New results of some existence theorems on nonlinear boundary value problems

IntroductionSeveralpapershavebeenwrittenaboutequationwhereLisalineardifferentialexpression,NacontinuouslyFrechetdifferentialoperatorsuchthatL N'(u)satisfiescertainconditions,andFanoperator,usuallywithboundedrange.AconcreteexampleiswhereLisaseconddifferentialoperator,gsatisfiessomeconditionsrelatingtoL,andfisbounded.AmainlyusedtechniqueisSchauderfixedpointtheorem.Byestablishingapriorbound,itisnaturaltoobtaintheexistCnceofthesolutions.Butthismethodcannotgettheuniquenessofthesolution,noristh…     

Pseudo Almost Periodic Solutions to a Class of Semilinear Differential Equations

This paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear differential equations involving the algebraic sum of two (possibly noncommuting) densely defined closed linear operators acting on a Hilbert space. Sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those semilinear equations are obtained. An erratum to this article is available at .     

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.     

ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS, WITH UNBOUNDED STOPPING TIMES AS TERMINAL AND WITH NON-LIPSCHITZ COEFFICIENTS, AND PROBABILISTIC INTERPRETATION OF QUASI-LINEAR ELLIPTIC TYPE INTEGRO- DIFFERENTIAL EQUATIO

Ｉｔ’ｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｓｏｌｕｔｉｏｎｓｏｆｂａｃｋｗａｒｄｓｔｏｃｈａｓｔｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ(ＢＳＤＥｓ)ｐｌａｙａｎｉｍｐｏｒｔａｎｔｒｏｌｅｉｎｔｈｅｆｉｎａｎｃｉａｌｍａｒｋｅｔ(ＳｅｅＲｅｆ.[1]).ＷｉｔｈｕｎｂｏｕｎｄｅｄｓｔｏｐｐｉｎｇｔｉｍｅｓτａｓｔｅｒｍｉｎａｌｓｕｎｄｅｒｔｈｅＬｉｐｓｃｈｉｔｚｃｏｎｄｉｔｉｏｎｓａｎｄｃｏｎｄｉｔｉｏｎｔｈａｔｔｅｒｍｉｎａｌｓｓａｔｉｓｆｙＥ｜ξ｜2ｅＫ(τ∧Ｔ)≤ｃ0<∞,0≤Ｔ<∞ｏｒξ=0,Ｒｅｆｓ…     

On the Euler Equation in an Unbounded Domain of the Plane

We discuss existence and uniqueness of solutions to the Euler Equation in an unbounded domain of the plane. We only assume the vorticity to be bounded, whereas in this kind of problems assumptions on its decreasing at infinity are usually made. The solution is obtained as limit of solutions to problems with compactly supported data. The existence of such limit physically means that the effects of far away fluid particles on the local evolution is negligible.     

On solutions of backward stochastic differential equations with jumps, with unbounded stopping times as terminal and with non-lipschitz coefficients, and probabilistic interpretation of quasi-linear elliptic type integro-differential equations

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non-Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi-linear elliptic type integro-differential equations is obtained. Foundation item: the National Natural Science Foundation of China (79790130); the Foundation of Zhongshan University Front Research Biography: Situ Rong (1935 −)     

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]…     

Problem of periodic solutions for neutral functional differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＣ(ｋ- 1)2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓ (ｋ -1 )＿ｔｈｏｒｄｅｒｃｏｎｔｉｎｕｏｕｓｄｉｆｆｅｒｅｎｔｉａｂｌｅａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　Ｃ2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　‖ｈ(ｔ)‖ =ｓｕｐｔ∈ [0 ,2π] ｜ｈ(ｔ) ｜ ,　　‖ｈ(ｔ)‖Ｐｋ- 1 =ｍａｘ‖ｈ(ｔ)‖ ,‖ｈ′(ｔ)‖ ,… ,‖ｈ(ｋ- 1) (ｔ)‖ ,　　ｘ(ｍ) (ｔ+ ·) (θ) =ｘ(ｍ) (ｔ+θ)　　θ∈Ｒ (ｍ =0 ,1 ,2 ,… ,ｋ-1 ) .Ｃｌｅａｒｌｙ ,ｘ(ｍ) (ｔ + ·) ∈Ｃ2π, …     

RESEARCH OF THE PERIODIC MOTION AND STABILITY OF TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｓｙｓｔｅｍｏｆｎｏｎｌｉｎｅａｒｏｓｃｉｌｌａｔｉｎｇ ,ｐｅｒｉｏｄｉｃｍｏｔｉｏｎｉｓｏｆｐｒｉｍｅｉｍｐｏｒｔａｎｃｅ .Ｂｕｔｅｘｉｓｔｅｎｃｅｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｉｓａｖｅｒｙｄｉｆｆｉｃｕｌｔｑｕｅｓｔｉｏｎ .Ｌｕｃｋｉｌｙｔｈｅｒｅｅｘｉｓｔｓｏｍｅｋｉｎｄｓｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｉｎａｃｔｕａｌｐｈｙｓｉｃａｌｓｙｓｔｅｍｓ .Ｔｈｅｒｅｆｏｒｅ ,ｗｅｕｓｕａｌｌｙｃｏｎｃｅｎｔｒａｔｅｄｏｕｒａｔｔｅｎｔｉｏｎｏｎｔ…     

Abstract Cauchy problem for fractional differential equations

In this paper, a generalized Darbo’s fixed-point theorem associated with Hausdorff measure of noncompactness is established. Then we apply this new variant fixed-point theorem to study some fractional differential equations in Banach spaces via the technique of measure of noncompactness. Many novel existence and uniqueness results for solutions are obtained under the more general conditions.     

Nonstationary ideal incompressible fluid flows: Conditions of existence and uniqueness of solutions

The problem of formulating minimal conditions on input data that can guarantee the existence and uniqueness of solutions of the boundary value problems describing non-one-dimensional ideal incompressible fluid flow is considered using as an example the initial boundary value problem in a space-time cylinder constructed on a bounded flow domain with the nonpenetration condition on its boundary (which corresponds to fluid flow in a closed vessel). The existence problems are considered only for plane flows, and the uniqueness issues for three-dimensional flows as well. The required conditions are obtained in the form of conditions specifying that the vorticity belongs to definite functional Orlicz spaces. The results are compared with well-known results. Examples are given of admissible types of singularities for which the obtained results are valid, which is a physical interpretation of these results. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 130–145, July–August, 2008.     

A Geometrical Approach to the Study¶of Unbounded Solutions¶of Quasilinear Parabolic Equations

In this article, we are interested in the existence and uniqueness of solutions for quasilinear parabolic equations set in the whole space ? ^N . We consider, in particular, cases when there is no restriction on the growth or the behavior of these solutions at infinity. Our model equation is the mean-curvature equation for graphs for which Ecker and Huisken have shown the existence of smooth solutions for any locally Lipschitz continuous initial data. We use a geometrical approach which consists in seeing the evolution of the graph of a solution as a geometric motion which is then studied by the so-called “level-set approach”. After determining the right class of quasilinear parabolic PDEs which can be taken into account by this approach, we show how the uniqueness for the original PDE is related to “fattening phenomena” in the level-set approach. Existence of solutions is proved using a local L ^∞ bound obtained by using in an essential way the level-set approach. Finally we apply these results to convex initial data and prove existence and comparison results in full generality, i.e., without restriction on their growth at infinity.     

Existence and uniqueness of periodic solutions for fourth-order nonlinear periodic systems

In this paper, we study the existence, uniqueness and stability of the periodic solutions for fourth-order nonlinear nonhomogeneous periodic systems with slowly changing coefficients by using the method of Liapunor Function. We obtain some sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions of these systems and estimate the extent to which the coefficients are allowed to change.     

2T-PERIODIC SOLUTION FOR m ORDER NEUTRAL TYPE DIFFERENTIAL EQUATIONS WITH TIME DELAYS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｎｅｕｔｒａｌｔｙｐｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎａｒｅｗｉｄｅｌｙｕｓｅｄｉｎｂｉｏｌｏｇｙ ,ｐｈｙｓｉｃｓａｎｄｃｈｅｍｉｓｔｒｙ .Ｔｈｅｓｔｕｄｙｏｆｔｈｅｎｅｕｔｒａｌｅｑｕａｔｉｏｎｈａｓｉｍｐｏｒｔａｎｔｓｉｇｎｉｆｉｃａｎｃｅｉｎｂｏｔｈｔｈｅｏｒｙａｎｄａｐｐｌｉｃａｔｉｏｎ .Ｉｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆｎｅｕｔｒａｌｅｑｕａｔｉｏｎａｓ∑ｍｉ=0[ａｉｘ(ｍ-ｉ…     

Global Existence and Large Time Behavior of Strong Solutions to the 2-D Compressible Nematic Liquid Crystal Flows with Vacuum

This paper is concerned with the strong solutions to the Cauchy problem of a simplified Ericksen-Leslie system of compressible nematic liquid crystals in two or three dimensions with vacuum as far field density. For strong solutions, some a priori decay rate (in large time) for the pressure, the spatial gradient of velocity field and the second spatial gradient of liquid crystal director field are obtained provided that the initial total energy is suitably small. Furthermore, with the help of the key decay rates, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) in two spatial dimensions.     

On two-dimensional large-scale primitive equations in oceanic dynamics(Ⅱ)

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially.Here the depth of the ocean is positive but not always a constant.By Faedo-Galerkin method and anisotropic inequalities,the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained.Moreover,by studying the asymptotic behavior of solutions for the above problem,the energy is exponential decay with time is proved.     

On two-dimensional large-scale primitive equations in oceanic dynamics （Ⅰ）

The initial boundary value problem for the two-dimensional primitive equations of large scale oceanic motion in geophysics is considered. It is assumed that the depth of the ocean is a positive constant. Firstly, if the initial data are square integrable, then by Fadeo-Galerkin method, the existence of the global weak solutions for the problem is obtained. Secondly, if the initial data and their vertical derivatives are all square integrable, then by Faedo-Galerkin method and anisotropic inequalities, the existerce and uniqueness of the global weakly strong solution for the above initial boundary problem are obtained.     

On two-dimensional large-scale primitive equations in oceanic dynamics （Ⅱ）

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially. Here the depth of the ocean is positive but not always a constant. By Faedo-Galerkin method and anisotropic inequalities, the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained. Moreover, by studying the asymptotic behavior of solutions for the above problem, the energy is exponential decay with time is proved.     

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele�CShaw Cell

We study the radial movement of an incompressible fluid located in a Hele–Shaw cell rotating at a constant angular velocity in the horizontal plane. Within an analytic framework, local existence and uniqueness of solutions is proved, and it is shown that the unique rotationally invariant equilibrium of the flow is unstable. There are, however, other time-independent solutions: using surface tension as a bifurcation parameter we establish the existence of global bifurcation branches consisting of stationary fingering patterns. The same results can be obtained by fixing the surface tension while varying the angular velocity. Finally, it is shown that the equilibria on a global bifurcation branch converge to a circle as the surface tension tends to infinity, provided they stay suitably bounded.     

Invariant submodels of the Westervelt model with dissipation

We study three-dimensional Westervelt model of nonlinear hydroacoustics with dissipation. We received all its invariant submodels. With the help of invariant solutions, we explored some wave processes, specifying their physical meaning. The boundary value problems describing these processes are reduced to the nonlinear integro-differential equations. We established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Also we considered the invariant solutions of rank 2 and 3. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters.