We consider a class of two-dimensional free boundary problems of type $div(a(X) \nabla u) = -div(\chi (u)H(X))$, where $H$ is a Lipschitz vector function satisfying $div(H(X))\geq 0$. We prove that the free boundary $\partial [u>0] \cap\Omega$ is represented locally by a family of continuous functions.

The Cauchy problem for a coupled Schrödinger and Benjamin-Ono system is shown to be globally well posed for a class of data without finite energy. The proof uses the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.

We develop a direct method for obtaining the blow-up rate of positive solutions of semilinear parabolic equations by using the theory of regularly varying functions. The method is applicable any time the blow-up set is a single point.

In this work, we discuss the existence, regularity and stability of solutions for some partial functional differential equations with infinite delay. We assume that the linear part generates an analytic semigroup on a Banach space $X$ and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.

We study a model of an antibiotic resistance in a hospital setting. The model connects two population levels - bacteria and patients. The bacteria population is divided into non-resistant and resistant strains. The bacterial strains satisfy ordinary differential equations describing the recombination and reversion processes producing the two strains within each infected individual. The patient population is divided into susceptibles, infectives infected with the non-resistant bacterial strain, and infectives infected with the resistant bacterial stain. The infective classes satisfy partial differential equations for the infection age densities of the two classes. We establish conditions for the existence of three possible equilibria for this model: (1) extinction of both infective classes, (2) extinction of the resistant infectives and endemicity of the non-resistant infectives, and (3) endemicity of both infective classes. We investigate the asymptotic behavior of the solutions of the model with respect to these equilibria.