Watersheds for solutions of nonlinear parabolic equations

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, 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.