GLOBAL SOLUTION OF THE INVERSE PROBLEM FOR ACLASS OF NONLINEAR EVOLUTION EQUATIONSOF DISPERSIVE TYPE

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Singular perturbation of initial value problem for a nonlinear second order systems

In this paper,the singular perturbation of initial value problem for nonlinearsecond order vector differential equationsε~rx″=f(t,x,x′,ε)x(0,ε)=a,x′(0,ε)=βis discussed,where r>0 is an arbitrary constant,ε>0 is a small parameter,x,f,aandβ∈R~n.Under suitable assumptions,by using the method of many-parameterexpansion and the technique of diagonalization,the existence of the solution of pertur-bation problem is proved and its uniformly valid asymptotic expansion of higher order isderived.     

Global solution of the inverse problem for a class of nonlinear evolution equations of dispersive type

The inverse problem for a class of nonlinear evolution equations of dispersive type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution was given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin. Contributed by Chen Yu-shu Foundation item: the National Natural Science Foundation of China (Significance 199990510); the National Key Basic Research Special Foundation of China (G1998020316); Liuhui Center for Applied Mathematics, Nankai University & Tianjin University Biography: Chen Fang-qi (1963-)     

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed. Foundation items: the National Natural Science Foundation of China (10071048); the “Hunfred Talents Project” by Chinese Academy of Sciences Biography: Mo Jia-qi (1937−)     

Well-posedness and Stability of a Multi-dimensional Tumor Growth Model

We study a moving boundary problem modeling the growth of in vitro tumors. This problem consists of two elliptic equations describing the distribution of the nutrient and the internal pressure, respectively, and a first-order partial differential equation describing the evolution of the moving boundary. An important feature is that the effect of surface tension on the moving boundary is taken into account. We show that this problem is locally well-posed for a large class of initial data by using analytic semi-group theory. We also prove that if the surface tension coefficient γ is larger than a threshold value γ _* then the unique flat equilibrium is asymptotically stable, whereas in the case γ  < γ _* this flat equilibrium is unstable.     

On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections

Mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections belongs to one of the most important and necessary steps while constructing numerous technical devices, as well as aviation and ship structural members. In first part of the paper fundamental hypotheses are formulated which allow us to derive Hamilton–Ostrogradsky equations. The latter yield equations governing shell behavior within the applied hypotheses and modified Pelekh–Sheremetev conditions. The aim of second part of the paper is to formulate fundamental hypotheses in order to construct coupled boundary problems of thermo-elasticity which are used in non-classical mathematical models for multi-layer shallow shells with initial imperfections. In addition, a coupled problem for multi-layer shell taking into account a 3D heat transfer equation is formulated. Third part of the paper introduces necessary phase spaces for the second boundary value problem for evolutionary equations, defining the coupled problem of thermo-elasticity for a simply supported shallow shell. The theorem regarding uniqueness of the mentioned boundary value problem is proved. It is also proved that the approximate solution regarding the second boundary value problem defining condition for the thermo-mechanical evolution for rectangular shallow homogeneous and isotropic shells can be found using the Bubnov–Galerkin method.     

Evolution of slowly modulated wave train on porous sea bed

In this paper, the problem of evolution of slowly modulated wave train on porous sea bed is investigated with the method of multiple scales. For the sea water in the upper region, the classical potential theory is used while the fluid motion in the porous sea bed is described by Darcy’s law. The equations of the first and second order modulations of wave amplitude are derived by using matching conditions on the sea bed. The corresponding solutions are found and seepage pressures are also given at the same time.     

Singularly perturbed solution to semilinear reaction diffusion equations with two parameters

A class of singularly perturbed initial boundary value problems for semilinear reaction diffusion equations with two parameters is considered, Under suitable conditions and using the theory of differential inequalities, the existence and the asymptotic behavior of the solution to the initial boundary value problem are studied.     

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.     

Spectrum Analysis of Some Kinetic Equations

We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with \({\gamma\geqq-2}\). As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate \({t^{-5/4}}\)) as \({t\to\infty}\) to that of the compressible Navier–Stokes equations for initial data around an equilibrium state.     

Existence and uniqueness of solutions for initial value problems of second order ordinary differential equations in Banach spaces

IntroductionLet(E,l'I)denotearealBanachspacewithapartialorderintroducedbyanormalconeKofE.Inthispaper,weshallconsiderthefollowinginitialvalueproblemofsecondorderordinarydifferentialequation(IVP):wheref:JxEZ~EandJ=LO,TiforT>0.AfunctionxeCI(J,E)issaidtobeasolutionofIVP(I)ifithasabsolutelycontinuousfirstderivativeandsatisfies(I)fora.e.teJ.Theuseofmonotonemethodsinthestudyoftheinitial(boundary)valueproblemsofordinarydiffferentialequationshasrecentlybeenquiteextensive(see,forexample[I~8]…     

Large-Time Behavior of Solutions to Hyperbolic-Elliptic Coupled Systems

We are concerned with the asymptotic behavior of a solution to the initial value problem for a system of hyperbolic conservation laws coupled with elliptic equations. This kind of problem was first considered in our previous paper. In the present paper, we generalize the previous results to a broad class of hyperbolic-elliptic coupled systems. Assuming the existence of the entropy function and the stability condition, we prove the global existence and the asymptotic decay of the solution for small initial data in a suitable Sobolev space. Then, it is shown that the solution is well approximated, for large time, by a solution to the corresponding hyperbolic-parabolic coupled system. The first result is proved by deriving a priori estimates through the standard energy method. The spectral analysis with the aid of the a priori estimate gives the second result.     

Exact solutions for some non-conservative hyperbolic systems

In this paper, we study initial value problem for some non-conservative hyperbolic systems of partial differential equations of first order. The first one is the Riemann problem for a model in elastodynamics and the second one the initial value problem for a system which is a generalization of the Hopf equation. The non-conservative products which appear in the equations do not make sense in the classical theory of distributions and are understood in the sense of Volpert (Math. USSR Sb. 2 (1967) 225). Following Lax (Comm. Pure Appl. Math. 10 (1957) 537) and Dal Maso et al. (J. Math. Pures Appl. 74 (1995) 483), we give an explicit solution for the Riemann problem for the elastodynamics equation. The coupled Hopf equation is studied using a generalization of the method of Hopf (Comm. Pure Appl. Math. 3 (1950) 201).     

CAUCHY PROBLEM OF ONE TYPE OF ATMOSPHERE EVOLUTION EQUATIONS

One type of evolution atmosphere equations was discussed. It is found that according to the stratification theory, （i） the inertial force has no influence on the criterion of the well-posed Cauchy problem; （ii） the compressibility plays no role on the well-posed condition of the Cauchy problem of the viscid atmosphere equations, but changes the well-posed condition of the viscid atmosphere equations; （iii） this type of atmosphere evolution equations is ill-posed on the hyperplane t = 0 in spite of its compressibility and viscosity; （iv） the Cauchy problem of compressible viscosity atmosphere with still initial motion is ill-posed.     

Second Order Abstract Neutral Functional Differential Equations

In this paper we are concerned with a class of second order abstract neutral functional differential equations with finite delay in a Banach space. We establish the existence of mild and classical solutions for the nonlinear equation, and we show that the map defined by the mild solutions of the linear equation is a strongly continuous semigroup of bounded linear operators on an appropriate space. We use this semigroup to establish a variation of constants formula to solve the inhomogeneous linear equation.     

On the nonlinear theory of the wing in a plane unsteady flow

The main aspects of the nonlinear theory of the wing in a plane unsteady fluid flow are generalized on the basis of the author’s previous results. An initial-boundary problem for complex velocity is formulated. A system of differential equations with conditions at points of vortex wake shedding is presented, which allows a large class of problems to be solved correctly. The Cauchy problem is solved by using a standard discretization procedure. The boundary-value problem is reduced at each time step to singular integral equations of the first and second kind. The accuracy of solving these equations by the method of discrete vortices and by the method of panels is compared. Specific features of pressure calculations in the case of a separated flow around the airfoil contour are discussed     

Global Weak Solutions to the Equations of Compressible Flow of Nematic Liquid Crystals in Two Dimensions

We consider weak solutions to a simplified Ericksen–Leslie system of two-dimensional compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity \({|\nabla\mathbf{d}|^2\mathbf{d}}\) in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.