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

Kosuke Ono

doi:10.3934/dcds.2003.9.651

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2003, Volume 9, Issue 3: 651-662. Doi: 10.3934/dcds.2003.9.651

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

Kosuke Ono^1,

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

Received: February 28, 2002

Revised: December 31, 2002

Published: April 2003

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Abstract

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.

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Mathematics Subject Classification: 35L15, 35L70, 35L05, 35B40.

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