We will establish an existence and regularity theory for weak solutions of a class of singular parabolic equations associated with Dirichlet data, whose prototype is u_t - div(|∇u|^{p-2}∇u) = 0 (1 < p < 2)

A thermistor is an electric circuit device made of ceramic material whose electric conductivity depends on the temperature. If the only heat source is the electric heating, the temperature and the electric potential satisfy a nonlinear elliptic system which is also degenerate if the electric conductivity is not uniformly bounded from above or away from zero. Under general boundary conditions, we establish existence and Hölder continuity of solutions of such a nonlinear nonuniformly elliptic system. When the elechic conductivity linearly depends on the temperature, we provide a non-uniqueness and non-existence example.

A phase change problem with Joule's heat ing describes tbe processes of electric heating in a conducting material. It is modeled as a coupled system of nonlinear partial differential equations with quadratic growth in the gradient. We establish the existence of a weak solution for the problem in two dimensions.

Consider in this paper the existence, uniqueness and asymptotic behaviour of the solutions of the mixed problem for a class of degenerate quasilincar parabolic equations and properties of the corresponding stationary solutions.

The initial value problem for a nonlinear evolution system with singular integral differential terms is studied. Dy means of a priori estimates of the solutions and Lcray-Schauder's fixed point theorem, we demonstrate the existence and uniqueness theorems of the generalized and classical global solutions to the problem.

We study the Calm-Hilliard equation with nonlinear principal part \frac{∂u}{∂t} + D[m(u)(kD^3u - DA(u))] = 0 The existence of classical solutions is established by means of the method based on Campanato spaces and the energy estimates. The corresponding uniqueness is also proved.