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Remarks on the semilinear wave equations. (English) Zbl 0965.35096

Summary: We study the semilinear wave equation of the form: \[ \square u+ u_t+|u_t|^{p-1} u_t= |u|^{q-1}u. \] When \(1< q\leq p\), a solution exists globally in all time and when \(1< p< q\), the local solution blows up in finite time for negative initial energy.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs