Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory. (English) Zbl 1099.35049
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)
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 |