Porous medium equation with a blow-up nonlinearity and a non-decreasing constraint

Abstract

The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (possibly nonunique) approximate solutions is established and it also enables us to obtain a local solution as a limit of approximate ones.1 a.e. in Ω. Let w, u, v ∈ H 1 (Ω) be minimizers of J w 0 , J u 0 , and J v 0 , respectively. Then, w ∨ v, w ∧ v ∈ H 1 (Ω). Moreover, by minimality we particularly haveBy using the fact ( comp princ cond comp princ cond 4.3) with a = w, a 0 = w 0 , b = v, and b 0 = v 0 , we getThus,w := w ∧ v andṽ := w ∨ v also minimize J w 0 and J v 0 , respectively, and w ≤ṽ. By using ( comp princ cond comp princ cond 4.3) again with the choice a = u, a 0 = u 0 , b =ṽ, b 0 = v 0 and by arguing as above, we deduce that v 1 := u ∨ṽ and u 1 := u ∧ṽ minimize J v 0 and J u 0 , respectively. Furthermore, ( comp princ cond comp princ cond