Abstract
In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.
Highlights
Introduction and preliminariesLet T := R/2πZ be the one-dimensional torus, let Q := (0, T )×T and M > 0
In this article we study the existence and uniqueness of the solution of the following heat equation involving a memory term:
In many problems arising in mathematical physics such as flow of fluid through fissured rocks, diffusion process of gas in a transparent tube, heat conduction in materials, and viscoelasticity, one may encounter memory effects that are relevant from a physical point of view and can be modeled by nonlocal terms
Summary
Let T := R/2πZ be the one-dimensional torus, let Q := (0, T )×T and M > 0. In this article we study the existence and uniqueness of the solution of the following heat equation involving a memory term: ut(t, x) − uxx(t, x) + M u(0, x) = u0(x). In [3], a modified Fourier’s law which is independent of the present value of the temperature gradient, but depends on its history, is introduced to correct the unphysical property of instantaneous propagation for the heat equation. By following [1, 5] (see, [2]) has been proposed a linearized theory for the heat transfer in isotropic media in which the heat flux depends both on the present value of the temperature gradient and its history being given by. A particular and simplified case of this modified Fourier’s law is used for the equation with memory term (1).
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