Small data solutions for semilinear wave equations with effective damping

Marcello D'Abbicco

doi:10.3934/proc.2013.2013.183

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2013, Volume 2013: 183-191. Doi: 10.3934/proc.2013.2013.183 open access

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Small data solutions for semilinear wave equations with effective damping

Marcello D'Abbicco^1,

1.
Department of Mathematics, University of Bari, Bari, 70124

Received: August 31, 2012

Published: October 2013

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Abstract

We consider the Cauchy problem for the semi-linear damped wave equation
$ u_{tt} - \Delta u + b(t)u_t = f(t,u),\qquad u(0,x) = u_0(x),\qquad u_t(0,x) = u_1(x). $
We prove the global existence of small data solution in low space dimension, and we derive $(L^m\cap L^2)-L^2$ decay estimates, for $m\in[1,2)$. We assume that the time-dependent damping term $b(t)>0$ is effective, that is, the equation inherits some properties of the parabolic equation $b(t)u_t - \Delta u = f(t,u)$.

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Mathematics Subject Classification: 35L71.

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References

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