Linear parabolic equations with strong singular potentials
Using an extension of a recent method of Cabré and Martel (1999), we extend the blow-up and existence result in the paper of Baras and Goldstein (1984) to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials. This problem has been left open in Baras and Goldstein. These potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail. Even in the special case when the leading operator is the Laplacian, we extend a recent result in Cabré and Martel from bounded smooth domains to unbounded nonsmooth domains.
Transactions of the American Mathematical Society
Goldstein, J., & Zhang, Q. (2003). Linear parabolic equations with strong singular potentials. Transactions of the American Mathematical Society, 355 (1), 197-211. https://doi.org/10.1090/S0002-9947-02-03057-X