Asymptotics for the nonlinear dissipative wave equation

Matsuyama, Tokio

doi:10.1090/S0002-9947-02-03147-1

Asymptotics for the nonlinear dissipative wave equation
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by Tokio Matsuyama PDF

Trans. Amer. Math. Soc. 355 (2003), 865-899 Request permission

Abstract:

We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like $a_1 (1+\vert x \vert )^{-\delta } \vert u_t \vert ^{\beta } u_t$ $(a_1$, $\beta$, $\delta >0)$ and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.

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