Integro-differential equations of hyperbolic type with positive definite kernels

Abstract

The main purpose of this work is to study the damping effect\ud of memory terms associated with singular convolution kernels on\ud the asymptotic behavior of the solutions of second order evolution\ud equations in Hilbert spaces. For kernels that decay exponentially at\ud infinity and possess strongly positive definite primitives, the exponential\ud stability of weak solutions is obtained in the energy norm.\ud It is also shown that this theory applies to several examples of\ud kernels with possibly variable sign, and to a problem in nonlinear\ud viscoelasticity

On integro‐differential inclusions with operator‐valued kernels

“…The authors of assumed that the integrated kernel defines a positive definite convolution operator on . However, according to [, Proposition 2.2] (a), this condition also implies (iii) for d = 0. By hypothesis (iii), we also cover kernels of bounded variation. Indeed, if is a function of strong bounded variation (see [, Definition 3.2.4]), then .…”

“…which yields (iii) for d = 0 according to (a). The authors of assumed that the integrated kernel defines a positive definite convolution operator on . However, according to [, Proposition 2.2] (a), this condition also implies (iii) for d = 0.…”

“…In the same way, the imaginary part is defined as Im M :D 1 2ı .M M /. In this terminology, condition (7) means that there exists c > 0 such that for each z 2 B C .r, r/, the estimate…”

“…In order to show the solvability condition (7) for the material laws M 0 and M 1 , we have to show that there exist r 1 , c > 0 with r 1 Ä r such that for all z 2 B C .r 1 , r 1 /,…”

“…Topics such as well-posedness, regularity and the asymptotic behaviour of solutions were studied by several authors, even for semilinear versions of (1) or (2) (e.g. [6][7][8] for the hyperbolic and [9][10][11] for the parabolic case).…”