On the Hölder Regularity of Signed Solutions to a Doubly Nonlinear Equation. Part III

Abstract

We establish the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is $$ \begin{align*} & \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad 1<p<2,\quad 0<p-1<q. \end{align*}$$The proof exploits the space expansion of positivity for the singular, parabolic $p$-Laplacian and employs the method of intrinsic scaling by carefully balancing the double singularity.

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

“…N.Liao in [30] has effected a direct proof of Hölder regularity for parabolic p-Laplace equation by using the expansion of positivity. It has also been used to give the first proofs of Hölder regularity for signchanging solutions of doubly nonlinear equations of porous medium type in [4,9,31].…”

Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

“…The statements hold trivially for r = 0, so assume r > 0. Set R 0 = R, ω 0 = ω, recall that a and η 0 are defined in (17) and (15), respectively, and define R n , ω n and Q n as in Proposition 2.5. Next, let r ∈ (0, R] and pick n ∈ N such that…”

“…More recently, intrinsic scaling has been applied successfully to doubly nonlinear parabolic equations whose prototype is ∂ t (|u| p−2 u) = div(|∇u| p−2 u), p > 1, see [3,2,17]. This equation is a combination of the porous medium equation and the p-Laplace equation.…”

Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

“…This difference is due to the assumption (R1) on the reaction term. Indeed, we define a by (17) to make sure (12) is satisfied in each step of the iterative scheme and condition (12) is introduced due to this growth assumption. We could have picked a proportional to η m−1 0 and checked whether (12) holds at each step, but (17) simplifies the arguments.…”

“…More recently, intrinsic scaling has been applied successfully to doubly nonlinear parabolic equations whose prototype is ∂ t (|u| p−2 u) = div(|∇u| p−2 u), p > 1, see [2,3,17]. This equation is a combination of the porous medium equation and the p-Laplace equation.…”