Uniform boundedness for weak solutions of quasilinear parabolic equations
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- by Karthik Adimurthi and Sukjung Hwang PDF
- Proc. Amer. Math. Soc. 148 (2020), 653-665 Request permission
Abstract:
In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form \[ u_t - \operatorname {div} \mathcal {A}(x,t,\nabla u) = 0, \] 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.References
- Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
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
- MR Author ID: 851320
- Email: karthikaditi@gmail.com, kadimurthi@snu.ac.kr
- Sukjung Hwang
- Affiliation: Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea
- MR Author ID: 1144003
- Email: sukjung_hwang@yonsei.ac.kr, sukjungh@gmail.com
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 653-665
- MSC (2010): Primary 35B45, 35K59
- DOI: https://doi.org/10.1090/proc/14667
- MathSciNet review: 4052202