Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory

Abstract

In this paper, we investigate a reaction-diffusion equation ut−duxx = au+ ∫ t 0 u p(x,τ)dτ+k(x) with double free boundaries. We study blowup phenomena in finite time and asymptotic behavior of time-global solutions. Our results show if ∫ h0 −h0 k(x)ψ1dx is large enough, then the blowup occurs. Meanwhile we also prove when T∗<+∞, the solution must blow up in finite time. On the other hand, we prove that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial datum is small sufficiently. AMS Subject Classifications: 35K20, 35R35, 92B05 Chinese Library Classifications: O175