Dung Le

Global existence for a class of triangular parabolic systems on domains of arbitrary dimension

A class of triangular parabolic systems given on bounded domains of R n with arbitrary n is investigated. Sufficient conditions on the structure of the systems are found to assure that weak solutions exist globally.

Bacterial Wall Attachment in a Flow Reactor

A mathematical model of microbial growth for limiting nutrients in a fully three dimensional flow reactor which accounts for the colonization of the reactor wall surface by the microbes is studied analytically. It can be viewed as a model of the large intestine or of the fouling of a commercial bioreactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each. The effects of adding an antimicrobial agent to the reactor are examined with and without wall growth.

Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy

for all φ ∈ C c (Q,R). Here, we write dz = dxdt and W 1,0 2 (Q,R ) is the set of vector-valued functions whose spatial weak derivatives are in L(Q). It is now a classical result (see [10, 16]) for the scalar case (m = 1) that bounded weak solutions to (1.1) are Hölder continuous and if A,F are sufficiently smooth, then bounded weak solutions are classical. On the other hand, in the case of systems of equations (i.e. m > 1), one cannot generally expect bounded weak solutions of (1.1) to be Hölder continuous everywhere (see counterexamples, e.g., in [5]). Partial regularity theory for (1.1), when A is linear in Du, was considered in [2, 3], where it was shown that bounded weak solutions are Hölder continuous on a full measure subset of Q and the Hausdörff dimension of their singular sets could be estimated. In a recent work, see [11, 14], we introduced the so-called linear A-heat approximation technique to investigate partial and everywhere regularity for degenerate parabolic systems. However, as pointed out in [1], it is crucial to establish everywhere Hölder continuity for bounded weak solutions for systems and to control their Hölder norms in order to discuss further important questions in applications (e.g.; see [18]) such as global existence and dynamics of the solutions. Works in these directions were reported in [6, 7, 8, 12, 17, 20] for systems satisfying

Global Existence Results for Near Triangular Nonlinear Parabolic Systems

Abstract We study the global existence and regularity of weak solutions to strongly coupled parabolic systems whose diffusion matrices are almost triangular.

Regularity of solutions to a class of cross diffusion systems

Regularity of bounded solutions to a class of strongly coupled parabolic systems is investigated. Conditions on the structure of the systems are found to assure that bounded solutions are Holder continuous. The theory is then applied to the general Shigesada--Kawasaki--Teramoto model in population dynamics.

Persistence for a class of triangular cross diffusion parabolic systems

Abstract The purpose of this paper is to investigate the dynamics of a class of triangular parabolic systems given on bounded domains of arbitrary dimension. In particular, the existence of global attractors and the persistence property will be established.

Strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains

Nonnegativity of weak solutions of parabolic and elliptic equations on nonsmooth domains is established. Strong positivity of weak solutions to elliptic equations is proved via a boundary weak Harnack inequality.

Dynamics of a bio-reactor model with chemotaxis

Long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio-reactors with chemotaxis will be studied. In particular, we show that the dynamical system possesses a global attractor and that it is strongly uniformly persistent if the trivial steady state is unstable. Using a result of Smith and Waltman on perturbation of global attractors, we also show that the positive steady state is unique and globally attracting.

Regularity for Fully Nonlinear P-Laplacian Parabolic Systems: the Degenerate Case

Abstract This paper introduces new nonlinear heat approximation and L∞ preserving homotopy techniques to investigate regularity properties of bounded weak solutions of strongly coupled p-Laplacian parabolic systems which consist of more than one equation defined on a domain of any dimension. The main results imply everywhere Hölder continuity of bounded weak solutions and the global existence of strong solutions to nonlinear p-Laplacian systems.