Hölder regularity for degenerate parabolic equations with variable exponents

In this paper, we discuss a class of degenerate parabolic equations with variable exponents. By using the Steklov average and Young's inequality, we establish energy and logarithmicestimates for solutions to these equations. Then based on the intrinsic scaling method, we provethat local weak solutions are locally continuous.

Asymptotic Properties of Solutions to the Cauchy Problem for Degenerate Parabolic Equations with Inhomogeneous Density on Manifolds

We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ | y | a for $$a \in (-1,1)$$ a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial _t - \Delta _x)^s$$ ( ∂ t - Δ x ) s for $$s \in (0,1)$$ s ∈ ( 0 , 1 ) . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ a = 0 ).

Radon Measure-valued Solutions for Nonlinear Strongly Degenerate Parabolic Equations with Measure Data

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time.