In this paper we prove existence, uniqueness and regularity of the solution to a mixed initial-boundary value problem for a semilinear uniformly parabolic equation with principal part in divergence form, in a noncylindrical space-time domain. We assume only mild regularity on the coefficients and on the non-cylindrical part of the lateral boundary (on which Dirichlet data are given). Also, we assume only mild regularity on the Dirichlet data. We consider two different situations, one with a bounded domain and one with an unbounded domain. This problem is of interest in combustion theory. In that situation, the noncylindrical part of the lateral boundary may be considered as an approximation of a flame front. The second order part of the equation is the Laplace operator. In particular, the results in this paper are used in [8] to prove the uniqueness of a ``limit'' solution to the combustion problem.

The transient drift-diffusion model describing the charge transport in semiconductors is investigated in the case that the currents are prescribed. It is shown that the solutions of the drift-diffusion system coupled with Maxwell's equations converge to the solution of the drift-diffusion system coupled with Poisson's equation if the magnetic susceptibility tends to zero. Furthermore it is shown that the densities converge to the thermal equilibrium state for $t\rightarrow\infty$ provided that the boundary conditions are compatible with the thermal equilibrium.

We prove a boundary estimate for the gradient of a weak solution to a partial differential equation of $p$-Laplacian type with coefficients of vanishing mean oscillation. Combining the boundary estimate with a known interior estimate we obtain a global higher integrability result. The main result generalizes known results for linear equations to a nonlinear case. Our method is based on choosing the right test function, and, in particular, we do not use any representation formulas for solutions.

A new approach to deriving trace-type formulas is given. In this way, a new representation for the densities of the first integrals of the Korteweg-de Vries equation are found in terms of spectral and scattering data of the associated Schrödinger operator. We also utilize our method to improve many already-known results on the KdV invariants and associated trace formulas.