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 *R ^{k}*,

**Download Technical Report:** Pdf (91
KB)