Properties of solutions to porous medium problems with different sources and boundary conditions

By Tongxing Li, N. Pintus, Giuseppe Viglialoro. Zeitschrift für Angewandte Mathematik und Physik, 2018

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In this paper, we study nonnegative and classical solutions $$u=u(\mathbf{x},t)$$u=u(x,t) to porous medium problems of the type where $$\Omega $$Ω is a bounded and smooth domain of $${\mathbb {R}}^N$$RN, with $$N\ge 1$$N≥1, $$I=(0,t^*)$$I=(0,t∗) is the maximal interval of existence of u, $$m>1$$m>1 and $$u_0(\mathbf{x})$$u0(x) is a nonnegative 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 ($$\Diamond $$◊) are established. Additionally, in the three-dimensional setting and when blow-up occurs, lower bounds for the blow-up time $$t^*$$t∗ are also derived.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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