In this paper we study nonnegative and classical solutions u = u(x, t) to porous medium problems of the typewhere Ω is a bounded and smooth domain of R N , with N ≥ 1, I = (0, t * ) is the maximal interval of existence of u, m > 1 and u 0 (x) is a nonngative and sufficiently regular function. The problem is equipped with different boundary conditions and depending on such boundary conditions as well as on the expression of the source g, global existence and blow-up criteria for solutions to (♦) are established. Additionally, in the three dimensional setting and when blow-up occurs, lower bounds for the blow-up time t * are also derived.2010 Mathematics Subject Classification. 35K55, 35K57, 35A01, 74H35.

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In this note, we establish some oscillation criteria for certain higher-order quasi-linear neutral differential equation. These criteria improve those results in the literature. Some examples are given to illustrate the importance of our results.