Inverse Problem for a Two-Dimensional Diffusion Equation in a Domain with Free Boundary

Abstract

We establish conditions for the existence and uniqueness of a smooth solution to the inverse problem for a two-dimensional diffusion equation with unknown time-dependent leading coefficient in a domain with free-boundary. The equation of unknown boundary is given in the form of the product of a known function of space variables and an unknown time-dependent function. Problems with free boundaries for equations of the parabolic type serve as mathematical models for the description of various processes [1]. In the one-dimensional case where the equation of unknown boundary is specified by a function only of time, these problems are fairly completely investigated with the use of various methods both for linear and quasilinear equations of the parabolic type (see, e.g., [2‐7]). In domains with free boundaries, the inverse problems for one- and two-dimensional parabolic equations containing unknown time-dependent coefficients were also considered in [8‐16]. Thus, in particular, in [13], the inverse problem for the heat conduction equation ut = a(t) u +f(x1;x2;t) with unknown coefficient a(t) was investigated in a two-dimensional domain of the form 0 < x1 < l(t), 0 < x2 < h(t); where the functions l = l(t) and h = h(t) are unknown. In a similar domain, the conditions for the existence and uniqueness of solution of the inverse problem for the parabolic equation ut = a1(t)ux1x1 +a2(t)ux2x2 +b(x1;x2;t)ux1 +c(x1;x2;t)ux2 +d(x1;x2;t)u +f(x1;x2;t) with unknown coefficients a1(t) and a2(t) were established in [14]. In [15], the inverse problem for the heatconduction equation with unknown leading coefficient was studied in a two-dimensional domain specified by the conditions l1(t) < x1 < l2(t) and h1(t) < x2 < h2(t); where lk(t); hk(t); k = 1; 2; are unknown. In the present paper, we consider the inverse problem for the diffusion equation in a two-dimensional domain of the general form with unknown moving boundary. The motion of the boundary is described by an unknown function of time. This approach enables one to simulate the processes for which the most important problem is to describe their dynamics, which is typical, e.g., of the processes studied in medicine, ecology, etc.