American Institute of Mathematical Sciences

May  2003, 9(3): 651-662. doi: 10.3934/dcds.2003.9.651

Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations

 1 Department of Mathematical and Natural Sciences, The University of Tokushima, 1-1 Minamijosanjima-cho, Tokushima 770-8502, Japan

Received  March 2002 Revised  January 2003 Published  February 2003

We study the global existence and asymptotic behavior of solutions to the Cauchy problem for the semilinear dissipative wave equations: $\square u + \partial_t u = |u|^{\alpha+1}$, $u|_{t=0}=\varepsilon u_0 \in H^1 \cap L^1$, $\partial_t u |_{t=0} = \varepsilon u_1 \in L^2 \cap L^1$ with a small parameter $\varepsilon>0$. When $N\le 3$ and $2/N<\alpha \le 2/[N-2]^+$, we show the global solvability and derive the sharp rates of the solutions.
Citation: Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651
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