Free Boundary Problem for a Non-Linear System of Parabolic Equations, Including One with Reversed Time

Abstract

The problem under consideration has its origin in the theory of transfer of non-electrolytes through thick deformable membranes, (L. Rubinstein 1974, 1980) and represents a strongly simplified model for a pure-diffusion formalism of that theory. The membran’s shape is determined here by a second order parabolic equation with the coefficient of diffusivity proportional to the difference of concentrations in appartments divided by a membrane. This difference changes its sign along a line which has to be determined in the course of the problem solution. Using Gevrey’s coordinate transform (M. Gevrey, 1914) one reduces the problem to the system of non-linear Volterra integral equations of the second kind and one linear Fredholm equation of the first kind with a symmetric kernel. The solution of the latter, if exists or not exists, may be represented with a prescribed accuracy in L₂ (E. Goursat, 1915). All other equations are solvable in Holder norms. The solution of the system of integral equations, understood in such a restrictive sense, is constructed by means of some contraction mapping.

The full version of the paper to appear in “Annali di Matematica pura ed applicata.”