On the Fisher‐KPP model with nonlocal nonlinear sources

Abstract

AbstractThe Cauchy problem considered in this paper is the following: where . When the coefficient remains positive, (1) is analogous to It is well known that when , the local solution of (2) blows up in finite time as long as the initial value is nontrivial. The main aim of this paper is to obtain sufficient conditions for global existence of solutions to (1) for and thus it forms a contrast to (2). The exponent produces an exact balance in the mass‐invariant scaling of diffusion and growth in (1). In particular, we prove that if , and where is the sharp constant of the inequality , then (1) admits a global classical solution. Moreover, the long time behavior of the solution is also discussed. In addition, we prove that when , the problem is solvable globally in time without any restriction on the initial data.