Shang, Haifeng; Song, Mengmeng Local and global existence for evolutionary \(p\)-Laplacian equation with nonlocal source. (English) Zbl 1424.35236 Differ. Integral Equ. 32, No. 3-4, 139-168 (2019). The authors investigate the existence and nonexistence of solutions for the parabolic equation \[ u_t-\Delta_p u=u^\alpha|\nabla u|^{l}\ \Big(\int_{{\mathbb R}^N}K(y)u^{q}(y,t)\,dy\Big)^{(r-1)/q}\quad\text{ in }{\mathbb R}^N\times(0,T) \] subject to initial condition \(u(x,0)=u_0(x)\). The authors derive first some useful a priori estimates which are further used to establish the local and global existence of a solution. Further, nonexistence results are obtained. In particular, a Fujita type critical exponent is deduced. Reviewer: Marius Ghergu (Dublin) Cited in 1 Document MSC: 35K92 Quasilinear parabolic equations with \(p\)-Laplacian 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B33 Critical exponents in context of PDEs 35B44 Blow-up in context of PDEs 35K15 Initial value problems for second-order parabolic equations 35K65 Degenerate parabolic equations 35R09 Integro-partial differential equations Keywords:parabolic problem; \(p\)-Laplacian equation; existence and nonexistence; Fujita critical exponent PDFBibTeX XMLCite \textit{H. Shang} and \textit{M. Song}, Differ. Integral Equ. 32, No. 3--4, 139--168 (2019; Zbl 1424.35236)