Nonlinear problems of the thermal conductivity equation

Abstract

The asymptotic behavior of solutions of parabolic equations at infinite times has been investigated for various cases [1–6]. Two initial boundary-value problems are considered in this paper. The solution of the thermal conductivity equation with a nonlinear right-hand side is found, including also nonlinear boundary conditions. It is shown that the solution of the corresponding problem tends either to a stable, steady-state solution, or to a periodic solution, depending on the initial values of the functions and constants appearing in the conditions of the problem. Other papers [7, 8] are devoted to finding the periodic solutions of these two problems encountered in hydrodynamics (diffusion, underground hydrodynamics), and to studying the asymptotic behavior of the corresponding initial boundary problems.