Local and global existence for evolutionary $$p$$-Laplacian equation with nonlocal source.(English)Zbl 1424.35236

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.

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