## 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}, 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 R^{1} when f(z)/z is bounded. The decay rate is exponential.

Original language | English |
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Pages (from-to) | 170-177 |

Number of pages | 8 |

Journal | WSEAS Transactions on Mathematics |

Volume | 17 |

State | Published - 2018 |

## Keywords

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