Intrinsic scaling for PDEs with an exponential nonlinearity

Abstract

We consider strongly degenerate equations in divergence form of the typewhere the exponential nonlinearity satisfies the condition 0 < γ − ≤ γ(x, t) ≤ γ + . We show, by means of intrinsic scaling, that weak solutions are locally continuous.

Parabolic Equations with Anisotropic Nonstandard Growth Conditions

“…We also refer to the papers [1,2,3,9] for a discussion of the regularity properties of weak solutions of the systems of equations with nonstandard growth conditions (see also the references therein to the previous work on this issue). The continuity properties of solutions of a parabolic equation with variable exponent of nonlinearity are studied in [13].…”

Entropy solutions for the $p(x)$-Laplace equation

“…[31]), the flow in porous media (cf. [4] and [21]), and problems in the calculus of variations involving variational integrals with nonstandard growth (cf. [31], [27], and [1]).…”

On a Two-Sidedly Degenerate Chemotaxis Model With Volume-Filling Effect

“…e., u n is the solution of (10) associated with u n and the solution v n of (9). The objective is to show that u n converges to Θ(u) in L 2 (Q T ).…”

“…This will reflect the fact that the diffusion processes in the equation evolve in a time scale determined instant by instant by the solution itself, so that, loosely speaking, it can be regarded as the heat equation in its own intrinsic time-configuration. For a modern account of the theory and related matters, see the updated survey [7]; recent applications in the context of the modelling of phase transitions or the porous media equation with variable exponent of nonlinearity are to be found in [8] and [9], respectively.…”