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On a wave equation with supercritical interior and boundary sources and damping terms. (English) Zbl 1244.35092

In this article the following system is investigated:

u tt -Δu+g 0 (u t )=|u| p-1 uinΩ×[0,), ν u+u+g(u t )=|u| k-1 uonΓ×[0,),u(0)=u 0 H 1 (Ω),u t (0)=u 1 L 2 (Ω),(1)

where Ω 3 is a bounded open set with sufficiently smooth boundary Γ, the maps g 0 (s) and g(s) are monotone and represent the interior and boundary dissipation, the Nemytski operators |u| p-1 u, p>1, |u| k-1 u, k>1, model the interior and boundary sources respectively.

The authors give conditions for g 0 , g and p, k under which the problem (1) has unique global weak solution u𝒞([0,),H 1 (Ω)). The exponential and algebraic uniform decay rates of the finite energy are established. The authors prove a blow up result for weak solutions of (1) with nonnegative initial energy.

MSC:
35L71Semilinear second-order hyperbolic equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35B35Stability of solutions of PDE
35L20Second order hyperbolic equations, boundary value problems
35B33Critical exponents (PDE)
35B40Asymptotic behavior of solutions of PDE
35B44Blow-up (PDE)