Class Of Quasilinear Parabolic Systems Research Articles

AN EXISTENCE RESULT FOR QUASILINEAR PARABOLIC SYSTEMS WITH LOWER ORDER TERMS

In this paper we prove the existence of weak solutions for a class of quasilinear parabolic systems, which correspond to diffusion problems, in the form where Ω is a bounded open domain of be given and The function v belongs to is in a moving and dissolving substance, the dissolution is described by f and the motion by g. We prove the existence result by using Galerkin’s approximation and the theory of Young measures.

Lorentz estimates for a class of nonlinear parabolic systems

In this paper we obtain the following local Lorentz estimatesB(|F|)∈Llocγ,q⇒B(|∇u|)∈Llocγ,qfor anyγ>1and0<q≤∞ of the weak solutions for a class of quasilinear parabolic systemsut−div(a(|∇u|)∇u)=div(a(|F|)F), where B(t)=∫0tτa(τ)dτ for t≥0.

Global attractors in stronger norms for a class of parabolic systems with nonlinear boundary conditions

For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal L p -regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for obtaining gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level.

Volume Filling Effect in Modelling Chemotaxis

The oriented movement of biological cells or organisms in response to a chemical gra- dient is called chemotaxis. The most interesting situation related to self-organization phenomenon takes place when the cells detect and response to a chemical which is secreted by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many particularized mod- els have been proposed to describe the aggregation phase of this process. Most of efforts where concentrated, so far, on mathematical models in which the formation of aggregate is interpreted as finite time blow-up of cell density. In recently proposed models cells are no more treated as point masses and their finite volume is accounted for. Thus, arbitrary high cell densities are precluded in such description and a threshold value for cells density is a priori assumed. Different modeling approaches based on this assumption lead to a class of quasilinear parabolic systems with strong nonlinearities including degenerate or singular diffusion. We give a survey of analytical results on the existence and uniqueness of global-in-time solutions, their convergence to stationary states and on a possibility of reaching the density threshold by a solution. Unsolved problems are pointed as well.

Periodic solutions of quasilinear parabolic systems with nonlinear boundary conditions

This paper is concerned with the existence of maximal and minimal periodic solutions of a class of quasilinear parabolic systems with nonlinear boundary conditions. Our approach to the problem is based on the method of upper and lower solutions and its associated monotone iterations. An application is also made to the reaction–diffusion system of Lotka–Volterra competition model.

Blow-up rate and profile for a class of quasilinear parabolic system

This paper deals with positive solutions to a class of nonlocal and degenerate quasilinear parabolic system with null Dirichlet boundary conditions. The blow-up rate and blow-up profile are gained if the parameters and the initial data satisfy some conditions.

Classical solutions of drift–diffusion equations for semiconductor devices: The two-dimensional case

We regard drift–diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole currents is an integrable function. Hence, Gauss’ theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.

A class of quasilinear parabolic systems with cross-diffusion effects

This paper, using the duality technique and Holder's inequality, proves the global existence of solutions for a class of the quasilinear parabolic systems with Cross-diffusion effects and competition interaction on any smooth bounded domain in R N .

Dissipativity and global attractors for a class of quasilinear parabolic systems

We show that a-priori weak Lp dissipativity implies strong L ∞ dissipativity for a class of weakly coupled quasilinear parabolic systems satisfies general structure conditions. The existence of global attractors of general nonlinear reaction diffusion systems will be proven.

The Cauchy problem for a system of quasilinear parabolic equations of chemical kinetics type

The Cauchy problem for the class of quasilinear parabolic systems of reaction—diffusion type is considered. Under appropriate assumptions regarding the nonlinearities and the initial functions, the nonlocal solvability of this problem and the stabilization of its solution, uniform with respect to the space variables, are established.

An optimal regularity result for a class of quasilinear parabolic systems

The regularity results of Hildebrandt-Widman and Wiegner for elliptic systems are generalized to the parabolic case.