We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, ∆α/2 for α ∈ (0, 2). Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz depende...

We study a family of diffusion models for compounded risk reserves, which account for the investment income earned and for the inflation experienced on claim amounts. We are interested in the models which the dividend payments are paid from the risk reserves. After defining the process of conditional probability over finite time, the classical diffusion processes results turn the nonlinear stoc...

In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for model equation mixed hyperbolic-parabolic degeneration order in hyperbolicity domain. parabolic domain, is fractional diffusion equation, whereas hyperbolic domain it loaded one-speed transfer equation. We prove uniqueness and existence theorem propose an explicit solution...

We investigate stochastic homogenization for some degenerate quasilinear parabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium. Assuming that this reference diffusion operator is ergodic, we can prove the homogenizati...

We study traveling fronts of equations of the form u(tt)+phi(u)u(x)=u(xx)+f(u). A criterion for the transition from linear to nonlinear marginal stability is established for positive functions phi(u) and for any reaction term f(u) for which the usual parabolic reaction diffusion equation u(t)=u(xx)+f(u) admits a front. As an application, we treat reaction diffusion systems with transport memory.

This paper contains a study of the long time behavior of a diffusion process in a periodic potential. The first goal is to determine a suitable rescaling of time and space so that the diffusion process converges to some homogeneous limit. The issue of interest is to characterize the effective evolution equation. The main result is that in some cases large drifts must be removed in order to get ...

The problem of reconstructing the drift of a diffusion in R, d ≥ 2, from the transition probability density observed outside a domain is considered. The solution of this problem also solves a new inverse problem for a class of parabolic partial differential equations. This work considerably extends [2] in terms of generality, both concerning assumptions on the drift coefficient, and allowing fo...

This paper deals with a one-dimensional parabolic cross-diffusion system describing two-species models. The system under consideration is strongly coupled and the coefficients of the equations are allowed to be discontinuous. The existence and uniqueness of nonnegative global solutions are given when one of the cross-diffusion pressures is zero. Our approach to the problem is by an approximatio...

Qinghua Feng School of Science Shandong university of technology Zhangzhou Road 12#, Zibo , Shandong, 255049 China [email protected] Abstract: Based on the concept of decomposition, two alternating group explicit methods are constructed for 1D convection-diffusion equation with variable coefficient and 2D diffusion equations respectively. Both the two methods have the property of unconditional sta...

The spectrum of the simplest 1D convection-diffusion operator is a discrete subset of the negative real axis, but the pseudospectra are regions in the complex plane bounded approximately by parabolas. Put another way, the norm of the resolvent is exponentially large as a function of the Pdclet number throughout a certain approximately parabolic region. These observations have a simple physical ...