Abstract
The purpose of this paper is to investigate the existence, uniqueness, and dynamics of a nonlinear reaction-diffusion equation with a nonlocal boundary condition which is motivated by a model problem arising from quasi-static thermoelasticity. The method of upper and lower solutions is used to obtain some existence-comparison results for both the time-dependent problem and its corresponding steady-state problem. A sufficient condition for the uniqueness of a steady-state solution is given. The comparison and uniqueness results are used to show the dynamical behavior of time-dependent solutions as well as their monotone convergence to a steady-state solution. Also given is a necessary and sufficient condition for the convergence of time-dependent solutions in relation to a steady-state solution which is not explicitly known. These results lead to the global stability of a steady-state solution for some special cases, including the model problem from thermoelasticity.
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