We prove comparison results between continuous and dis-continuous viscosity sub-and super-solutions of the generalized Dirichlet problem for quasilinear elliptic and parabolic equations. The main consequence of these results is the uniqueness of continuous solutions of such problems, when they exist.

In this paper we investigate symmetry properties of positive solution of quasilinear parabolic problems in the whole space. As the main result, we prove that if the problem converges exponentially to a symmetric one, then the solution converges to the space of symmetric functions. We also show, that this result does not hold true, if the convergence is not exponential.

An initial boundary value problem of convection-diffusion type for a singularly perturbed quasilinear parabolic equation is considered on an interval. For this problem we construct ε-uniformly convergent difference schemes (nonlinear iteration-free schemes and their iterative variants) based on the domain decomposition method, which allow us to implement sequential and parallel computations on ...

In this paper we investigate the nature of the adapted ;solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an "ordinary" sense over an arbitrarily prescribed time duration, via a direct "Four Step Scheme". Using this scheme, we further...

We study the asymptotic behaviour near nite blow-up time t = T of the solutions to the one-dimensional degenerate quasilinear parabolic equation u t = (u u x) x + u in R (0; T); > 0; 1 < < + 1; with bounded, nonnegative, compactly supported initial data. This parameter range corresponds to global blow-up where u(x; t) ! 1 as t ! T ? for any x 2 R. We prove that the rescaled function f(; t) = (T...

We consider quasilinear parabolic equations on RN satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.

We prove the reduction principle and study other attractivity properties of the center and center-unstable manifolds in the vicinity of a steady-state solution for quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains.

We study degenerate quasilinear parabolic systems in two different domains, which are connected by a nonlinear transmission condition at their interface. For a large class of models, including those modeling pollution aggression on stones and chemotactic movements of bacteria, we prove global existence, uniqueness and stability of the solutions.

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic–parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small L1 ∩ H3 perturbations, converging time-asymptotically to a translate of...

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a g...