Watersheds for solutions of nonlinear parabolic equations

Cima Joseph, Derrick William, Kalachev Leonid

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)170-177
Number of pages8
JournalWSEAS Transactions on Mathematics
StatePublished - 2018


  • Key–Words: Nonlinear parabolic equations
  • Nodal solutions
  • Positive solutions


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