Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Absorption and Nonlinear Nonlocal Boundary Conditions

In this paper we consider classical solutions of an initial boundary value problem for a~system of semilinear parabolic equations with absorption and nonlinear nonlocal boundary conditions. Nonlinearities in equations and boundary conditions may not satisfy the Lipschitz condition. To prove the existence of a solution we regularize the original problem. Using the Schauder--Tikhonov fixed point theorem, the existence of a local solution of regularized problem is proved. It is shown that the limit of solutions of the regularized problem is a maximal solution of the original problem. Using the properties of a maximal solution, a comparison principle is proved. In this case, no additional assumptions are made when nonlinearities in absorption do not satisfy the Lipschitz condition. Conditions are found under which solutions are positive functions. The uniqueness of the solution is established. It is shown that the trivial solution $(0, 0 )$ may not be unique.