Qualitative properties of solutions to a generalized Fisher-KPP equation

TL;DR

This paper studies a generalized Fisher-KPP equation, constructing stationary solutions, analyzing their asymptotic behavior, and classifying solutions as either decaying or blowing up, with decay rates for solutions tending to zero.

Contribution

It introduces new stationary solutions and characterizes their behavior, providing a detailed classification of solution dynamics for the generalized Fisher-KPP equation.

Findings

Stationary solutions constructed for different parameter regimes.

Stationary solutions act as separatrices between decay and blow-up.

Decay rates for solutions approaching zero as time tends to infinity.

Abstract

The following Fisher-KPP type equation $$ u_t=Ku_{xx}-Bu^q+Au^p, \quad (x,t)\in\real\times(0,\infty), $$ with $p>q>0$ and $A$, $B$, $K$ positive coefficients, is considered. For both $p>q>1$ and $p>1$, $q=1$, we construct stationary solutions, establish their behavior as $|x|\to\infty$ and prove that they are separatrices between solutions decreasing to zero in infinite time and solutions presenting blow-up in finite time. We also establish decay rates for the solutions that decay to zero as $t\to\infty$.