4,918 research outputs found
On a drift-diffusion system for semiconductor devices
In this note we study a fractional Poisson-Nernst-Planck equation modeling a
semiconductor device. We prove several decay estimates for the Lebesgue and
Sobolev norms in one, two and three dimensions. We also provide the first term
of the asymptotic expansion as .Comment: to appear in Annales Henri Poincar\'
Coagulation reaction in low dimensions: Revisiting subdiffusive A+A reactions in one dimension
We present a theory for the coagulation reaction A+A -> A for particles
moving subdiffusively in one dimension. Our theory is tested against numerical
simulations of the concentration of particles as a function of time
(``anomalous kinetics'') and of the interparticle distribution function as a
function of interparticle distance and time. We find that the theory captures
the correct behavior asymptotically and also at early times, and that it does
so whether the particles are nearly diffusive or very subdiffusive. We find
that, as in the normal diffusion problem, an interparticle gap responsible for
the anomalous kinetics develops and grows with time. This corrects an earlier
claim to the contrary on our part.Comment: The previous version was corrupted - some figures misplaced, some
strange words that did not belong. Otherwise identica
Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB)
equations, i.e. scalar conservation laws with diffusive-dispersive
regularization. We review the existence of traveling wave solutions for these
two classes of evolution equations. For classical equations the traveling wave
problem (TWP) for a local KdVB equation can be identified with the TWP for a
reaction-diffusion equation. In this article we study this relationship for
these two classes of evolution equations with nonlocal diffusion/dispersion.
This connection is especially useful, if the TW equation is not studied
directly, but the existence of a TWS is proven using one of the evolution
equations instead. Finally, we present three models from fluid dynamics and
discuss the TWP via its link to associated reaction-diffusion equations
Hyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions
Under consideration is the hyperbolic relaxation of a semilinear
reaction-diffusion equation on a bounded domain, subject to a dynamic boundary
condition. We also consider the limit parabolic problem with the same dynamic
boundary condition. Each problem is well-posed in a suitable phase space where
the global weak solutions generate a Lipschitz continuous semiflow which admits
a bounded absorbing set. We prove the existence of a family of global
attractors of optimal regularity. After fitting both problems into a common
framework, a proof of the upper-semicontinuity of the family of global
attractors is given as the relaxation parameter goes to zero. Finally, we also
establish the existence of exponential attractors.Comment: to appear in Quarterly of Applied Mathematic
Traveling waves for a bistable equation with nonlocal-diffusion
We consider a single component reaction-diffusion equation in one dimension
with bistable nonlinearity and a nonlocal space-fractional diffusion operator
of Riesz-Feller type. Our main result shows the existence, uniqueness (up to
translations) and stability of a traveling wave solution connecting two stable
homogeneous steady states. In particular, we provide an extension to classical
results on traveling wave solutions involving local diffusion. This extension
to evolution equations with Riesz-Feller operators requires several technical
steps. These steps are based upon an integral representation for Riesz-Feller
operators, a comparison principle, regularity theory for space-fractional
diffusion equations, and control of the far-field behavior
Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations
In this paper we consider a -dimensional () parabolic-elliptic
Keller-Segel equation with a logistic forcing and a fractional diffusion of
order . We prove uniform in time boundedness of its solution
in the supercritical range , where is an explicit
constant depending on parameters of our problem. Furthermore, we establish
sufficient conditions for , where
is the only nontrivial homogeneous solution. Finally, we
provide a uniqueness result
On the fractional Fisher information with applications to a hyperbolic-parabolic system of chemotaxis
We introduce new lower bounds for the fractional Fisher information. Equipped
with these bounds we study a hyperbolic-parabolic model of chemotaxis and prove
the global existence of solutions in certain dissipation regimes
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