Critical exponent for the semilinear wave equation with time-dependent damping

Jiayun Lin; Kenji Nishihara; Jian Zhai

doi:10.3934/dcds.2012.32.4307

Article Contents

2012, Volume 32, Issue 12: 4307-4320. Doi: 10.3934/dcds.2012.32.4307

This issue Previous Article Some results on perturbations of Lyapunov exponents Next Article Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps

Critical exponent for the semilinear wave equation with time-dependent damping

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R.
2.
Faculty of Political Science and Economics, Waseda University, Tokyo 169-8050
3.
Department of Mathematics, Zhejiang University, Hangzhou 310027

Received: May 2011

Revised: May 2012

Published: December 2012

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Abstract

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{ll} u_{tt} - \Delta u + b(t)u_t=|u|^{\rho}, & (t,x) \in \mathbb{R}^+ \times \mathbb{R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x), & x \in \mathbb{R}^N. \end{array}\right. (*) $$ When $b(t)=b_0(t+1)^{-\beta}$ with $b_0>0$ and $-1 < \beta <1$ and $\int_{{\bf R}^N} u_i(x)\,dx >0\,(i=0,1)$, we show that the time-global solution of ($*$) does not exist provided that $1<\rho \leq \rho_F(N):= 1+2/N$ (Fujita exponent). On the other hand, when $\rho_F(N)<\rho<\frac{N+2}{[N-2]_+}:= \left\{ \begin{array}{ll} \infty & (N=1,2), \\ (N+2)/(N-2) & (N \ge 3), \end{array} \right.$ the small data global existence of solution has been recently proved in [K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that $0 \le \beta<1$. We can prove the small data global existence even if $-1<\beta<0$. Thus, we conclude that the Fujita exponent $\rho_F(N)$ is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109--114].

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Mathematics Subject Classification: Primary: 35L05, 35L70; Secondary: 35B33.

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