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 ∞.