In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary, or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point c...

We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definiti...

We consider self-similar potential flow for compressible gas with polytropic pressure law. Self-similar solutions arise as large-time asymptotes of general solutions, and as exact solutions of many important special cases like Mach reflection, multidimensional Riemann problems, or flow around corners. Self-similar potential flow is a quasilinear second-order PDE of mixed type which is hyperboli...

The steady state of the quasilinear convection-diffusion-reaction equation ut −∇(D(u)∇u) + b(u)∇u+ c(u) = 0 (1) is studied. Depending on the ratio between convection and diffusion coefficients, equation (1) ranges from parabolic to almost hyperbolic. From a numerical point of view two main difficulties can arise related with the existence of layers and/or the non smoothness of the coefficients....

We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t→ +∞. The case where b(t) ∼ (1 + t) with p < 1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and t...

In this paper, we study the stability of quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The exponential mean square and pathwise solutions established. Moreover, under certain hypothesis on perturbations, can be derived, without utilizing stability.

"Using the theory of Young measures, we prove existence solutions to a strongly quasilinear parabolic system \[\frac{\partial u}{\partial t}+A(u)=f,\] where $A(u)=-\text{div}\,\sigma(x,t,u,Du)+\sigma_0(x,t,u,Du)$, $\sigma(x,t,u,Du)$ and $\sigma_0(x,t,u,Du)$ are satisfy some conditions $f\in L^{p'}(0,T;W^{-1,p'}(\Omega;\R^m))$."