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Existence and asymptotic stability for evolution problems on manifolds with damping and source terms. (English) Zbl 1073.35168

Summary: One considers the nonlinear evolution equation with source and damping terms

u tt +Au+g(u t )=|u| ρ uon𝛤×(0,),

where 𝛤 is a compact manifold. We prove the global existence of solutions making use of the potential well method. Furthermore, we study the asymptotic behaviour of solutions adapting the ideas introduced by P. Martinez in [ESAIM, Control Optim. Calc. Var. 4, 419–444 (1999; Zbl 0923.35027)], when the frictional damping mechanism g on the manifold does not necessarily have a polynomial growth near the origin.

35L70Nonlinear second-order hyperbolic equations
58J45Hyperbolic partial differential equations on manifolds