TY - JOUR
T1 - Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory
AU - Huang , Jiahui
AU - Yuan , Junli
AU - Zhao , Yan
JO - Journal of Partial Differential Equations
VL - 3
SP - 249
EP - 260
PY - 2020
DA - 2020/06
SN - 33
DO - http://doi.org/10.4208/jpde.v33.n3.5
UR - https://global-sci.org/intro/article_detail/jpde/17073.html
KW - Nonlinear memory, free boundary, blowup, asymptotic behavior.
AB - <p style="text-align: justify;">In this paper, we investigate a reaction-diffusion equation $u_t-du_{xx}=au+\int_{0}^{t}u^p(x,\tau){\rm d}\tau+k(x)$ with double free boundaries. We study blowup phenomena in finite time and asymptotic behavior of time-global solutions. Our results show if $\int_{-h_0}^{h_0}k(x)\psi_1 {\rm d}x$ is large enough, then the blowup occurs. Meanwhile we also prove when $T^*&lt;+\infty$, 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.</p>