Damped wave equation with super critical nonlinearities. (English) Zbl 1224.35281

Hayashi, Nakao; Kaikina, Elena I.; Naumkin, Pavel I.

Differ. Integral Equ. 17, No. 5-6, 637-652 (2004).

The authors consider the Cauchy problem for the semilinear damped wave equation \(\partial ^2_t u + \partial _t u - \Delta u = N(u)\), where essentially \(| N(U)| \leq C | u| ^{\rho }\), with \(\rho > 1 + n/2\). Assuming that the initial data are sufficiently small and smooth, they establish the global existence and decay of unique classical solution. The proof is based on a number of technical Fourier-type estimates of the linear part and the fixed point argument.

Reviewer: Dalibor Pražák (Praha)

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

damped wave equation; global existence for small data; supercritical nonlinearity; decay estimates