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A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms. (English) Zbl 0812.35081
For the equations \[ u_{tt}- \Bigl( \alpha+ 2\beta \int_ \Omega| \nabla u(t,y) |^ 2 dy \Bigr) \Delta u+ \delta u_ t= \mu u^ 3, \] \(x\in \Omega\subset \mathbb{R}^ 3\), smoothly bounded, \(t>0\), with Dirichlet boundary conditions and initial conditions, \(\alpha>0\), \(\beta\geq 0\), \(\delta,\mu\in \mathbb{R}\), global existence and blow-up results, resp., are proved. If \(\alpha>0\), \(\beta\geq 0\), \(\delta>0\), \(\mu>0\), if the initial data belong to the associated stable set, and if \(\mu C_ 4^ 4 >2\beta\) (where \(C_ p\) is the constant in \(\| u\|_ p\leq C_ p\| \nabla u\|_ 2)\), then a global smooth solution exists, provided the initial data are sufficiently small. If \(\alpha>0\), \(\beta\geq 0\), \(\delta\geq 0\), \(\mu>0\), if \(\mu C_ 4^ 4> 2\beta\), and if the data belong to the unstable set, then the local solution blows up in finite time. Also for the case \(\mu C_ 4^ 4\leq 2\beta\), a global existence result is presented.
Reviewer: R.Racke (Konstanz)

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