TY - JOUR

T1 - Watersheds for solutions of nonlinear parabolic equations

AU - Joseph, Cima

AU - William, Derrick

AU - Leonid, Kalachev

N1 - Publisher Copyright:
© 2018 World Scientific and Engineering Academy and Society. All rights reserved.

PY - 2018

Y1 - 2018

N2 - 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 exponential.

AB - 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 exponential.

KW - Key–Words: Nonlinear parabolic equations

KW - Nodal solutions

KW - Positive solutions

UR - http://www.scopus.com/inward/record.url?scp=85052069638&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85052069638

SN - 1109-2769

VL - 17

SP - 170

EP - 177

JO - WSEAS Transactions on Mathematics

JF - WSEAS Transactions on Mathematics

ER -