Quasilinear Parabolic Systems Research Articles

A general decay result in a quasilinear parabolic system with viscoelastic term

We consider a quasilinear parabolic system of the form A ( t ) | u t | m − 2 u t − Δ u + ∫ 0 t g ( t − s ) Δ u ( x , s ) d s = 0 , for m ≥ 2 , A ( t ) a bounded and positive definite matrix, and g a continuously differentiable decaying function and establish a general decay result from which the usual exponential and polynomial decay results are only special cases.

Regularity and uniqueness in quasilinear parabolic systems

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.

Global existence and nonexistence for some degenerate and strongly coupled quasilinear parabolic systems

This paper deals with positive solutions of degenerate and strongly coupled quasilinear parabolic systems u t = v α Δ u + u ( a 1 − b 1 u + c 1 v ) , v t = u β Δ v + v ( a 2 + b 2 u − c 2 v ) with null Dirichlet boundary condition and positive initial conditions describing a cooperating two-species Lotka–Volterra model with cross-diffusion, where the constants a i , b i , c i > 0 for i = 1 , 2 and α, β are non-negative. The local existence of positive classical solutions is proved. Moreover, the authors proved that the solutions are global if intra-specific competition of the species are strong, whereas the solutions may blow up if the inter-specific cooperation are strong and α , β ⩽ 1 .

A Quasilinear Parabolic System with Nonlocal Boundary Condition

We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. (2009), which considered the special case , and Wang et al. (2007), which studied the single equation.

Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system

The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$ u_t = v^p \left( {\Delta u + a\int_\Omega {udx} } \right),v_t = u^q \left( {\Delta v + b\int_\Omega {vdx} } \right) $$ with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution (u, v) to this problem. Moreover, a necessary and sufficient condition for the non- global existence of the solution is obtained. Under some further conditions on the initial data, we get criteria for the finite time blow-up of the solution.

Probabilistic representation of solutions for quasi-linear parabolic PDE via FBSDE with reflecting boundary conditions

A probabilistic representation of the solution (in the viscosity sense) of a quasi-linear parabolic PDE system with non-lipschitz terms and a Neumann boundary condition is given via a fully coupled forward-backward stochastic differential equation with a reflecting term in the forward equation. The extension of previous results consists on the relaxation on the Lipschitz assumption on the drift coefficient of the forward equation, using a previous result of the authors.

Model of chemotaxis with threshold density and singular diffusion

A quasilinear singular parabolic system corresponding to recent models of chemotaxis in which (1) there is an impassable threshold for the density of cells and (2) the diffusion of cells becomes singular (fast or superdiffusion) when the density approaches the threshold. It is proved that for some range of parameters describing the relation between the diffusive and the chemotactic component of the cell flux there are global-in-time classical solutions which in some cases are separated from the threshold uniformly in time. Global-in-time weak solutions in the case of fast diffusion and the set of stationary states are studied as well. The applications of the general results to particular models are shown.

Solutions to integro-differential systems arising in Hydrodynamic models for charged transport in semiconductors

This work is devoted to the study of strong and global solutions for a quasilinear parabolic integro-differential system arising in Hydrodynamic models for charged transport in semiconductors. The existence results are proven by using a priori estimates and fixed point theorems.

Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source

This paper deals with a class of localized and degenerate quasilinear parabolic systems $$u_t=f(u)(\Delta u+av(x_0,t)),\qquad v_t=g(v)(\Delta v+bu(x_0,t))$$ with homogeneous Dirichlet boundary conditions. Local existence of positive classical solutions is proven by using the method of regularization. Global existence and blow-up criteria are also obtained. Moreover, the authors prove that under certain conditions, the solutions have global blow-up property.

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i ( u i ) may have the property D i ( 0 ) = 0 for some or all i = 1 , … , N , and the boundary condition is u i = 0 . Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.

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.

A degenerate and strongly coupled quasilinear parabolic system with crosswise diffusion for a mutualistic model

This paper deals with positive solutions of degenerate and strongly coupled quasilinear parabolic system u t = v α Δ u + u ( a 1 − b 1 u l + c 1 v s ) , v t = u β Δ v + v ( a 2 + b 2 u p − c 2 v q ) with null Dirichlet boundary condition describing a cooperating model with crosswise diffusion, where the constants a i , b i , c i > 0 ( i = 1 , 2 ) , α , β ≥ 0 and l , s , p , q ≥ 1 . Local existence of positive classical solution is proved. Moreover, it will be proved that the solution is global if intra-specific competitions of the species are strong, whereas the solution may be nonglobal if the inter-specific cooperation is strong and 0 < α ≤ s , 0 < β ≤ p with α , β < 2 .

Elliptic and parabolic inequalities with point singularities on the boundary

It is shown that various quasilinear elliptic and parabolic differential inequalities and systems of such inequalities defined on bounded domains, and which have point singularities on the boundary do not have solutions. The method of nonlinear capacity is used in the proof. Examples show that the conditions obtained by this method cannot be improved in the class of problems under consideration.Bibliography: 14 titles.

Asymptotic behavior of solutions to a quasilinear nonuniform parabolic system modelling chemotaxis

In this paper, we study a quasilinear nonuniform parabolic system modelling chemotaxis and taking the volume-filling effect into account. The results on the existence of a unique global classical solution was obtained in Cieślak (2007) [4]. However, the convergence to equilibrium was not considered in that paper. In this paper, we first obtain the crucial uniform boundedness of the solution. Then with the help of a suitable non-smooth Simon–Łojasiewicz approach we obtain the results on convergence to equilibrium and the decay rate.

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.

A class of nonlocal and degenerate quasilinear parabolic system not in divergence form

In this paper, we investigate the positive solution of the nonlocal and degenerate quasilinear parabolic system not in divergence form u t = f ( u ) ( Δ u + a ∫ Ω v d x ) , v t = g ( v ) ( Δ v + b ∫ Ω u d x ) with null Dirichlet boundary conditions. The blow-up and global existence criteria are obtained. Furthermore, we prove that under certain conditions the solutions have global blow-up.

Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems

This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: u1t = f1(u2)(�u1 + a1u1), � � � ,u(n 1)t = fn 1(un)(�un 1 + an 1un 1), unt = fn(u1)(�un+anun) with homogenous Dirichlet boundary condition and posi- tive initial condition, where ai (i = 1,2, ��� ,n) are positive constants and fi (i = 1,2, ��� ,n) satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that: (i) when min{a1, � � � , an} ≤ �1 then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when min{a1, � � � , an} > �1, and the initial datum (u10, � � � , un0) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where �1 is the first eigenvalue of −� in with homogeneous Dirichlet boundary condition.

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.

Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms

This paper deals with the bounded and blowup solutions of the quasilinear parabolicsystem $u_t = u^p ( \Delta u + a v) + f(u, v, Du, x)$ and$v_t = v^q ( \Delta v + b u) + g(u, v, Dv, x)$ with homogeneous Dirichlet boundarycondition. Under suitable conditions on the lower order terms $f$ and $g$,it is shown that all solutions are bounded if $(1+c_1) \sqrt{ab} \lambda_1$, where $\lambda_1$ is the first eigenvalueof $-\Delta $ in $\Omega$ with Dirichlet data and $c_1 > -1$ related to $f$ and $g$.