On finite-time blow-up for a nonlocal parabolic problem arising from shear bands in metals

Abstract

Results on finite-time blow-up of solutions to the nonlocal parabolic problem\[{ut=Δu+δeu(∫Ωeu)p amp;in Ω×(0,T), 0>p>1, δ>0,u(x,t)=0,amp;(x,t)∈∂Ω×(0,T),u(x,0)=u0(x)⩾0,amp;x∈Ω\begin {cases} u_t=\Delta u +\delta \dfrac {e^u}{\left (\int _{\Omega }e^{u}\right )^p} \ \ & \text {in $\Omega \times (0,T)$, $0>p>1$, $\delta >0$},\\ u(x,t)=0, & (x,t)\in \partial \Omega \times (0,T),\\ u(x,0)=u_0(x)\geqslant 0,& x\in \Omega \end {cases}\]are established. They extend some known results to higher dimensions.