The paper deals with a degenerate and singular parabolic equation with nonlinear boundary condition. We first get the behavior of the solution at infinity, and establish the critical global existence exponent and critical Fujita exponent for the fast diffusive equation, furthermore give the blow-up set and upper bound of the blow-up rate for the nonglobal solutions.

This paper considers the optimal control of a degenerate parabolic partial differential equation governing a di usive population with logistic growth terms. Assuming this population causes damage to forest and agricultural land, the optimal control is the trapping rate and the cost functional is a combination of the damage and trapping costs. We prove existence, uniqueness, and regularity resul...

The null controllability of parabolic operators in bounded domains, with both boundary or locally distributed controls, is a well-established property, see, e.g., (Bensoussan et al., 1993) and (Fattorini, 1998). Such a property brakes down, however, for degenerate parabolic operators even when degeneracy occurs on ”small” subsets of the space domain, such as subsets of the boundary. This talk w...

We deal with the blowup properties of the solution to the degenerate and singular parabolic system with nonlocal source and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained. Furthermore, under certain conditions it is prove...

In this paper we study controllability properties of linear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of 'regional null controllability', showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate. keywords: l...

We study a seawater intrusion problem in a confined aquifer. This process can be formulated as a coupled system of partial differential equations which includes an elliptic and a degenerate parabolic equation. Existence results of weak solutions, under realistic assumptions, are established through time discretization combined with parabolic regularization.