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Uniform blow up profiles for diffusion equations with nonlocal source and nonlocal boundary. (English) Zbl 1065.35150

Summary: Long time behavior of solutions to semilinear parabolic equations with nonlocal nonlinear source \(u_t- \Delta u= \int_\Omega g(u)\,dx\) in \(\Omega\times(0, T)\) and with nonlocal boundary condition \(u(x, t)= \int_\Omega f(x, y)u(y, t)\,dy\) on \(\partial\Omega\times(0, T)\) is studied. The authors establish local existence, global existence and nonexistence of solutions and discuss the blow up properties of solutions. Moveover, they derive the uniform blow up estimates for \(g(s)= s^p\) \((p> 1)\) and \(g(s)= e^s\) under the assumption \(\int_\Omega f(x, y)\,dy< 1\) for \(x\in\partial\Omega\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
45K05 Integro-partial differential equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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