Abstract
The wave equation with a source term is considered u tt−Δu=|u| ρu in Ω×(0,+∞). We prove the existence and uniform decay rates of the energy by assuming a nonlinear feedback β( u t ) acting on the boundary provided that β has necessarily not a polynomial growth near the origin. To obtain the existence of global solutions we make use of the potential well method combined with the Faedo–Galerkin procedure and constructing a special basis. Furthermore, we prove that the energy of the system decays uniformly to zero and we obtain an explicit decay rate estimate adapting the ideas of Lasiecka and Tataru (Differential Integral Equations 6 (3) (1993) 507) and Patrick Martinez (ESAIN: Control, Optimisation Calc. Var. 4 (1999) 419). The resulting problem generalizes Martinez results and complements the works of Lasiecka and Tataru (1993) and Vitillaro (Glasgow Math. J. 44 (2002) 375).
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