Regularity results for a class of doubly nonlinear very singular parabolic equations

Functional inequalities and applications to doubly nonlinear diffusion equations

“…What is more, then evolution equation (1) requires different methods for 1 < p < 2 and 2 < p being called singular and degenerate, respectively. Studies on solutions to equations like (1) attract deep attention of various groups developing their theory from different points of view [2,4,14,39,40,41,49,54,60,67]. The issue of convergence of solutions to a self-similar profile to nonlinear diffusion equations has been studied e.g.…”

“…For an initial datum u 0 ∈ L 1 ( N ), existence and uniqueness to (1) is settled, see [54,2]. Moreover, in the range or parameters under consideration, we have that u(t) ∈ C 1,α ( N ) for some α ∈ (0, 1), see [67,39,40,34] and mass is conserved, i.e., N u(t, x) d x = N u 0 (x) d x for t > 0, For further information about basic properties of solutions to (1) we refer to the monographs [35], [64, Part III] and references therein.…”

Functional inequalities and applications to doubly nonlinear diffusion equations

“…What is more, then evolution equation (1) requires different methods for 1 < p < 2 and 2 < p being called singular and degenerate, respectively. Studies on solutions to equations like (1) attract deep attention of various groups developing their theory from different points of view [2,4,14,39,40,41,48,53,59,66]. The issue of convergence of solutions to a self-similar profile to nonlinear diffusion equations has been studied e.g.…”

“…For an initial datum u 0 ∈ L 1 ( N ), existence and uniqueness to (1) is settled, see [53,2]. Moreover, in the range or parameters under consideration, we have that u(t) ∈ C 1,α ( N ) for some α ∈ (0, 1), see [66,39,40,34] and mass is conserved, i.e., N u(t, x) d x = N u 0 (x) d x for t > 0, For further information about basic properties of solutions to (1) we refer to the monographs [35], [63, Part III] and references therein.…”

Parabolic Harnack Estimates for anisotropic slow diffusion

“…We refer to Chapter 4 of the book [4] and the references therein for an account of the applications. Even if the theory is quite complete, especially in the degenerate and singular supercritical case (for the exact definition of these technical terms, see for example [32]), some important questions still remain open and are the object of intense research.…”