Initial-boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermodiffusion in solid body (Q1978087)

Summary: We prove a theorem about existence, uniqueness and regularity of the solution to an initial-boundary value problem for the nonlinear coupled parabolic system \[ c(\theta_1,\theta_2) \partial_t\theta_1- a^1_{\alpha\beta}(\theta_1, \theta_2,\nabla\theta_1, \nabla\theta_2) {\partial^2\theta_1\over\partial x_\alpha\partial x_\beta}+ d(\theta_1, \theta_2,\nabla\theta_1, \nabla\theta_2){\partial\theta_2\over \partial t}= Q_1, \] \[ n(\theta_1,\theta_2) \partial_t\theta_2- a^2_{\alpha,\beta}(\theta_1, \theta_2,\nabla\theta_1, \nabla\theta_2) {\partial^2\theta_2\over \partial x_\alpha\partial x_\beta}+ d(\theta_1,\theta_2, \nabla\theta_1, \nabla\theta_2) {\partial\theta_1\over\partial t}= Q_2. \] Such a system appears in the thermodiffusion in solid bodies. In our proof we use an energy method, methods of Sobolev spaces, semigroup theory and the Banach fixed point theorem.