Entire solutions to advective Fisher-KPP equation on the half line

Consider the advective Fisher-KPP equation u(t) = u(xx) - beta u(x)+ f(u) on the half line [0, infinity) with Dirichlet boundary condition at x = 0. In a recent paper [10], the authors considered the problem without advection (i.e., beta = 0) and constructed a new type of entire solution U(x, t), which, under the additional assumption f '' (u) <= 0, is concave and U(infinity, t) = 1 for all t is an element of R. In this paper, we consider the equation with advection and without the additional assumption f '' (u) <= 0. In case beta = 0, using a quite different approach from [10] we construct an entire solution (U) over tilde which is similar as U in the sense that (U) over tilde(infinity, t) equivalent to 1 and (U) over tilde(., t) is asymptotically flatas t -> -infinity, but different from U in the sense that it does not have to be concave. Our result reveals that the asymptotically flat (as t -> -infinity) property rather than the concavity is more essential for such entire solutions. In case beta < 0, we construct another new entire solution <(U)over cap> which is completely different from the previous ones in the sense that (U) over tilde(infinity, t) increases from 0 to 1 as t increasing from -infinity to infinity. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this paper, the advective Fisher-KPP equation on the half line with Dirichlet boundary conditions is considered. By constructing new types of entire solutions under different assumptions and analyzing the properties based on the value of beta, it is found that the essential property for such entire solutions is the asymptotically flat property as t approaches negative infinity, rather than concavity.