WSEAS Transactions on Mathematics

Print ISSN: 1109-2769
E-ISSN: 2224-2880

Volume 17, 2018

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Volume 17, 2018

Watersheds for Solutions of Nonlinear Parabolic Equations

AUTHORS: Joseph Cima, William Derrick, Leonid Kalachev

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ABSTRACT: In this paper we describe a technique that we have used in a number of publications to find the “watershed” under which the initial condition of a positive solution of a nonlinear reaction-diffusion equation must lie, so that this solution does not develop into a traveling wave, but decays into a trivial solution. The watershed consists of the positive solution of the steady-state problem together with positive pieces of nodal solutions ( with identical boundary conditions). We prove in this paper that our method for finding watersheds works in Rk , k ≥ 1, for increasing functions f(z)/z. In addition, we weaken the condition that f(z)/z be increasing, and show that the method also works in R1 when f(z)/z is bounded. The decay rate is exponentia

KEYWORDS: Nonlinear parabolic equations, positive solutions, nodal solution


[1] S. Chen and W. Derrick, Global existence and blow-up of solutions for a semilinear parabolic system, Rocky Mountain J. Math. 29, 1999, pp. 449–457.

[2] W. Derrick, L. Kalachev and J. Cima, Characterizing the domains of attraction of stable stationary solutions of semilinear parabolic equations, Int. J. Pure and Appl. Math. 11, 2004, pp. 83– 102.

[3] W. Derrick, L. Kalachev and J. Cima, Collapsing Heat Waves, Math. Comput. Modeling 46, 2007, pp. 612–624.

[4] P. Fife, Dynamics of internal layers and diffusion interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics No. 53, SIAM, Philadelphia 1988.

[5] R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 1937, pp. 353–369

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 17, 2018, Art. #22, pp. 170-177

Copyright © 2018 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0

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