The University of Montana
Department of Mathematical Sciences

Technical report #13/2013

Watersheds for Solutions of Parabolic Systems

J. A. Cima
Department of Mathematics
University of North Carolina, Chapel Hill, NC 27599
cima@email.unc.edu

W. R. Derrick
Department of Mathematical Sciences
University of Montana, Missoula, MT 59802
derrick@mso.umt.edu

L. V. Kalachev
Department of Mathematical Sciences
University of Montana, Missoula, MT 59802
kalachev@mso.umt.edu

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 ƒ(z) ⁄ z. In addition, we weaken the condition that ƒ(z) ⁄ z be increasing, and show that the method also works in R1 when f(z) ⁄ z is bounded. The decay rate is exponential.

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