Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1333-1365.doi: 10.1007/s10473-021-0419-3

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REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM

Jianfeng ZHOU1, Zhong TAN2   

  1. 1. School of Mathematical Sciences, Peking University, Beijing 100871, China;
    2. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
  • Received:2020-04-15 Online:2021-08-25 Published:2021-09-01
  • Supported by:
    The first author is partially supported by the Postdoctoral Science Foundation of China (2019TQ0006); the second author is partially supported by the National Natural Science Foundation of China (11726023, 11531010).

Abstract: We study the regularity of weak solutions to a class of second order parabolic system under only the assumption of continuous coefficients. We prove that the weak solution $u$ to such system is locally Hölder continuous with any exponent $\alpha\in(0,1)$ outside a singular set with zero parabolic measure. In particular, we prove that the regularity point in $Q_T$ is an open set with full measure, and we obtain a general criterion for the weak solution to be regular in the neighborhood of a given point. Finally, we deduce the fractional time and fractional space differentiability of $D u$, and at this stage, we obtain the Hausdorff dimension of a singular set of $u$.

Key words: Parabolic system, regularity, weak solution, Hausdorff dimension

CLC Number: 

  • 35D30
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