In this paper, we consider the nonlinear parabolic equation with an integro-differential term. By using classical inequalities and the Moser iteration technique, we establish the estimates for u and ∇u. Then we prove an inequality of Poincaré type. As a byproduct of our proof, we derive a Campanato type growth estimate for u which follows from L^∞ estimates of ∇u. Besides, the Hölder continuity of solution is presented by the isomorphism theorem.

integro-differential equation, Hölder continuity, Moser iteration.

Received: May 19, 2024; Accepted: June 21, 2024; Published: July 2, 2024

How to cite this article: Zongqing Yang and Junhui Xie, Hölder regularity of solutions for certain degenerate parabolic integro-differential equation, Advances in Differential Equations and Control Processes 31(3) (2024), 379-395. https://doi.org/10.17654/0974324324021

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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