Quasi Solution of a Nonlinear Inverse Parabolic Problem

Abstract

In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to $$ \aleph (u):\equiv u_{t}-\nabla \cdot (F(x,\nabla u)) $$ where the function F is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function F. At the first step of the methodology, we give a stability result corresponding to connectivity of F and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the last step, we conclude that the nonlinear inverse parabolic problem has at least one quasi solution in that class of functions.