A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions

Intrinsic scaling for PDEs with an exponential nonlinearity

“…In the recent paper [1], it is proved, under less restrictive assumptions on γ, that there exists a unique solution to the initial boundary value problem associated with this equation and that the solution is bounded. It is also shown that the solution is nonnegative if the initial data is nonnegative; that is why the assumption u(x, t) ∈ [0, 1] a.e.…”

“…The main feature in equation (1) is clearly the exponential nonlinearity that makes it extremely degenerate. Recently, Antontsev and Shmarev [1] obtained results on the existence and uniqueness of weak solutions of (1), together with some localization properties. Under appropriate assumptions, we prove in this paper that weak solutions are locally continuous.…”

A two-obstacle problem with variable exponent and measure data

“…Such problems are interesting from the purely mathematical point of view. Moreover, they have potential applications in various fields such as flow through porous media [1], thermorheological fluids [2], image processing [5,10], and especially electrorheological fluids (an essential class of non-Newtonian fluids), which have been used not only in fast-acting hydraulic valves and clutches, brakes, and shock absorbers, but also in some new fields such as accurate abrasive polishing, robotics, and space technology [3,18].…”

Study of solutions to a class of certain parabolic systems with variable exponents

“…The case of a single equation of the type (2) has been studied in [4][5][6]20] and the authors established the existence and uniqueness results, in [20], the authors use the difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint to p i (x).…”