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

Consider the advective Fisher-KPP equation u t = u x x − β u x + f (u) on the half line [ 0 , ∞) with Dirichlet boundary condition at x = 0. In a recent paper [10] , the authors considered the problem without advection (i.e., β = 0) and constructed a new type of entire solution U (x , t) , which, under the additional assumption f ″ (u) ≤ 0 , is concave and U (∞ , t) = 1 for all t ∈ R. In this paper, we consider the equation with advection and without the additional assumption f ″ (u) ≤ 0. In case β = 0 , using a quite different approach from [10] we construct an entire solution U ˜ which is similar as U in the sense that U ˜ (∞ , t) ≡ 1 and U ˜ (⋅ , t) is asymptotically flat as t → − ∞ , but different from U in the sense that it does not have to be concave. Our result reveals that the asymptotically flat (as t → − ∞) property rather than the concavity is more essential for such entire solutions. In case β < 0 , we construct another new entire solution U ˆ which is completely different from the previous ones in the sense that U ˆ (∞ , t) increases from 0 to 1 as t increasing from −∞ to ∞. [ABSTRACT FROM AUTHOR]