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1997 Global existence and finite-time blow-up for a class of nonlocal parabolic problems
J. W. Bebernes, A. A. Lacey
Adv. Differential Equations 2(6): 927-953 (1997).


An analysis of the nonlocal parabolic equation $$ u_t-\Delta u=\frac{\delta f(u)}{(\int_\Omega f(u))^p},\quad x\in\Omega, t>0, \tag"(P)" $$ and its associated steady state equation $$ -\Delta u=\frac{\delta f(u)}{(\int_\Omega f(u))^p}, \quad x\in \Omega, \tag"(S)" $$ with Dirichlet boundary conditions is given, assuming $\Omega\subset\Bbb R^n$ is a smooth bounded domain, $\delta>0$, $p\ge0$, and $f$ is positive and locally Lipschitz continuous. Existence-nonexistence results are proven for (S) when $f(u)=e^u$ or $e^{-u}$ and $\Omega$ is a ball or star-shaped domain. For $f(u)\ge c>0$, $n=1$, and $p\ge1$, we prove that (P) has a global bounded solution for any nonnegative initial data $u_0(x)$ and any $\delta>0$. For $f(u)=e^u$, $n=2$, $\Omega=B_1(0)$, $u_0(x)$ radially symmetric, nonnegative, if $p>1$, (P) has a unique, globally bounded solution for any $\delta>0$. If $p=1$, $0<\delta<8\pi$, (P) again has a bounded global solution. For $f(u)=e^u$, $n=1$ or 2, if $p<1$ and $\delta>\delta^*$ where $\delta^*$ is critical value for existence-nonexistence for (S), then the solution $u$ of (P) blows up in finite time.


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J. W. Bebernes. A. A. Lacey. "Global existence and finite-time blow-up for a class of nonlocal parabolic problems." Adv. Differential Equations 2 (6) 927 - 953, 1997.


Published: 1997
First available in Project Euclid: 22 April 2013

zbMATH: 1023.35512
MathSciNet: MR1606347

Primary: 35K55
Secondary: 35B05, 35J60

Rights: Copyright © 1997 Khayyam Publishing, Inc.


Vol.2 • No. 6 • 1997
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