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Time decay of solutions of semilinear strongly damped generalized wave equations. (English) Zbl 0753.35011

(Author’s abstract.) We study the asymptotic behavior in time of the solutions of dissipative perturbations of wave-type equations in \(\mathbb{R}^ N\), \(u_{tt}+Bu_ t+Au+C(u)=0\), with commuting positive operators \(A\), \(B\) and a power-like nonlinearity \(G(u)\). First we give some (pseudo) conformal invariants of the linear operator in the equation. This allows us to derive optimal decay rates for the solutions of the linearized problems. We then prove some decay estimates for the nonlinear problems using the tools of scattering theory and the aforementioned conformal invariants.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
35L30 Initial value problems for higher-order hyperbolic equations
35L05 Wave equation
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