Initial-boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermodiffusion in solid body
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.
homogeneous Dirichlet boundary conditions
scientific article; zbMATH DE number 1453208