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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)

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
nonlocal term; global small solution; blow-up; stable set