Our study of abstract quasi-linear parabolic problems in time-weighted Lp-spaces, begun in [17], is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reactiondiffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method pres...

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the so-called generalized Kimura diffusion operators [8, 9, 6]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scaleinvariant Harnack inequality fo...

Kinetics and selected variables (temperature, particle size and time) for extraction of Terminalia Catappa L Kernel Oil (TCKO) were investigated using solvent extraction. Kinetic models studied were: parabolic diffusion, power law, hyperbolic, Elovich and pseudo-second-order. In ascending order, the best-fitted models at the optimum temperature and oil yield were Elovich’s model, hyperbolic...

Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf3)C. We consider non-hermitian re...

Abstract. A (Patlak-) Keller-Segel model in two space dimensions with an additional crossdiffusion term in the equation for the chemical signal is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical substance. This allows one to prove, for arbitrarily small cross diffusion, the global existence of ...

We establish the Harnack inequality for advection-diffusion equations with divergencefree drifts of low regularity. While our previous work [IKR] considered the elliptic case, here we treat the more challenging parabolic problem by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale-invariant spaces.

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.

The theory of maximal monotone operators is utilized to prove the existence of weak periodic solutions for a coupled nonlinear parabolic-elliptic system.The diffusion term in the parabolic equation contains a Leray-Lions operator.This system may be regarded as a generalized version of the evolution thermistor model.

In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator ∂t − ∂x(c∂x) where the diffusion coefficient c has a jump. As a consequence of this Carleman estimate, we deduce consistent nullcontrollability results for classes of semi-linear parabolic equations.

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection– diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.