## Lifespan Estimate of Damped Semilinear Wave Equation in Exterior Domain with Neumann Boundary Condition 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 , and is also different from the zero Dirichlet boundary value problem on half line for critical power, compared to the result in .

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

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

## 4 定理2.1的证明

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

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

$\begin{equation} \left(C_2\varepsilon+Y_1(R)+Y_2(R)\right)^p\le C_3R(Y_1'(R)+Y_2'(R)), \end{equation}$

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

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