×

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.

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

Citations:

Zbl 1099.35024
PDFBibTeX XMLCite
Full Text: DOI