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Convergence of minimizers for the p-Dirichlet integral.

Stephan Luckhaus — 1993

Mathematische Zeitschrift

Existence and regularity of weak solutions to the Dirichletproblem for semilinear elliptic systems of higher order.

Stephan Luckhaus — 1979

Journal für die reine und angewandte Mathematik

Phase Transition in multicomponent systems.

Stephan Luckhaus; Augusto Visintin — 1983

Manuscripta mathematica

Homogenization of a boundary condition for the heat equation

Ján Filo; Stephan Luckhaus — 2000

Journal of the European Mathematical Society

An asymptotic analysis is given for the heat equation with mixed boundary conditions rapidly oscillating between Dirichlet and Neumann type. We try to present a general framework where deterministic homogenization methods can be applied to calculate the second term in the asymptotic expansion with respect to the small parameter characterizing the oscillations.

Extension of a regularity result concerning the dam problem

Gianni Gilardi; Stephan Luckhaus — 1991

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

One proves, in the case of piecewise smooth coefficients, that the time derivative of the solution of the so called dam problem is a measure, extending the result proved by the same authors in the case of Lipschitz continuous coefficients.

The continuum reaction-diffusion limit of a stochastic cellular growth model

Stephan Luckhaus; Livio Triolo — 2004

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A competition-diffusion system, where populations of healthy and malignant cells compete and move on a neutral matrix, is analyzed. A coupled system of degenerate nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The healthy cells move much slower than the malignant ones, such that no diffusion for their density survives in the limit. The malignant cells may locally accumulate, while for the healthy ones an exclusion rule is considered....

On an evolutionary nonlinear fluid model in the limiting case

Stephan Luckhaus; Josef Málek — 2001

Mathematica Bohemica

We consider the two-dimesional spatially periodic problem for an evolutionary system describing unsteady motions of the fluid with shear-dependent viscosity under general assumptions on the form of nonlinear stress tensors that includes those with $p$ -structure. The global-in-time existence of a weak solution is established. Some models where the nonlinear operator corresponds to the case $p = 1$ are covered by this analysis.

Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems

Stephan Luckhaus; Yoshie Sugiyama — 2006

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the following reaction-diffusion equation: $(KS) \{\begin{matrix} u_{t} = \nabla \cdot (\nabla u^{m} - u^{q - 1} \nabla v), & x \in ℝ^{N}, 0 < t < \infty, 0 = Δ v - v + u, & x \in ℝ^{N}, 0 < t < \infty, u (x, 0) = u_{0} (x), & x \in ℝ^{N}, \end{matrix}$ where $N \geq 1, m > 1, q \geq max {m + \frac{2}{N}, 2}$ . In [Sugiyama, Nonlinear Anal. 63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of $q \geq max {m + \frac{2}{N}, 2}$ , the above problem (KS) is solvable globally in time for “small $L^{\frac{N (q - m)}{2}}$ data”. Moreover, the decay of the solution (u,v) in $L^{p} (ℝ^{N})$ was proved. In...