Lin, Zhigui; Liu, Yurong Uniform blow up profiles for diffusion equations with nonlocal source and nonlocal boundary. (English) Zbl 1065.35150 Acta Math. Sci., Ser. B, Engl. Ed. 24, No. 3, 443-450 (2004). 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 \(x\in\partial\Omega\). Cited in 19 Documents 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 Keywords:local existence; global existence and nonexistence; blow up estimates PDFBibTeX XMLCite \textit{Z. Lin} and \textit{Y. Liu}, Acta Math. Sci., Ser. B, Engl. Ed. 24, No. 3, 443--450 (2004; Zbl 1065.35150) Full Text: DOI