Translator Disclaimer
December 2011 Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Time-dependent Damping
Kenji NISHIHARA
Tokyo J. Math. 34(2): 327-343 (December 2011). DOI: 10.3836/tjm/1327931389

Abstract

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{@{}ll} u_{tt} - \Delta u + b(t)u_t=f(u)\,, & (t,x) \in {\bf R}^+ \times {\bf R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x)\,, & x \in {\bf R}^N\,. \end{array}\right. {(*)} $$ hen $b(t)=(t+1)^{-\beta}$ with $0\le \beta <1$, the damping is effective and the solution $u$ to ($*$) behaves as that to the corresponding parabolic problem. When $f(u)=O(|u|^{\rho})$ as $u \to 0$ with $1<\rho < \frac{N+2}{[N-2]_+}$(the Sobolev exponent), our main aim is to show the time-global existence of solutions for small data in the supercritical exponent $\rho>\rho_F(N):=1+2/N$. We also obtain some blow-up results on the solution within a finite time, so that the smallness of the data is essential to get global existence in the supercritical exponent case.

Citation

Download Citation

Kenji NISHIHARA. "Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Time-dependent Damping." Tokyo J. Math. 34 (2) 327 - 343, December 2011. https://doi.org/10.3836/tjm/1327931389

Information

Published: December 2011
First available in Project Euclid: 30 January 2012

zbMATH: 1242.35174
MathSciNet: MR2918909
Digital Object Identifier: 10.3836/tjm/1327931389

Rights: Copyright © 2011 Publication Committee for the Tokyo Journal of Mathematics

JOURNAL ARTICLE
17 PAGES


SHARE
Vol.34 • No. 2 • December 2011
Back to Top