Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process

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dc.creatorQilin, Liu
dc.creatorFei, Liang
dc.creatorYuxiang, Li
dc.date2008-10-14
dc.date.accessioned2026-07-07T10:09:59Z
dc.date.available2026-07-07T10:09:59Z
dc.descriptionIn this paper, we consider the asymptotic behavior of the nonlocal parabolic problem \[ u_{t}=Δu+\displaystyle\frac{λf(u)}{\big(\int_Ωf(u)dx\big)^{p}}, x\in Ω, t>0, \] with homogeneous Dirichlet boundary condition, where $λ>0, p>0$, $f$ is nonincreasing. It is found that: (a) For $0<p\leq1$, $u(x,t)$ is globally bounded and the unique stationary solution is globally asymptotically stable for any $λ>0$; (b) For $1<p<2$, $u(x,t)$ is globally bounded for any $λ>0$; (c) For $p=2$, if $0<λ<2|\partialΩ|^2$, then $u(x,t)$ is globally bounded, if $λ=2|\partialΩ|^2$, there is no stationary solution and $u(x,t)$ is a global solution and $u(x,t)\to\infty$ as $t\to\infty$ for all $x\inΩ$, if $λ>2|\partialΩ|^2$, there is no stationary solution and $u(x,t)$ blows up in finite time for all $x\inΩ$; (d) For $p>2$, there exists a $λ^*>0$ such that for $λ>λ^*$, or for $0<λ\leqλ^*$ and $u_0(x)$ sufficiently large, $u(x,t)$ blows up in finite time. Moreover, some formal asymptotic estimates for the behavior of $u(x,t)$ as it blows up are obtained for $p\geq2$.
dc.description20pages
dc.identifierhttps://arxiv.org/abs/0810.2521
dc.identifierhttp://arxiv.org/abs/0810.2521
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/171493
dc.subjectAnalysis of PDEs
dc.subject35B35, 35B40, 35K60
dc.titleAsymptotic behavior of a nonlocal parabolic problem in Ohmic heating process
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