Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Dynamics of reaction-diffusion equations with nonlocal boundary conditions

Author: C. V. Pao
Journal: Quart. Appl. Math. 53 (1995), 173-186
MSC: Primary 35K57; Secondary 35B40, 35Q72
DOI: https://doi.org/10.1090/qam/1315454
MathSciNet review: MR1315454
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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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DOI: https://doi.org/10.1090/qam/1315454
Article copyright: © Copyright 1995 American Mathematical Society

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