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Zbl 1139.35057
Wang, Yulan; Mu, Chunlai; Xiang, Zhaoyin
Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary. (English)
[J] Bound. Value Probl. 2007, Article ID 64579, 12 p. (2007). ISSN 1687-2770/e

Summary: This paper considers the properties of positive solutions for a nonlocal equation $$u_t(x,t)=\Delta u+\int_{\Omega}u^q(y,t)\,dy -ku^p(y,t),\quad \text{in}\quad \Omega\times (0,T),$$ with nonlocal boundary condition $$u(x,t)=\int_\Omega f(x,y)u(y,t)dy,\quad \text{on}\quad \partial\Omega\times(0,T),$$ and initial condition $$u(x,0)=u_0(x),\quad \text{for}\quad x\in \Omega$$ where $p,q\geq 1,\ k>0,$ and $\Omega\subset\Bbb R^n$ is a bounded domain with smooth boundary $\partial \Omega$. Conditions for the existence and nonexistence of global positive solutions are given. Moreover, we establish uniform blow-up estimates for the blow-up solution.

MSC 2000:: *35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions of PDE
35K20 Second order parabolic equations, boundary value problems

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