Summary
The problem under consideration has its origin in the theory of transfer of nonelectrolytes through thick deformable membranes [1], [2] and represents a strongly simplified model for a pure-diffusion formalism of that theory. The membrane's shape is determined here by a second order parabolic equation with the coefficient of diffusivity proportional to the difference of concentrations in compartments 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 [3] one reduces the problem to the system of nonlinear 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[4]. All other equations are solvable in Hölder norms. The solution of the system of integral equations, understood in such a restrective sense, is constructed by means of some contraction mapping.
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L. Rubinstein,Passive transfer of low-molecular nonelectrolytes across deformable semipermeable membranes, I. Equations of convective-diffusion transfer of nonelectrolytes across deformable membranes of a large curvature, Bull. Math. Biol., (4)36 (1974), pp. 365–377.
L. Rubinstein,On the equations of convective-diffusion transfer of low-molecular nonelectrolytes across deformable semipermeable membranes of a large curvature. In Free boundary problems, Proc. Sem. held in Pavia Sept.–Octob., 1979, vol. 2, pp. 507–538, Roma, 1980.
M. Geverey,Sur les équations aux dérivées partielles du type parabolique (suite), J. de Math. pures et appl.,10 (1914), p. 106.
E.Goursat,Cours d'analyse mathématique, 5-th ed. III, G-V, Paris.
M. Gevrey,Sur les équations aux dérivées partielles du type parabolique, J. de Math. pure et appl.,9 (1913), pp. 305–475.
A. Tichonov,Sur l'équation de la chaleur à plusieurs variables, Le bulletin de l'univ. d'état de Moscou, Série Intern. sA.v. 1, F. 9, 1938.
L.Rubinstein,The Stefan problem, Transi. Math. Monogr.27, Am. Math. Soc., 1971.
L. Rubinstein,On the forced convection in a plane layer with an axial symmetry, Dokl. Ak. Nauk, SSSR,135 (3) (1960), pp. 553–555.
G. N.Watson,A treatise on the theory of Bessel functions, Cambridge, 1944.
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Rubinstein, L. Free boundary problem for a nonlinear system of parabolic equations, including one with reversed time. Annali di Matematica pura ed applicata 135, 27–71 (1983). https://doi.org/10.1007/BF01781061
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DOI: https://doi.org/10.1007/BF01781061