Electron. J. Diff. Equ., Vol. 2015 (2015), No. 288, pp. 1-24.

Holder continuity of bounded weak solutions to generalized parabolic p-Laplacian equations II: singular 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 $1<g_0 \leq g_1 \le 2$. Through this generalization in the setting from Orlicz spaces, we provide a uniform proof with a single geometric setting that a bounded weak solution is locally Holder continuous with some degree of commonality between degenerate and singular types. By using geometric characters, our proof does not rely on any of alternatives which is based on the size of solutions.

Submitted July 17, 2015. Published November 19, 2015.
Math Subject Classifications: 35B45, 35K67.
Key Words: Quasilinear parabolic equation; singular 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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