This paper considers quadratic and super-quadratic reaction-diffusion systems for reversible chem... more This paper considers quadratic and super-quadratic reaction-diffusion systems for reversible chemistry, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.
We consider a model system consisting of two reaction-diffusion equations, where one species diff... more We consider a model system consisting of two reaction-diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via nonlinear reversible Robin-type boundary conditions for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division. Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, we show explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate.
Mathematical Models and Methods in Applied Sciences, 2013
We present and discuss a mathematical model for the operation of bilayer organic photovoltaic dev... more We present and discuss a mathematical model for the operation of bilayer organic photovoltaic devices. Our model couples drift-diffusion-recombination equations for the charge carriers (specifically, electrons and holes) with a reaction-diffusion equation for the excitons/polaron pairs and Poisson's equation for the self-consistent electrostatic potential. The material difference (i.e. the HOMO/LUMO gap) of the two organic substrates forming the bilayer device are included as a work-function potential.
We study the large-time behaviour of a non-local evolution equation for the density of particles ... more We study the large-time behaviour of a non-local evolution equation for the density of particles or individuals subject to an external and an interaction potential. In particular, we consider interaction potentials which are singular in the sense that their first derivative is discontinuous at the origin.
Journal de Mathématiques Pures et Appliquées, 2010
In this paper, we are concerned with a class of multi-dimensional balance laws with a nonlocal di... more In this paper, we are concerned with a class of multi-dimensional balance laws with a nonlocal dissipative source which arise as simplified models for the hydrodynamics of radiating gases. At first we introduce the energy method in the setting of smooth perturbations and study the stability of constants states. Precisely, we employ Fourier space analysis to quantify the energy dissipation rate and recover the optimal time-decay estimates for perturbed solutions via an interpolation inequality in Fourier space. As application, the developed energy method is used to prove stability of smooth planar waves in all dimensions n ≥ 2, and also to show existence and stability of time-periodic solutions in the presence of the time-periodic source. Optimal rates of convergence of solutions towards the planar waves or time-periodic states are also shown provided initially L 1 -perturbations.
The Aizenman-Bak model for reacting polymers is considered for spatially inhomogeneous situations... more The Aizenman-Bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a nondegenerate size-dependent coefficient. Both the break-up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels. We demonstrate that the entropy-entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension, but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non-degenerate diffusion in dimension one. We finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial. Up to our knowledge, this is the first result of explicit equilibration rates for spatially inhomogeneous coagulationfragmentation models.
We present a new a-priori estimate for discrete coagulation-fragmentation systems with size-depen... more We present a new a-priori estimate for discrete coagulation-fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global L 2 bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations.
Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-brea... more Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.
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Papers by K. Fellner