Title

Intrinsic decay rate estimates for semilinear abstract second order equations with memory

Abstract

Semilinear abstract second order equation with a memory is considered. The memory kernel g(t) is subject to a general assumption, introduced for the first time in Alabau-Boussouira and Cannarsa (C. R. Acad. Sci. Paris Ser. I 347, 867–872, 2009), g′ ≤ –H(g), where the function H(∙) ∈ C1(R+) is positive, increasing and convex with H(0) = 0. The corresponding result announced in Alabau-Boussouira and Cannarsa (C. R. Acad. Sci. Paris Ser. I 347, 867–872, 2009) (with a brief idea about the proof) provides the decay rates expressed in terms of the relaxation kernel in the case relaxation kernel satisfies the equality g′ = –H(g) (Theorem 2.2 in Alabau-Boussouira and Cannarsa, C. R. Acad. Sci. Paris Ser. I 347, 867–872, 2009). In the case of inequality g′ ≤ –H(g), Alabau-Boussouira and Cannarsa (C. R. Acad. Sci. Paris Ser. I 347, 867–872, 2009) claims uniform decay of the energy without specifying the rate (Theorem 2.1 in Alabau-Boussouira and Cannarsa, C. R. Acad. Sci. Paris Ser. I 347, 867–872, 2009). The result presented in this paper establishes the decay rate estimates for the general case of inequality g′ ≤ –H(g). The decay rates are expressed (Theorem 2) in terms of the solution to a given nonlinear dissipative ODE governed by H(s). Applications to semilinear elasto-dynamic systems with memory are also provided.

Publication Title

Springer INdAM Series

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