The global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model for two competing species is proved. The main feature of the model is that the diffusion matrix is nonsymmetric and generally not positive definite and that the nondiagonal matrix elements (the cross-diffusion terms) are allowed to be "large." The ideas of the existence proof are a careful approximation of the cross-diffusion terms using finite differences and the use of an entropy inequality yielding a priori estimates.
Maxwell-Stefan systems describing the dynamics of the molar concentrations of a gas mixture with an arbitrary number of components are analyzed in a bounded domain under isobaric, isothermal conditions. The systems consist of mass balance equations and equations for the chemical potentials, depending on the relative velocities, supplemented with initial and homogeneous Neumann boundary conditions. Global-in-time existence of bounded weak solutions to the quasilinear parabolic system and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to the singular Maxwell-Stefan diffusion matrix, the cross-diffusion coupling, and the lack of standard maximum principles. Key ideas of the proofs are the Perron-Frobenius theory for quasi-positive matrices, entropy-dissipation methods, and a new entropy variable formulation allowing for the proof of nonnegative lower and upper bounds for the concentrations.2000 Mathematics Subject Classification. 35K55, 35A01, 35B40, 35Q79.
The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H 1 estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler-Galerkin approximation, discrete entropy estimates, and L 1 weak compactness arguments. Furthermore, employing the entropy-entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the crossdiffusion constants are.
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