## Lifespan Estimate of Damped Semilinear Wave Equation in Exterior Domain with Neumann Boundary Condition

Zhao Jinglei,1, Lan Jiacheng,1, Yang Shanshan,2

Abstract

This paper concerns about the upper bound of lifespan estimate to damped semilinear wave equations in exterior domain with vanishing Neumann boundary condition. We find that the initial boundary value problem with Neumann boundary condition admits the same upper bound of lifespan as that of the Cauchy problem in $\mathbb{R}^n (n\ge 1)$. This fact is different from the zero Dirichlet boundary value problem in 2-D exterior domain for lifespan estimate, compared to the corresponding result in [6], and is also different from the zero Dirichlet boundary value problem on half line for critical power, compared to the result in [16].

Keywords： Lifespan ; Damped semilinear wave equations ; Neumann boundary condition ; Exterior problem

Zhao Jinglei, Lan Jiacheng, Yang Shanshan. Lifespan Estimate of Damped Semilinear Wave Equation in Exterior Domain with Neumann Boundary Condition. Acta Mathematica Scientia[J], 2021, 41(4): 1033-1041 doi:

## 1 引言

$$$\left\{ \begin{array}{ll} u_{tt} - \Delta u +u_t = |u_t|^p, \quad (t, x)\in[0, T)\times B_1^c, \\ u(0, x) = \varepsilon f(x), \quad u_t(0, x) = \varepsilon g(x), \quad x\in B_1^c, \\ \partial_ru(t, x)|_{\partial B_1^c} = 0, \\ \end{array} \right.$$$

## 4 定理2.1的证明

$\begin{eqnarray} && \varepsilon\left(\int_{B_1^c}(f(x)+g(x)){\rm d}x\right)+ \int_0^{T_{\varepsilon}}\int_{B_1^c}|u|^p\Phi(t, x){\rm d}x{\rm d}t{}\\ & = &\int_0^{T_{\varepsilon}}\int_{B_1^c}u\Phi_{tt}{\rm d}x{\rm d}t-\int_0^{T_{\varepsilon}}\int_{B_1^c}u\Phi_{t}{\rm d}x{\rm d}t-\int_0^{T_{\varepsilon}}\int_{B_1^c}u\Delta\Phi {\rm d}x{\rm d}t{}\\ &\triangleq&I+II+III. \end{eqnarray}$

$\begin{eqnarray} Y_2(R)& = &\int_0^{T_{\varepsilon}}\int_{B_1^c}|u|^p\bigg(\int_0^R\eta^{2p'}(t/\sigma)\theta^{*2p'}(|x|^2/\sigma)\sigma^{-1}{\rm d}\sigma\bigg){\rm d}x{\rm d}t{}\\ & = &\int_0^{T_{\varepsilon}}\int_{B_1^c}|u|^p\bigg(\int_{\frac{|x|^2}{R}}^\infty\eta^{2p'}(s)\theta^{*2p'}\big(\frac{|x|^2s}{t}\big)s^{-1}{\rm d}s\bigg){\rm d}x{\rm d}t{}\\ &\le& \log2\int_0^{T_{\varepsilon}}\int_{B_1^c}|u|^p\eta_R^{2p'}(t)\theta_R^{2p'}(x){\rm d}x{\rm d}t. \end{eqnarray}$

$$$\left(C_2\varepsilon+Y_1(R)+Y_2(R)\right)^p\le C_3R(Y_1'(R)+Y_2'(R)),$$$

$$$\left(C_2\varepsilon+\overline{Y}(R)\right)^p\le C_3R\overline{Y}'(R),$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Crispo F , Maremonti P .

An interpolation inequality in exterior domains

Rend Semi Mate Univ Padova, 2004, 112, 11- 39

Fujita H .

On the blowing up of solutions of the Cauchy problem for utu+u1+α

Journal of the Faculty of Science University Tokyo Sect I, 1966, 13, 109- 124

Geng J B , Yang Z Z , Lai N A .

Blow-up and lifespan estimates for initial boundary value problems for semilinear Schrödinger equations on half-line

Acta Math Sci, 2016, 36A (6): 1186- 1195

Huang S J , Meng X W .

Improved ordinary differential inequality and its application to semilinear wave equations

Acta Math Sci, 2020, 40A (5): 1319- 1332

Ikeda M , Ogawa T .

Lifespan of solutions to the damped wave equation with a critical nonlinearity

Journal of Differential Equations, 2016, 261 (3): 1880- 1903

Ikeda M , Sobajima M .

Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain

Journal of Mathematical Analysis and Applications, 2019, 470, 318- 326

Ikehata R .

Global existence of solutions for semilinear damped wave equation in 2-D exterior domain

Journal of Differential Equations, 2004, 200, 53- 68

Ikehata R .

Two dimensional exterior mixed problem for semilinear damped wave equations

Journal of Mathematical Analysis and Applications, 2005, 301, 366- 377

Lai N A , Takamura H , Wakasa K .

Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent

Journal of Differential Equations, 2017, 263, 5377- 5394

Lai N A , Yin S L .

Finite time blow-up for a kind of initial-boundary value problem of semilinear damped wave equation

Mathematical Methods in the Applied Sciences, 2017, 40 (4): 1223- 1230

Lai N A , Zhou Y .

The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions

Journal De Mathematiques Pures Et Appliquees, 2018, 123, 229- 243

Li T , Zhou Y .

Breakdown of solutions to $□u+u_t=|u|.{1+α}$

Discrete Contin Dyn Syst, 1995, 1 (4): 503- 520

Li Y C .

Classical solutions for fully nonlinear wave equations with dissipation

Chinese Annals of Mathematics Series A, 1996, 17 (4): 451- 466

Lin Y H , Jiang H B , Yin S L .

Global existence for damped semilinear wave equations outside obstacles in high dimensions(in Chinese)

Scientia Sinica Mathematica, 2018, 48 (4): 507- 518

Nishihara K .

Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application

Mathematische Zeitschrift, 2003, 244 (3): 631- 649

Nishihara K , Zhao H J .

Existence and nonexistence of time-global solutions to damped wave equation on half-line

Nonlinear Analysis: Theory Methods Applications, 2005, 61 (6): 931- 960

Ogawa T , Takeda H .

Non-existence of weak solutions to nonlinear damped wave equations in exterior domains

Nonlinear Analysis: Theory Methods Applications, 2009, 70 (10): 3696- 3701

Todorova G , Yordanov B .

Critical exponent for a nonlinear wave equation with damping

Journal of Differential Equations, 2001, 174 (2): 464- 489

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