+2)^{2}}\), \(a_{j+1}=a_{j}q+n(q-1)\), \(b_{j+1}=b_{j}q+2\). Therefore, we conclude

$$\begin{aligned} &a_{j}=q^{j-1}\bigl((n-1)p/2+n\bigr)-n, \\ &b_{j}=q^{j-1}\bigl(n+1+2/(q-1)\bigr)-2/(q-1), \\ &D_{j}\geq \frac{C_{28}D_{j-1}^{q}}{q^{2(j-1)}}, \end{aligned}$$

where \(C_{28}=\frac{C_{24}m(0)}{(n+1+2/(q-1))^{2}}\). A straightforward calculation gives rise to

$$F_{0}(t)\geq (t+R)^{n}t^{-\frac{2}{q-1}}\exp \bigl(q^{j-1}J(t)\bigr),$$

where \(J(t)\geq \log (t^{1+2/(q-1)-(n-1)p/2}D_{1})-C_{29}\), \(t\geq R\). As a consequence, we deduce the lifespan estimate

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

When \(n=2\), inserting (7.10) into (7.9) leads to

$$F_{0}(t)\geq \frac{C_{27}m(0)D_{j}^{q}}{(b_{j}q+2)^{2}}(t+R)^{-2(q-1)-a_{j}q}t^{b_{j}q+2}.$$

Similar to the derivation in the case \(n \geq 3\), we have \(J(t)\geq \log (t^{1+2/(q-1)-p/2}D_{1})-C_{30}\). Hence, we arrive at the lifespan estimate

$$ T(\varepsilon )\leq C\varepsilon ^{-\frac{2p(q-1)}{2q+2-p(q-1)}}.$$

When \(n=1\), inserting (7.10) into (7.9), we acquire

$$F_{0}(t)\geq \frac{C_{27}m(0)D_{j}^{q}}{(b_{j}q+2)^{2}}(t+R)^{-a_{j}q-2(q-1)}t^{b_{j}q+2}.$$

A simple computation indicates

$$ a_{j}=2q^{j-1}-2,\qquad b_{j}= \biggl(2+\frac{2}{q-1}\biggr)q^{j-1}-\frac{2}{q-1}, \qquad D_{j} \geq \frac{C_{31}D_{j-1}^{q}}{q^{2(j-1)}}.$$

It is deduced from (7.10) that \(F_{0}(t)\geq (t+R)^{2}t^{-\frac{2}{q-1}}\exp (p^{j-1}J(t))\), where \(J(t)\geq \log (D_{1}t^{\frac{2}{q-1}})-C_{32}\) (\(t\geq R\)). Therefore, we derive the lifespan estimate

$$ T(\varepsilon )\leq C\varepsilon ^{-\frac{p(q-1)}{2}}.$$

The proof of Theorem 1.6 is finished. □

7.2 Proof of Theorem 1.7

Similar to the derivation in the proof of Theorem 1.6, we achieve

$$F_{0}(t)>C_{23}\varepsilon ^{p}t^{n+1-(n-1)\frac{p}{2}}.$$

It holds that \(n+1-(n-1)\frac{p}{2}<1\) when \(p>\frac{2n}{n-1}\). Taking advantage of (2.4) and (7.2) leads to

$$ F_{0}(t)\geq C_{33}\varepsilon t, $$
(7.11)

where \(C_{33}=m(0)\int _{\Omega ^{c}}u_{1}(x)\phi _{0}(x)\,dx>0\). Inserting (7.11) into (7.8) gives rise to

$$F_{0}(t)\geq C_{34}\varepsilon ^{q} (t+R)^{-n(q-1)}t^{q+2},$$

where \(C_{34}=\frac{C_{24}C_{33}^{q}m(0)}{q(q+1)}>0\). Assume that

$$ F_{0}(t)\geq \tilde{D}_{j}(t+R)^{-\tilde{a}_{j}}t^{\tilde{b}_{j}} $$
(7.12)

with \(\tilde{D}_{1}=C_{34}\varepsilon ^{q}\), \(\tilde{a}_{1}=n(q-1)\), \(\tilde{b}_{1}=q+2\). Combining (7.8) and (7.12), we conclude

$$F_{0}(t)\geq \frac{C_{24}m(0)\tilde{D}_{j}^{q}}{(q\tilde{b}_{j}+2)^{2}}(t+R)^{-q \tilde{a}_{j}-n(q-1)}t^{q\tilde{b}_{j}+2}.$$

It follows that

$$F_{0}(t)\geq (t+R)^{n}t^{-\frac{2}{q-1}}\exp \bigl(q^{j-1}\tilde{J}(t)\bigr),$$

where \(\tilde{J}(t)\geq \log (t^{q+2+2/(q-1)-nq}\tilde{D}_{1})-C_{35}\), \(t\geq R\). Thus, we obtain the lifespan estimate

$$ T(\varepsilon )\leq C\varepsilon ^{-\frac{q-1}{q+1-n(q-1)}}.$$

The proof of Theorem 1.7 is completed.

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Acknowledgements

The author Sen Ming would like to express his sincere thanks to Professor Yi Zhou for his guidance and encouragement during the postdoctoral study in Fudan University. The author Sen Ming also would like to express his sincere thanks to Professors Han Yang and Ning-An Lai for their helpful suggestions and discussions.

Funding

The project is supported by the Fundamental Research Program of Shanxi Province (No. 20210302123021, No. 20210302123045, and No. 20210302123182), the Program for the Innovative Talents of Higher Education Institutions of Shanxi Province, the Innovative Research Team of North University of China (No. TD201901), and the National Natural Science Foundation of P. R. China (No. 11601446).

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Ming, S., Du, J., Su, Y. et al. Blow-up of solution to semilinear wave equations with strong damping and scattering damping. Bound Value Probl 2022, 52 (2022). https://doi.org/10.1186/s13661-022-01634-w

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