Electron. J. Diff. Equ., Vol. 2015 (2015), No. 287, pp. 1-32.

Holder continuity of bounded weak solutions to generalized parabolic p-Laplacian equations I: degenerate case

Sukjung Hwang, Gary M. Lieberman

Abstract:
Here we generalize quasilinear parabolic p-Laplacian type equations to obtain the prototype equation
$$ u_t - \hbox{div} \Big(\frac{g(|Du|)}{|Du|} Du\Big) = 0, $$
where g is a nonnegative, increasing, and continuous function trapped in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with $2\le g_0 \leq g_1 < \infty$. Through this generalization in the setting from Orlicz spaces, we provide a proof for the Holder continuity of such solutions which has much in common with that proof of Holder continuity of solutions of singular equations.

Submitted July 17, 2015. Published November 19, 2015.
Math Subject Classifications: 35B45, 35K65.
Key Words: Quasilinear parabolic equation; degenerate equation; generalized structure; a priori estimate; Holder continuity.

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Sukjung Hwang
Center for Mathematical Analysis and Computation
Yonsei University, Seoul 03722, Korea
email: sukjung_hwang@yonsei.ac.kr
Gary M. Lieberman
Department of Mathematics
Iowa State University
Ames, IA 50011, USA
email: lieb@iastate.edu

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