Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2014

AUTHORS

Irena Lasiecka , Xiaojun Wang

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. More... »

PAGES

271-303

References to SciGraph publications

Book

TITLE

New Prospects in Direct, Inverse and Control Problems for Evolution Equations

ISBN

978-3-319-11405-7
978-3-319-11406-4

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-319-11406-4_14

DOI

http://dx.doi.org/10.1007/978-3-319-11406-4_14

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1040325893


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