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 \(H(\cdot ) \in C^{1}(R^{+})\) 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.
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- 1.
In this work, we use \(\mathbb{R}_{+}\) for [0, ∞), which is different from (0, ∞). Hence \(C^{1}(\mathbb{R}_{+})\) is for C 1([0, ∞)), meaning that it contains functions whose first derivatives can be continuously extended to the boundary. Similar to \(C^{2}(\mathbb{R}_{+})\) and so on.
- 2.
Here we can take s(0) = E(0) for notational convenience, since \(E(T)\leq E(0)\).
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Acknowledgements
Research of I. Lasiecka and X. Wang partially supported by DMS Grant 0104305 and AFOSR Grant FA9550-09-1-0459.
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Lasiecka, I., Wang, X. (2014). Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory. In: Favini, A., Fragnelli, G., Mininni, R. (eds) New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Springer INdAM Series, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-11406-4_14
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