Small data solutions for semilinear wave equations with effective damping

Pages: 183 - 191, Issue special, November 2013

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Marcello D'Abbicco - Department of Mathematics, University of Bari, Bari, 70124, Italy (email)

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 $b(t)u_t - \Delta u = f(t,u)$.

Keywords:  Semilinear waves, effective damping, decay estimates, small data, global existence.
Mathematics Subject Classification:  35L71.

Received: September 2012; Published: November 2013.