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Posted December 17, 2025
Last modified February 8, 2026

Applied Analysis Seminar Questions or comments?

Tuoc Phan, University of Tennessee–Knoxville
On Lin type Hessian estimates for solutions to a class of singular-degenerate parabolic equations

We disscuss a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in a Muckenhoupt class. We present new results on weighted F.-H. Lin type estimates of the Hessian matrices of solutions. As examples, we demonstrate that the results are applicable to equations whose leading coefficients are of logistic-type singularities, as well as those are of polynomial blow-up or vanishing with sufficiently small exponents. A central component of the approach is the development of local quantitative lower estimates for solutions, which are interpreted as the mean sojourn time of sample paths, a stochastic-geometric perspective that generalizes the seminal work of L. C. Evans. By utilizing intrinsic weighted cylinders and perturbation arguments alongside with parabolic ABP estimates, we effectively manage the operator's degeneracies and singularities. We also briefly address regularization and truncation strategies that ensure our estimates are robust. We conclude with a discussion of future applications and related developments in the field.

Posted January 12, 2026

Applied Analysis Seminar Questions or comments?

Daniel Massatt, New Jersey Institute of Technology
TBA

Event contact: Stephen Shipman

Posted January 11, 2026

Applied Analysis Seminar Questions or comments?

Zhiyuan Geng, Purdue University
TBA