Blow-up and asymptotic behavior of solutions for reaction-diffusion equations with free boundaries

We study the blow-up phenomena and the asymptotic behavior of time-global solutions for reaction-diffusion equations ut=uxx+au+up (a∈R and p>1), with free boundaries and initial data σϕ(x). We give a sharp threshold value σ⁎=σ⁎(ϕ,a,p)≥0 such that the solution blows up in finite time when σ>σ⁎, vanishes (i.e. u→0 as t→∞) when σ<σ⁎, and when σ=σ⁎, it converges as t→∞ to 0 or to an evenly decreasing positive stationary solution, depending on whether a≥0 or a<0. Moreover, when blow-up happens, we show that the blow-up set is compact in the occupying domain of initial data and the free boundaries keep bounded.