Existence and decay of solutions of a viscoelastic equation with a nonlinear source

Abstract In a bounded domain, we consider u tt - Δ u + ∫ 0 t g ( t - τ ) Δ u d τ = | u | γ u , where γ > 0 , and g is a nonnegative and decaying function. We prove a local existence theorem and show, for certain initial data and suitable conditions on g and γ , that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g .

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