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Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary. (English) Zbl 1139.35057

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 \(\Omega\subset\mathbb R^n\) 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:

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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References:

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