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.