Local Hoelder continuity for doubly nonlinear parabolic equations

Abstract

We give a proof of the Hölder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincaré inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods. 2000 Mathematics Subject Classification. Primary 35B65. Secondary 35K65, 35D10. Key words and phrases. Hölder continuity, Caccioppoli estimates, intrinsic scaling, Harnack's inequality. Research of JMU supported by CMUC/FCT and project UTAustin/MAT/0035/2008. 1 2 KUUSI, SILJANDER AND URBANO two complementary cases: Case I : 0 ≤ ess inf u << ess osc u and Case II : u p−2 u t ≈ Cu t .In large scales, i.e., in Case I, the scaling property of the equation dominates and, consequently, the reduction of the oscillation follows immediately from Harnack's inequality. In small scales, on the other hand, the oscillation is already very small and thus the solution itself is between two constants, the infimum and the supremum, whose difference is negligible. Correspondingly, the nonlinear time derivative term, which formally looks like u p−2 u t , behaves like u t and we end up with a p-parabolic type behavior. However, also in this case, we still need to modify the known arguments. In particular, the energy estimates are not available in the usual form and we need to use modified versions as in [5], [15] and [25].Our argument also applies to doubly nonlinear equations of p-Laplacian type that are of the formwith A(x, t, ·, ·) satisfying the usual structure assumptions. For expository purposes, we only consider (1.1). Very recently, a direct geometric method to obtain local Hölder continuity for parabolic equations has been developed in [6] and [9]. Despite the effort, the general picture remains unclear. Preliminaries

The Fisher-KPP problem with doubly nonlinear diffusion

“…For the Porous Medium case (p = 2) we refer to [51,52], while for the p-Laplacian case we suggest [19,41] and the references therein. Finally, in the doubly nonlinear setting, we refer to [31,45,53] and, for the "pseudo-linear" case, [36]. The results obtained in these works showed the Hölder continuity of the solution of problem (1.1).…”

Bistable reaction equations with doubly nonlinear diffusion

“…For what concerns the regularity, we refer to [53,54] for the Porous Medium setting, while for the p-Laplacian case we suggest [21,41] and the references therein. Finally, in the doubly nonlinear setting, we refer to [33,48,55] and, for the "pseudo-linear" case, [37]. Finally, we mention [21,54,56,57] for a proof of the Comparison Principle, which will be an essential technical tool in the proofs of the PDEs part.In order to fix the notations and avoid cumbersome expressions in the rest of the paper, we introduce the constant γ := m(p − 1) − 1, which will play an important role in our study.…”

Parabolic weighted norm inequalities and partial differential equations

“…In contrast with the earlier attempts, our point of view is related to Moser's work on the parabolic Harnack inequality in [24] and [25]. More precisely, in the regularity theory for the doubly nonlinear parabolic PDEs of the type (1.1) ∂(|u| p−2 u) ∂t − div(|∇u| p−2 ∇u) = 0, 1 < p < ∞, (see [11], [13], [15], [29]), there is a condition (Definition 3.2) that plays a role identical to that of the classical Muckenhoupt condition in the corresponding elliptic theory. Starting from the parabolic Muckenhoupt condition…”