Existence and behavior of positive solutions for a class of parabolic reaction-diffusion equations

The two-point boundary value problem y′ = λ2 y -p, y(0) = a > 0, y(l) =b>0 (∗) for p > 0 is known to have 0, 1, or 2 positive solutions, depending on p and the magnitude of A. Therefore it is reasonable to expect the behavior as time increases of the solution of the related parabolic problem uxx -ut = λ2, u(0,t) = a, u(l,t) = b, u(x,0) = c(x) (∗∗) to depend on λ in an interesting way. Basically, we show (as a special case of more general results) that for suitable initial data c positive solutions of (∗∗) exist for all t > 0 only for sufficiently small λ and, when they exist for all t > 0, tend to some solution of(∗) as t → ∞. But only the maximal solution of (∗) has the property that solutions of (∗∗) tend to it for initial data c(x) in a two-sided neighborhood of y(x)