Blowup and vanishing of a free boundary problem with a nonlocal reaction term

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.

Keywords:

• Nonlocal reaction term
• free boundary
• blowup
• vanishing

Mathematics Subject Classifications:

• Primary: 35K20
• 35R35
• Secondary: 92B05

Acknowledgments

The research was partially supported by PRC grant NSFC 11771174 and 11301207.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research supported by National Science Foundation of China [grant numbers 11771174 and 11301207).