Blow-up for semilinear parabolic equations with nonlinear memory (Q1880223)

The authors consider the nonlocal semilinear parabolic equation \[ u_t-\Delta u =u^q\int_0^t u^p(x,t)\,ds, \quad x\in \Omega, t>0, \] under homogeneous Dirichlet boundary condition. For several ranges of nonnegative parameters \(p\), \(q\) the authors determine the existence and nonexistence of global solutions, they give criteria for blow-up and estimate the blow-up rate. One of the questions the authors leave open for \(q=0\), namely, the existence of a solution that is eventually monotone in time, was answered in affirmative by \textit{P. Souplet} [Z. Angew. Math. Phys. 55, No. 1, 28--31 (2004; Zbl 1099.35024)].