Abstract Consider a double degenerate parabolic equation arising from the electrorheological fluids theory and many other diffusion problems. Let $v_{\varepsilon }$ v ε be viscous solution of equation. By showing that $|\nabla v_{\varepsilon }|\in L^{\infty }(0,T; L_{\m...

in Hilbert space and their realizations in function spaces as initial-boundary value problems for partial differential equations which may contain degenerate or singular coefficients. The Cauchy problem consists of solving (1.1) subject to the initial condition Jdu(0) = h. We are concerned with the case where the solution is given by an analytic semigroup; it is this sense in which the Canchy p...

Using the maximum principle for semicontinuous functions (Differential Integral Equations 3 (1990), 1001–1014; Bull. Amer. Math. Soc. (N.S) 27 (1992), 1–67), we establish a general ‘‘continuous dependence on the nonlinearities’’ estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with timeand space-dependent nonlinearities. Our result generalizes a result by Souga...

We prove comparison results between viscosity sub and supersolutions of degenerate elliptic and parabolic equations associated to, possibly non-linear, Neumann boundary conditions. These results are obtained under more general assumptions on the equation (in particular the dependence in the gradient of the solution) and they allow applications to quasilinear, possibly singular, elliptic or para...

The quasilinear degenerate evolution equation of parabolic type d(Mv) dt + L(Mv)v = F (Mv), 0 < t ≤ T considered in a Banach space X is written, putting Mv = u, in the form du dt + A(u)u 3 F (u), 0 < t ≤ T , where A(u) = L(u)M−1 are multivalued linear operators in X for u ∈ K, K being a bounded ball ‖u‖Z < R in another Banach space Z continuously embedded in X. Existence and uniqueness of the l...

This paper is addressed to establishing controllability and observability for some forward linear stochastic complex degenerate/singular Ginzburg-Landau equations. It sufficient establish appropriate inequalities the corresponding backward The key prove Carleman estimates of operators. Compared with existing deterministic results, it necessary overcome difficulties caused by coefficients random...

We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L1-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, fi...

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case doubly hyperbolic equation, propose a generalization formulation prove result without any structure