On quasilinear parabolic equations involving weighted p-Laplacian operators

Abstract

Abstract. In this paper we consider the initial boundary value problem for a class of quasilinear parabolic equations involving weighted p-Laplacian operators in an arbitrary domain, in which the conditions imposed on the non-linearity provide the global existence, but not uniqueness of solutions. The long-time behavior of the solutions to that problem is considered via the concept of global attractor for multi-valued semiflows. The obtained results recover and extend some known results related to the p-Laplacian equations. Mathematics Subject Classification (2000). 35B41, 35K65, 35D05.

Pullback attractor for differential evolution inclusions with infinite delays

“…Let f n ∈ P F (u n ) such that u n (r) = S(r − τ )φ τ n (0) + W τ ( f n )(r). Since {u n } is bounded, one can see that {f n } is integrably bounded by using (F) (3). Thanks to the compactness of S( · ), {W τ ( f n )} is relatively compact which implies that {u n } is relatively compact as well.…”

“…Thanks to the theories of attractors for multivalued semiflows/processes given in [7,8,10,20,21], one can find a global attractor for semiflows/processes governed by solutions of DIs, which is a compact set attracting all solutions as the time goes to infinity in some contexts. Some deployments of these theories for differential inclusions and differential equations without uniqueness can be found in [2,3,[8][9][10]18,24,25].…”

“…If X is a finite dimensional space, (F)(4) can be removed since it follows from (F)(3). Specifically, in this case…”

Blow-up analysis for parabolic p-Laplacian equations with a gradient source term

“…The blow-up solutions of parabolic p-Laplician equations have been studied by many authors (see, for instance, [1][2][3][4][5][6][7][8][9][10]). In this work, we research the blow-up solutions of the following parabolic p-Laplacian equations with a gradient source term:…”

Caccioppoli-type estimates and Hardy-type inequalities derived from weighted p-harmonic problems

“…Derivation of the Hardy inequalities on the basis of supersolutions to p-harmonic differential problems can be found in papers by D'Ambrosio [22][23][24] and Barbatis et al [5,6]. Other interesting results linking the existence of solutions in elliptic and parabolic PDEs with Hardy type inequalities are presented in [2,4,36,61,62], see also references therein. We refer also to the contribution by the third author [56], where instead of the nonweighted p-Laplacian in (1.1) one deals with the A-Laplacian: A u = div A (|∇u|) |∇u| 2 ∇u , involving a function A from an Orlicz class.…”