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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

${u}_{t}\left(x,t\right)={\Delta }u+{\int }_{{\Omega }}{u}^{q}\left(y,t\right)\phantom{\rule{0.166667em}{0ex}}dy-k{u}^{p}\left(y,t\right),\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{1.em}{0ex}}{\Omega }×\left(0,T\right),$

with nonlocal boundary condition

$u\left(x,t\right)={\int }_{{\Omega }}f\left(x,y\right)u\left(y,t\right)dy,\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{1.em}{0ex}}\partial {\Omega }×\left(0,T\right),$

and initial condition

$u\left(x,0\right)={u}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{1.em}{0ex}}x\in {\Omega }$

where $p,q\ge 1,\phantom{\rule{4pt}{0ex}}k>0,$ and ${\Omega }\subset {ℝ}^{n}$ 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 of PDE 35K20 Second order parabolic equations, initial boundary value problems
##### References:
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