Non-homogeneous nonlinear damped wave equations in unbounded domains.(English)Zbl 0728.35071

Author’s abstract: The author presents a global existence theorem of solutions for the following Cauchy problem: $u_{tt}-\partial_ ia_{ik}\partial_ ku+u_ t=f(t,x,u,u_ t,\nabla u,\nabla u_ t,\nabla^ 2u)\text{ in } \{t>0,\quad x\in \Omega \},$
$u(0,x)=u^ 0(x),\quad u_ t(0,x)=u^ 1(x)\text{ on } \{t=0,\quad x\in \Omega \}.$ The domain $$\Omega$$ equals $$R^ 3$$ or $$\Omega$$ is an exterior domain in $$R^ 3$$ with smoothly bounded star-shaped complement. In the latter case the boundary condition is of Dirichlet type, i.e., $$u|_{\partial \Omega}=0$$. The main theorem is obtained for small data $$\{u^ 0,u^ 1\}$$ under certain conditions on the coefficients $$a_{ik}$$. The $$L^ p-L^ q$$ decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.
Reviewer: M.Tsuji (Kyoto)

MSC:

 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations
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