Souplet, Philippe Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory. (English) Zbl 1099.35049 Z. Angew. Math. Phys. 55, No. 1, 28-31 (2004). The author considers the nonlocal semilinear parabolic equation \[ u_t-\Delta u =\int_0^t u^p(x,t)\,ds, \quad x\in \Omega, t>0, \] under homogeneous Dirichlet boundary condition. For solutions that are monotone in time the blow-up rate is known to be the same as for the ODE \(u_t=u^p\). The author proves the existence of the monotone solutions. Reviewer: Peter Poláčik (Minneapolis) Cited in 21 Documents MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations Keywords:nonlocal parabolic equation; nonlinear memory; blow-up rate; monotonicity Citations:Zbl 1099.35024 PDFBibTeX XMLCite \textit{P. Souplet}, Z. Angew. Math. Phys. 55, No. 1, 28--31 (2004; Zbl 1099.35049) Full Text: DOI