Uniqueness for multidimensional kernel determination problems from a parabolic integro-differential equation

Abstract

We study two problems of determining the kernel of the integral terms in a parabolic integro-differential equation. In the first problem the kernel depends on time t and x = (x 1 , . . . , xn) spatial variables in the multidimensional integro-differential equation of heat conduction. In the second problem the kernel it is determined from one dimensional integro-differential heat equation with a time-variable coefficient of thermal conductivity. In both cases it is supposed that the initial condition for this equation depends on a parameter y = (y 1 , . . . , yn) and the additional condition is given with respect to a solution of direct problem on the hyperplanes x = y. It is shown that if the unknown kernel has the form k(x, t) =[?] i=o N a i (x)b i (t), then it can be uniquely determined.

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

“…Partial differential equations describing the dynamics of viscoelastic materials have enormous implications to applications of these materials in engineering and scientific communities; the governing equations incorporate hereditary terms to stress that the aftereffect in the materials can not be neglected (see References [3][4][5][6]). In theoretical study or engineering applications, the aftereffect of some materials could be neglected for sufficiently large time, while the aftereffect of the other materials could last in infinitely long time periods.…”