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Global existence for semilinear damped wave equations in the scattering case. (English) Zbl 1424.35237

The authors study the Cauchy problem to the equation \(u_{tt}-\Delta_{g}u+\mu(1+t)^{-\beta}u_t=| u_t|^p,\) where \(\beta >1\) and \((\mathbb R^n,g)\) is a non-trapping asymptotically Euclidean manifold. Under some assumptions on \(\beta, p, \mu\) and the manifold \((\mathbb R^n,g)\) the global existence of the problem for small initial data is proved. The solution is obtained as a limit of the solutions to damped wave equations.

MSC:

35L05 Wave equation
35L15 Initial value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations