Mathematical analysis and stability of a chemotaxis model with logistic term

In this paper we study a non‐linear system of differential equations arising in chemotaxis. The system consists of a PDE that describes the evolution of a population and an ODE which models the concentration of a chemical substance. We study the number of steady states under suitable assumptions, the existence of one global solution to the evolution problem in terms of weak solutions and the stability of the steady states. Copyright © 2004 John Wiley & Sons, Ltd.     

Some Generalized Problems for Polyharmonic Functions

In the first section of this paper, some non-local boundary value problem for the polyharmonic equation in the plane is considered. This problem consists in determining solution of the polyharmonic equation satisfying some special non-local-type boundary condition on two curves. The existence theorem is proved. In the second section, an example for the case of the biharmonic equation is considered. In the third section, some non-local, non-linear problem of Riquier type is examined. The Riquier-type problem consists in determining the polyharmonic function in the plane whose value together with its successive Laplacians are prescribed on the boundary. The existence theorem is proved and an example for the case of the biharmonic equation is considered.     

Steady convection in the presence of an external magnetic field

The problem of the equilibrium of a liquid enclosed in a vessel heated from below has been considered by Sorokin [1], Iudovich and Ukhovskii [2] and Velt [3]. It has been established that if the Rayleigh number λ exceeds a certain critical value λ₀, then secondary steady flows arise in the liquid.

The stability of a conductive liquid heated from below has been studied by many authors. The most complete and general studies are those of Sorokin and Sushkin [4], whose paper contains the appropriate bibliography, and that of Shliomis [5]. The results of [4 and 5] make clear the physical picture of the phenomena associated with the heating of a conductive fluid and indicate the possible existence of secondary steady and periodic flows.

The existence of steady convective flows in a conductive liquid are proved below. Our study is based on the procedure set forth in [2].     

Stationary solutions to the equations of dynamics of mixtures of heat-conductive compressible viscous fluids

We consider the equations describing the three-dimensional steady motions of binary mixtures of heat-conductive compressible viscous fluids. An existence theorem for the boundary value problem that corresponds to flows in a bounded domain is proved in the class of weak generalized solutions.     

Liquid Flow in Partially Saturated Porous Media

We consider a picture for the filtration of a liquid in a partiallysaturated porous medium, leading to a two-phase one-dimensionalfree boundary problem of the following type: The liquid pressuresatisfies an elliptic equation in the saturated region and anon-linear parabolic equation in the unsaturated region, whilepressure and velocity are continuous across the interface. This scheme reduces to the study of the non-linear parabolicfree boundary problem in the unsaturated phase with cauchy dataprescribed on the free boundary, for such a problem existence,uniqueness and continuous dependence theorems are proved.     

轴对称压差系统的自相似解

考察了二维压差系统的轴对称活塞均匀膨胀而产生的自相似流动.在轴对称和自相似假设下,该问题可以简化为一个自治的非线性常微分方程组的自由边值问题.通过对常微分方程组的积分曲线性质的详细分析,建立该自由边值问题正光滑解的整体存在性.     

Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

We consider a strongly coupled PDE–ODE system that describes the influence of a slow and large vehicle on road traffic. The model consists of a scalar conservation law accounting for the main traffic evolution, while the trajectory of the slower vehicle is given by an ODE depending on the downstream traffic density. The moving constraint is expressed by an inequality on the flux, which models the bottleneck created in the road by the presence of the slower vehicle. We prove the existence of solutions to the Cauchy problem for initial data of bounded variation.     

A reaction–diffusion problem with a reaction front

We state a 1D model with quasi-stationary gas flows approximation for a carbon reactivity test in the production of silicon. The mathematical problem we formulate is a non-linear boundary value problem for a third-order ordinary differential equation with non-linear boundary conditions, which are non-local in time. We prove existence and uniqueness of a classical solution and provide a numerical example. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

CONCERNING TIME-PERIODIC SOLUTIONS OF THE NAVIER-STOKES EQUATIONS IN CYLINDRICAL DOMAINS UNDER NAVIER BOUNDARY CONDITIONS

The problem of the existence of time-periodic flows in infinite cylindrical pipes in correspondence to any given, time-periodic, total flux, was solved only quite recently in [1]. In this last reference we solved the above problem for flows under the non-slip boundary condition as a corollary of a more general result. Here we want to show that the abstract theorem proved in [1] applies as well to the solutions of the well known slip (or Navier) boundary condition (1.7) or to the mixed boundary condition (1.14). Actually, the argument applies for solutions of many other boundary value problems. This paper is a continuation of reference [1], to which the reader is referred for some notation and results.     

Asymptotic stability of a mathematical model of cell population

We consider a simplified system of a growing colony of cells described as a free boundary problem. The system consists of two hyperbolic equations of first order coupled to an ODE to describe the behavior of the boundary. The system for cell populations includes non-local terms of integral type in the coefficients. By introducing a comparison with solutions of an ODE's system, we show that there exists a unique homogeneous steady state which is globally asymptotically stable for a range of parameters under the assumption of radially symmetric initial data.     

Self-similar solutions of the Euler equations with spherical symmetry

We consider self-similar flows arising from the uniform expansion of a spherical piston and preceded by a shock wave front. With appropriate boundary conditions imposed on the piston surface and the spherical shock, the isentropic compressible Euler system is transformed into a nonlinear ODE system. We formulate the problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions.     

Inviscid and viscous stationary waves of gas flow through contracting-expanding nozzles

In this work we study steady states of one-dimensional viscous isentropic compressible flows through a contracting-expanding nozzle. Treating the viscosity coefficient as a singular parameter, the steady-state problem can be viewed as a singularly perturbed system. For a contracting-expanding nozzle, a complete classification of steady states is given and the existence of viscous profiles is established via the geometric singular perturbation theory. Particularly interesting is the existence of a maximal sub-to-super transonic wave and its role in the formation of other complicated transonic waves consisting of a sub-to-super portion.     

Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate

We revisit the situation of a thin liquid film driven up an inclined substrate by a thermally induced Marangoni shear stress against the opposing parallel component of gravity. In contrast to previous studies, we focus here on the meniscus region, in a case where the substrate is nearly horizontal. Our numerical simulations show that the time-dependent lubrication model for the film profile can reach a steady state in the meniscus region that is unlike the monotonic solutions investigated earlier. A systematic investigation of the steady states of the lubrication model is carried out by studying the phase space of the corresponding third-order ODE system. We find a rich structure of the phase space including multiple non-monotonic solutions with the same far-field film thickness. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of a Mutualistic Model with Diffusion

This article is concerned with a system of semilinear parabolic equations with no-flux boundary condition in a mutualistic ecological model. Stability result of the equilibrium about relevant ODE problem is proved by discussing its Jacobian matrix, we give two priori estimates and prove that the model is permanent when ε_1+ ε_2 ≠ 0. Moreover sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the model are obtained. Nonexistence of nonconstant positive steady states of the model is also given. When ε_1+ε_2= 0, grow up property is derived if the geometric mean of the interaction coefficients is greater than 1(α_1α_2 1),while if the geometric mean of the interaction coefficients is less than 1(α_1α_2 1), there exists a global solution. Finally, numerical simulations are given.     

On the Semiconductor System

In this paper, the semiconductor system is discussed. The existence and uniqueness or the global solution or the carrier transport problem are obtained. Under the condition that the width in some direction of the domain being sufficiently small, the existence and uniqueness of the solution of the steady states are proved. It is also proved that the solution of the carrier transport problem tends to the solution or the steady states problem exponentially when t goes to infinity.     

Numerical solution of the momentum and heat transfer equations for a hydromagnetic flow due to a stretching sheet of a non-uniform property micropolar liquid

A study of the hydromagnetic flow due to a stretching sheet and heat transfer in an incompressible micropolar liquid is made. Temperature-dependent thermal conductivity and a non-uniform heat source/sink render the problem analytically intractable and hence a numerical study is made using the shooting method based on Runge-Kutta and Newton-Raphson methods. The two problems of horizontal and vertical stretching are considered to implement the numerical method. The former problem involves one-way coupling between linear momentum and heat transport equations and the latter involves two-way coupling. Further, both the problems involve two-way coupling between the non-linear equations of conservation of linear and angular momentums. A similarity transformation arrived at for the problem using the Lie group method facilitates the reduction of coupled, non-linear partial differential equations into coupled, non-linear ordinary differential equations. The algorithm for solving the resulting coupled, two-point, non-linear boundary value problem is presented in great detail in the paper. Extensive computation on velocity and temperature profiles is presented for a wide range of values of the parameters, for prescribed surface temperature (PST) and prescribed heat flux (PHF) boundary conditions.     

NON-STATIONARY STOKES FLOWS UNDER LEAK BOUNDARY CONDITIONS OF FRICTION TYPE

1. IntroductionThe purpose of this paper is to consider the initial value problem for the Stokes flow undernonlinear boundary conditions of friction type, which will be described below in 52 togetherWith our motivations arising from applications, and to show that the solvability can be obtainedimmediately by means of the non-linear semigroup theory (NSG theory) which had originatedfrom the celebrated work by Y. Komura ([12J) in 1967 and was elaborated by many authors(for a concise explanatio…     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

Subsonic solutions to compressible transonic potential problems for isothermal self-similar flows and steady flows

We establish boundary and interior gradient estimates, and show that no supersonic bubble appears inside of a subsonic region for transonic potential flows for both self-similar isothermal and steady problems. We establish an existence result for the self-similar isothermal problem, and improve the Hopf maximum principle to show that the flow is strictly elliptic inside of the subsonic region for the steady problem.     

Global existence for a nonlinear theory of bubbly liquids

Global existence of smooth solutions is proved for an effective theory of bubbly liquids for either the initial value problem or initial boundary value problem in one dimension. This shows that the theory does not describe shock waves or bubble collapse. Since the analysis is not for the steady boundary value problem, there is no discussion of resonance. The proof uses a semilinear form of the equations to get local existence. A priori bounds resulting from energy conservation and a nonlinear Gronwall-like inequality are then derived to prove global existence.