Abstract
In this paper we study the self-improving property of the obstacle problem related to the singular porous medium equation by using the method developed by Gianazza and Schwarzacher (J. Funct. Anal. 277(12):1–57, 2019). We establish a local higher integrability result for the spatial gradient of the mth power of nonnegative weak solutions, under some suitable regularity assumptions on the obstacle function. In comparison to the work by Cho and Scheven (J. Math. Anal. Appl. 491(2):1–44, 2020), our approach provides some new aspects in the estimations of the nonnegative weak solution of the obstacle problem.
Highlights
Kinnunen and Lewis [12, 13] proved the higher integrability of weak solutions to parabolic systems of p-Laplacian type by using an intrinsic scaling method
In this paper we study the self-improving property of the obstacle problem related to the singular porous medium equation by using the method developed by Gianazza and Schwarzacher
There are two different approaches to the study of higher integrability of weak solutions to porous medium equations, one is the approach developed by Gianazza and Schwarzacher [9, 10], the other is the approach developed by Bögelein et al [1, 2]
Summary
Kinnunen and Lewis [12, 13] proved the higher integrability of weak solutions to parabolic systems of p-Laplacian type by using an intrinsic scaling method. Cho and Scheven [5, 6] proved the higher integrability of weak solutions to obstacle problems related to the porous medium equation and their proofs followed the approach in [1, 2]. The approach in [2] is effective in treating the higher integrability of obstacle problem for the singular porous medium equation, but it is natural to try to use the old idea in [10] to study the same problem To this end, the present work is intended as an attempt to follow the approach in [10] to establish a self-improving result for the obstacle problem. Throughout this paper, we compare our arguments with [2, 6, 10]
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