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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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