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Integro-differential equations of hyperbolic type with positive definite kernels. (English) Zbl 1218.45010

The main purpose of this paper is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in Hilbert spaces. For kernels that decay exponentially at infinity and possess strongly positive definite primitives, the exponential stability of weak solutions is obtained in the energy norm. It is also shown that this theory applies to several examples of kernels with possibly variable sign, and to a problem in nonlinear viscoelasticity. Finally, an application to a nonlinear wave equation with memory is given. This paper ends with Appendix A intended to assist the reader with some technical properties for linear systems.

MSC:

45K05 Integro-partial differential equations
45M10 Stability theory for integral equations
45M05 Asymptotics of solutions to integral equations
47J35 Nonlinear evolution equations
45G10 Other nonlinear integral equations
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