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

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|u|^{p}$\end{document}, derivative-type nonlinearity |ut|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|u_{t}|^{p}$\end{document}, and combined type nonlinearities |ut|p+|u|q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|u_{t}|^{p}+|u|^{q}$\end{document} 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.

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 have been studied extensively (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). It is worth noting that problem (1.2) admits the Fujita exponent p F (n) = 1 + 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) = ∞. If n ≥ 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 ≥ 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 ≥ 4) is demonstrated by taking advantage of the Kato lemma (see [15]). Han [4,5] investigates the nonexistence of a global solution and the upperbound 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 ≥ 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 ≤ 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) = ∞. When n ≥ 2, p G (n) = n+1 n-1 . Zhou [16] illustrates the blowup 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 (1.4) The behaviors of the solution are divided into four cases according to the decay rate β. When β ∈ (-∞, -1), the damping term is overdamping, which means that the solution dose not decay to zero when t → ∞. When β ∈ [-1, 1), the damping term is effective and the solution behaves like that of the heat equation. When β = 1, the damping term is scale invariant by imposing the scaling u(x, t) = u(λx, λ(1 + t) -1)(λ > 0). When β ∈ (1, ∞), the solution scatters to the free-wave equation when t → ∞. This is the scattering case. The corresponding nonlinear damped wave equation causes extensive attention (see detailed illustrations in [18][19][20][21][22][23][24][25][26][27][28][29][30]), where f (u, u t ) = |u| p , |u t | p , |u t | p + |u| q , respectively. In the scattering case β > 1, Lai et al. [24] derive the blowup 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 ≥ 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 β = 1, Imai et al. [20] establish the lifespan estimate of the solution to problem (1.5) with f (u, u t ) = |u| p when μ = 2 and 1 < p ≤ 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 ≤ p F (1) when μ = 2. Kato et al. [21] discuss the blow-up dynamics and lifespan estimate of the solution to problem (1.5) with μ = 2, β = 1, f (u, u t ) = |u| p (1 < p ≤ 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 μ 1 1+t u t and mass term μ 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 β = 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 [32][33][34][35][36] for more details.
Many scholars focus on studying the Cauchy problem for the wave equation with a strong damping term x ∈ R n , t > 0, (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 + 2 n . In addition, the existence of a global solution is obtained when p > 1 + 3 n-1 (n ≥ 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 testfunction 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 [24-26, 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 [4][5][6]17], where the semilinear wave equations without damping are discussed. We observe that Lai et al. [24][25][26] 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 [24][25][26] 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.
are nonnegative functions and u 0 does not vanish identically. If a solution u to prob-
where C > 0 is a constant independent of ε.
Theorem 1.7 Let n ≥ 2, μ > 0, and β > 1. Assume that the initial values satisfy the same conditions in Theorem 1.6. If p > 2n n-1 and 1 < q < 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(ε) satisfies where C > 0 is a constant independent of ε.

Remark 1.1 The restriction q < 2n
n-2 is necessary to guarantee the integrability of the nonlinear term |u| q in Theorem 1.6.

Preliminaries
In order to prove the main results, we present several related lemmas and the definition of a weak solution.
where C is a positive constant.
Employing the integration by parts in (2.2) and letting t → T, we obtain We present a multiplier In the case β > 1, we have (2.4)

Proof of Theorem 1.1
We set (3.1)

Multiplying (3.1) by m(t) and integrating over (0, t) yields
Replacing φ(x, t) by ψ 1 (x, t), we come to It follows that From straightforward computations, we observe which results in This completes the proof of Lemma 3.1. (2.2) with f (u, u t ) = |u| p and applying Lemma 2.1, we derive

Proof of Theorem 1.3
Utilizing Lemma 2.1, we obtain Thus, we arrive at According to (3.5) and (4.10), we conclude Similar to the discussion in Theorem 1.1, we derive Assume that As a result, we come to the lifespan estimate On the other hand, similar to the proof of Theorem 1.2, we have F 0 (t) ≥ C 14 ε(t + 1). (4.13) Inserting (4.13) into (4.11), we deduce Assume that  We deduce that (4.15) is stronger than (4.12) when 1 < p < 2, which is equivalent to This finishes the proof of Theorem 1.3.
The proof of Theorem 1.4 is completed.
This completes the proof of Lemma 7.1.
The proof of Theorem 1.6 is finished.

Proof of Theorem 1.7
Similar to the derivation in the proof of Theorem 1.6, we achieve F 0 (t) > C 23 ε p t n+1-(n-1) p 2 .
The proof of Theorem 1.7 is completed.