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January/February 2019 The sharp estimate of the lifespan for semilinear wave equation with time-dependent damping
Masahiro Ikeda, Takahisa Inui
Differential Integral Equations 32(1/2): 1-36 (January/February 2019).

Abstract

We consider the following semilinear wave equation with time-dependent damping. \begin{align*} \left\{ \begin{array}{ll} \partial_t^2 u - \Delta u + b(t)\partial_t u = |u|^{p}, & (t,x) \in [0,T) \times \mathbb R^n, \\ u(0,x)=\varepsilon u_0(x), u_t(0,x)=\varepsilon u_1(x), & x \in \mathbb R^n, \end{array} \right. \end{align*} where $n \in \mathbb N$, $p > 1$, $\varepsilon>0$, and $b(t) \approx (t+1)^{-\beta}$ with $\beta \in [-1,1)$. It is known that small data blow-up occurs when $1 < p < p_F$ and, on the other hand, small data global existence holds when $p > p_F$, where $p_F:=1+2/n$ is the Fujita exponent. The sharp estimate of the lifespan was well studied when $1 < p < p_F$. In the critical case $p=p_F$, the lower estimate of the lifespan was also investigated. Recently, Lai and Zhou [15] obtained the sharp upper estimate of the lifespan when $p=p_F$ and $b(t)=1$. In the present paper, we give the sharp upper estimate of the lifespan when $p=p_F$ and $b(t) \approx (t+1)^{-\beta}$ with $\beta \in [-1,1)$ by the Lai--Zhou method.

Citation

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Masahiro Ikeda. Takahisa Inui. "The sharp estimate of the lifespan for semilinear wave equation with time-dependent damping." Differential Integral Equations 32 (1/2) 1 - 36, January/February 2019.

Information

Published: January/February 2019
First available in Project Euclid: 11 December 2018

zbMATH: 07031707
MathSciNet: MR3909977

Subjects:
Primary: 35Q55, 37D10, 37K40, 37K45

Rights: Copyright © 2019 Khayyam Publishing, Inc.

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Vol.32 • No. 1/2 • January/February 2019
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