Hölder estimates for a natural class of equations of the type of fast diffusion

Abstract

517.9HSlder estimates for weak solutions of doubly nonlinear parabolic equations of the type of fast diffusion with coefficients satisfying only natural growth conditions and the monotonicity requirement are obtained. Bibliography: 17 titles.

Existence of very weak solutions of doubly nonlinear parabolic equations with measure data

“…For instance, Hölder regularity and Harnack's inequality for bounded weak solutions were established in [12,13,29,39] and [22,38], respectively. Besides, [28,30,35] are concerned with the asymptotic behavior of solutions of doubly nonlinear parabolic equations for certain values of the quantity p + m, and the local boundedness of the gradient was shown in [31] under the additional assumption that u is strictly positive.…”

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…We mention here, among many others, the papers [32], [33], [34], [53] and the monographs [17], [54]. In particular, we refer to the very recent monograph [18] for a comprehensive treatment of the Harnack inequality for non-negative solutions to p-Laplacian and porous medium equations.…”

“…Hölder continuity, from e.g. [33], [34]; we will use this regularity to show that the map which associates to any couple of functions…”

“…Since u ε , v ε are Hölder continuous in Q T , bounded in C(Q T ) uniformly in ε > 0 and the structure conditions of [33] and [34] are satisfied for the equations of system (2.1), whenever ε ∈ (0, ε 0 ), [33 …”

“…Here Ω is an open bounded domain of R N with smooth boundary ∂Ω, satisfying the property of positive geometric density, see [39], Q T := Ω × (0, T ), T > 0, τ i ∈ (0, +∞), the functions K i , a, and b belong to L ∞ (Q T ). The exponents p and q belong to the interval (1, 2), m > p, n > q and s m = |s| m−1 s. Setting Au := div(|∇u m | p−2 ∇u m ) and l := (m − 1)(p − 1), the operator Au becomes m p−1 div(|u| l |∇u| p−2 ∇u), which is the operator considered by Ivanov in [32], [33] and [34]. According to the classification proposed in these papers, we say that the first equation in (1.1) is of 1. slow diffusion type if m > Of course, analogous definition in terms of n and q can be given for the second equation in (1.1).…”