C^1,α regularity for quasilinear parabolic equations with nonstandard growth

2022/08/25 by Karthik Adimurthi, Adimurthi, Karthik, Suchandan Ghosh +3
Engineering · Mathematics · #35B65 #35K59 #35K92 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2208.12322

openalex publication_date 2022/08/25 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28

Abstract

In this paper, we obtain C^1,α estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being u_t - div (|∇ u|^p-2 ∇ u + a(t)|∇ u|^q-2 ∇ u) = 0, u_t - div (|∇ u|^p(t)-2 ∇ u) = 0. under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switching factor a(t) and the variable exponent p(t). Moreover, we are able to take any p ≤ q < ∞ to obtain the desired regularity.

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