We prove a theorem about existence, uniqueness and regularity of the solution to an initial-boundary value problem for a nonlinear coupled parabolic system consisting of two equations. Such a system appears in the thermodiffusion in solid body. In our proof we use an energy method, methods of Sobolev spaces, semigroup theory and the Banach fixed point theorem.
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Jerzy A. Gawinecki, P. Kacprzyk, B. Bar-Joseph, Initial-Boundary Value Problem for Some Coupled Nonlinear Parabolic System of Partial Differential Equations Appearing in Thermodiffusion in Solid Body. Z. Anal. Anwend. 19 (2000), no. 1 pp. 121–130