Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 19 (2010), pp. 207-220.

Bounded solutions of nonlinear parabolic equations

Nsoki Mavinga, Mubenga N. Nkashama

We are concerned with bounded solutions existing for all times for nonlinear parabolic equations with possibly nonlinear boundary conditions. A counterexample shows that, without an additional condition, a (weak) maximum principle does not hold for linear problems defined on the entire real line in time. We consider solutions bounded for all times and derive a (weak) maximum principle which is valid on the entire real line. Using comparison techniques, a priori estimates and approximation methods, we prove the existence and, in some cases, positivity and uniqueness of bounded solutions existing for all times.

Published September 25, 2010.
Math Subject Classifications: 35K55, 35K60.
Key Words: Nonlinear parabolic equations; nonlinear boundary conditions; sub and super-solutions; interpolation inequalities; a priori estimates; bounded solutions.

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Nsoki Mavinga
Department of Mathematics, University of Rochester
Rochester, NY 14627-0138, USA
email: mavinga@math.rochester.edu
Mubenga N. Nkashama
Department of Mathematics, University of Alabama at Birmingham
Birmingham, AL 35294-1170, USA
email: nkashama@math.uab.edu

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