ABSTRACT

In this paper, we investigate a reaction–diffusion equation $u_t-d{u}_{xx}=u(a+b{u}^{p-1}{\int }_{g\left(t\right)}^{h\left(t\right)}{u}^q\text{ d}x)$ with double free boundaries. We study blowup phenomena and asymptotic behavior of time-global solutions. For $u_0\left(x\right)=\sigma \varphi \left(x\right)$, when $a\ge {\lambda }_1,\sigma >0$, if $h_0\ge \frac{\pi }{2}\sqrt{\frac{d}{a}}$, then u will blow up in finite time. Meanwhile, we also prove when $T^{\ast }<+\mathrm{\infty }$, the solution must blow up in finite time. On the other hand, we discuss the vanishing of solutions. We prove that vanishing will occur as long as the initial datum are small sufficiently.

中文翻译：

---

具有非局部反应项的自由边界问题的爆炸和消失

摘要

在本文中，我们研究了一个反应-扩散方程${你}_{吨}-d{你}_{XX}=你(\mathrm{一个}+b{你}^{p-1}{\int }_{G\left(吨\right)}^{H\left(吨\right)}{你}^q\text{ d}X)$具有双重自由边界。我们研究时间全局解的爆破现象和渐近行为。为了${你}_0\left(X\right)=\sigma \phi \left(X\right)$， 什么时候$\mathrm{一个}\ge {\lambda }_1,\sigma >0$， 如果$H_0\ge \frac{\pi }{2}\sqrt{\frac{d}{\mathrm{一个}}}$，那么你会在有限的时间内爆炸。同时，我们也证明当${吨}^{*}<+\mathrm{\infty }$，解决方案必须在有限时间内爆炸。另一方面，我们讨论解决方案的消失。我们证明，只要初始数据足够小，就会发生消失。