The porous medium equation with variable exponent revisited
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Journal of Evolution Equations
The porous medium equation with variable exponent revisited Eurica Henriques
Abstract. The author presents a simplified proof for the local continuity of the weak solutions to the porous medium equation with variable positive bounded exponent γ (x, t) u t − ∇ · |u|γ (x,t) ∇u = 0, in T = × (0, T ], 0 < T < ∞. .
1. Introduction and main result In the past decades, several authors paid their attention and gave their contribution to the study of stationary and evolutionary equations presenting exponential nonlinearities. In this setting, the two most common parabolic equations, that appear in a natural way in continuum mechanics, concern mathematical models for the flow of electro-rheological fluids or fluids with temperature-dependent viscosity, and even for processes of filtration through inhomogeneous anisotropic media are u t − ∇ · |u|γ (x,t) ∇u = 0, a generalization of the porous medium equation and u t − ∇ · |∇u| p(x,t)−2 ∇u = 0, a generalization of the parabolic p-Laplacian equation. Just to state some of the works concerning regularity results, we refer to [4–6,10] (see the references therein). In this work, we are concerned with the first one; more specifically we are concerned with the proof of a more elegant and less demanding result on the local regularity of its weak solutions than the one presented in [8]. In [8] was obtained the local continuity Mathematics Subject Classification: 35B65, 35K55, 35K65 Keywords: Porous medium equation, Singular/degenerate PDE, Regularity theory, Intrinsic scaling. The research of the author was partially financed by Portuguese Funds through FCT (Fundao para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020.
E. Henriques
J. Evol. Equ.
of the weak solutions to the porous medium equation with variable exponent modeled by the equation (1.1) u t − ∇ · |u|γ (x,t) ∇u = 0, in T = × (0, T ], 0 < T < ∞ where is a bounded domain in R N , γ (x, t) is considered to be a bounded function 0 < γ − ≤ γ (x, t) ≤ γ + < ∞, (x, t) ∈ T and to satisfy the following regularity condition γ ∈ L ∞ 0, T ; W 1, p () , for some p > max {2, N } .
(1.2)
(1.3)
This regularity assumption on γ played an important role along the proof of the regularity result, especially when deducing the energy estimates. To a more detailed introduction to the study of the evolutionary pde (1.1), we invite the reader to take a look at [8] and at the references therein. The purpose of (and novelty presented in) this work is to show how one can avoid to assume (1.3) and even so recover the local continuity to the weak solutions of (1.1). Namely, the local regularity of u will be obtained asking only the variable nonlinearity function γ (x, t) to be a bounded positive function: Theorem 1. Assume the variable nonlinearity γ to be a positive bounded function satisfying (1.2). Then any locally bounded weak solution to (1.1) is locally continuous in T . The proof follows the iterative method devised by De Giorgi for elliptic equa
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