In this paper we present a necessary and sufficient condition to guarantee the complete reducibility for quasilinear hyperbolic systems and give some examples.

In this paper we prove the gradient estimates for fully nonlinear geometric equation using a normal perturbation techniques.

In this paper, we establish the existence of global self-similar solutions for the heat and convection-diffusion equations. This we do in some homogeneous Besov spaces using the theory of Besov spaces and the Strichartz estimates. Further, the structure of the self-similar solutions has also been established by using an equivalent norm for Besov spaces.

We consider a free boundary problem obtained from the asymptotic limit of a FitzHugh-Nagumo system, or more precisely, a slow-diffusion, fast-reaction equation governing a phase indicator, coupled with an ordinary differential equation governing a control variable ν. In the range (-1, 1), the v value controls the speed of the propagation of phase boundaries (interfaces) and in the mean time changes with dynamics depending on the phases. A new feature included in our formulation and thus made our model different from most of the contemporary ones is the nucleation phenomenon: a phase switch occurs whenever v elevates to 1 or drops to -1. For this free boundary problem, we provide a weak formulation which allows the propagation, annihilation, and nucleation of interfaces, and excludes interfaces from having (space- time) interior points. We study, in the one space dimension setting, the existence, uniqueness, and non-uniqueness of weak solutions. A few illustrating examples are also included.