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Uniform boundedness for weak solutions of quasilinear parabolic equations


Authors: Karthik Adimurthi and Sukjung Hwang
Journal: Proc. Amer. Math. Soc. 148 (2020), 653-665
MSC (2010): Primary 35B45, 35K59
DOI: https://doi.org/10.1090/proc/14667
Published electronically: October 28, 2019
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Abstract: In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form

$\displaystyle u_t - \operatorname {div} \mathcal {A}(x,t,\nabla u) = 0, $

where the nonlinearity $ \mathcal {A}(x,t,\nabla u)$ is modelled after the well-studied $ p$-Laplace operator. The question of boundedness has received a lot of attention over the past several decades with the existing literature showing that weak solutions in either $ \frac {2N}{N+2}<p<2$, $ p=2$, or $ 2<p$, are bounded. The proof is essentially split into three cases mainly because the estimates that have been obtained in the past always included an exponent of the form $ \frac {1}{p-2}$ or $ \frac {1}{2-p}$, which blows up as $ p \rightarrow 2$. In this note, we prove the boundedness of weak solutions in the full range $ \frac {2N}{N+2} < p < \infty $ without having to consider the singular and degenerate cases separately. Subsequently, in a slightly smaller regime of $ \frac {2N}{N+1} < p < \infty $, we also prove an improved boundedness estimate.

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Additional Information

Karthik Adimurthi
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea; and Tata Institute of Fundamental Research - Centre for Applicable Mathematics, Bangalore, Karnataka, India 560065
Email: karthikaditi@gmail.com, kadimurthi@snu.ac.kr

Sukjung Hwang
Affiliation: Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea
Email: sukjung_hwang@yonsei.ac.kr, sukjungh@gmail.com

DOI: https://doi.org/10.1090/proc/14667
Keywords: Boundedness, quasilinear parabolic equations, $p$-Laplace operators
Received by editor(s): January 7, 2019
Received by editor(s) in revised form: April 15, 2019
Published electronically: October 28, 2019
Additional Notes: The first author is the corresponding author
The first author was supported by the National Research Foundation of Korea grant NRF-2015R1A4A1041675.
The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2017R1D1A1B03035152).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2019 American Mathematical Society