Infinitely many solutions of p-Laplacian equations with limit subcritical growth

We discussed a class of p-Laplacian boundary problems on a bounded smooth domain,the nonlinearity is odd symmetric and limit subcritical growing at infinite.A sequence of critical values of the variational functional was constructed after the general- ized Palais-Smale condition was verified.We obtain that the problem possesses infinitely many solutions and corresponding energy levels of the functional pass to positive infinite. The result is a generalization of a similar problem in the case of subcritical.     

Existence and nonexistence of positive solutions of semilinear elliptic equation with inhomogeneous strong Allee effect

In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.     

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term简

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.     

Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere

We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is bounded and that the solutions are monotone decreasing. The upper bound, the value of the solution at zero, can be approximated analytically in terms of the physical parameters. We obtain solutions to the boundary value problem, using both the Taylor series （which work well for weak nonlinearity） and the b-expansion method （valid for strong nonlinearity）. From here, we are able to deduce the qualitative behavior of the solution profiles with a change in any one of the physical parameters.     

On the uniqueness of boundary integral equation for the exterior Helmholtz problem

From the point of view of energy analysis,the cause that the uniqueness of theboundary integral equation induced from the exterior Helmholtz problem does not hold isinvestigated in this paper.It is proved that the Sommerfeld’s condition at the infinity ischanged so that it is suitable not only for the radiative wave but also for the absorptive wavewhen we use the boundary integral equation to describe the exterior Helmholtz problem.Therefore,the total energy of the system is conservative.The mathematical dealings toguarantee the uniqueness are discussed based upon this explanation     

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method (LBIEM),singularities in the local boundary integrals need to be treated specially. In the current paper,local integral equations are adopted for the nodes inside the domain trod moving least square approximation (MLSA) for the nodes on the global boundary,thus singularities will not occur in the new al- gorithm.At the same time,approximation errors of boundary integrals are reduced significantly.As applications and numerical tests,Laplace equation and Helmholtz equa- tion problems are considered and excellent numerical results are obtained.Furthermore, when solving the Hehnholtz problems,the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions.Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.     

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method.The invariance of some sets under the flow of these problems and the vac- uum isolation of solutions are obtained by introducing a family of potential wells.Then the threshold result of global existence and nonexistence of solutions are given.Finally, the problem with critical initial conditions are discussed.     

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems

This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/planesymmetric acoustic wave problems.The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only.Moreover,a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived,and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating,translating and saving the multipole/local expansion coefficients of the image domain.The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems.As for exterior acoustic problems,the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method.Details on the implementation of the present method are described,and numerical examples are given to demonstrate its accuracy and efficiency.     

Reflection and radiation of a wave system at the open end of a submerged elastic pipe

The reflection and radiation of a wave system at the open end of a submerged semi-infinite elastic pipe are studied. This wave system consists of a flexural wave in the pipe, an acoustic surface wave in the fluid exterior to the pipe and an acoustic wave in the pipe’s interior. Fourier transform techniques are used to formulate this semi-infinite geometry problem rigorously as a Wiener-Hopf type equation. An approximate solution is obtained by using a perturbation method in which the ratio of the massdensities of the fluid and the pipe material is regarded as a small parameter. The calculation of the reflection coefficient is emphasized, and the polar plots of the radiation coefficient are also presented.     

REFLECTION AND RADIATION OF A WAVE SYSTEM AT THE OPEN END OF A SUBMERGED ELASTIC PIPE

The reflection and radiation of a wave system at the open end of a submerged.semi-infinite elastic pipe are studicd.This wave system consists of a flexural wave in the pipe,anacoustic surface wave in the fluid exterior to the pipe and an acoustit wave in the pipe’sinterior.Fourier transfrom techniques are used to formulate this semi-infinite geometryproblem rigorously as a Wiener-Hopf type equation.An approximate solution is obtainedby using a perturbation method in which the ratio of the massdensities of the fluid and thepipe material is regarded as a small parameter.The calculation of the reflection coefficientis emphasized,and the polar plots of the radiation coefficient are also presented.     

Existence and Asymptotic Behavior of Solutions to Semilinear Wave Equations with Nonlinear Damping and Dynamical Boundary Condition

Our aim in this article is to study a nonautonomous semilinear wave equation with nonlinear damping and dynamical boundary condition. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Then, by deriving an appropriate Lyapunov energy, we show that if the exponent in the ?ojasiewicz-Simon inequality is large enough (depending on the damping), then weak solutions converge to equilibrium.     

Diffusion-Induced Blowup in a Nonlinear Parabolic System

A two-component semilinear parabolic system on a bounded domain with Neumann boundary conditions is studied. It is shown that for a certain kind of nonlinearity, the blowup of solutions may occur when the diffusion coefficients are not equal, though the corresponding ODE possesses a globally stable equilibrium.     

Variational Methods in Semilinear Wave Equation with Nonlinear Boundary Conditions and Stability Questions

We present variational approach to the semilinear equation of the vibrating string \(x_{tt}(t,y)-{\Delta } x(t,y)+l(t,y,x(t,y))=0\) in bounded domain and certain type of nonlinearity on the boundary. To this effect we derive new dual variational method. Next the question of stability of solutions with respect to initial conditions is discussed.     

A Remark on the Convergence of Global and Bounded Solutions for a Semilinear Wave Equation on ∝ N

We give a new and easier proof for the result of Feireisl [Fe1] concerning the convergence of global and bounded solutions of the wave equation with small initial energy. We also prove that the convergence takes place with an exponential decay.     

Radial Symmetry for Elliptic Boundary-Value Problems on Exterior Domains

By the Alexandroff-Serrin method [2, 14] of moving hyperplanes we obtain radial symmetry for the domain and the solutions of on an exterior domain , subject to the overdetermined boundary conditions , on , at and in . In particular, the following conjecture from potential theory due to P. Gruber (cf. [11, 8]) is proved: Let or be a bounded smooth domain with a constant source distribution on and let be the induced single-layer potential. If is constant in , then is a ball. (Accepted January 29, 1996)     

波浪与外圆弧开孔壁双圆筒柱的相互作用

应用比例边界有限元法(SBFEM)研究了短峰波与圆筒外接圆弧开孔结构物的相互作用. 求解时将外接圆弧延伸构建一个虚拟圆, 该圆的孔隙影响系数可由矩阵G_0统一进行表达. 整个流场可划分成一个有限域和一个无限域. SBFEM只需对虚拟圆边界进行离散, 使空间维数降低一阶, 在圆的半径方向保持解析, 并且无限域处的辐射边界条件能自动满足. 利用变分原理推导SBFEM方程, 有限域和无限域分别采用贝塞尔函数和汉克尔函数作为基函数来求得对应域的解. 将计算结果与解析解和其他数值方法进行了比较, 验证了该方法是一种用很少单元便能得到精确结果的高效算法. 进一步研究了诸如短峰波波向、结构的几何、材料参数等因素对结构所受波浪载荷及绕射波轮廓的影响, 并进行了分析.      

Long-Time Behavior of a Coupled Heat-Wave System Arising in Fluid-Structure Interaction

This paper is devoted to the analysis of the long-time behavior of a coupled wave-heat system in which a wave and a heat equation evolve in two bounded domains, with natural transmission conditions at a common interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. This model may be viewed as a simplified version of linearized models which arise in fluid-structure interaction. First, we show the strong asymptotic stability of solutions to this system. Then, based on the construction of ray-like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface, we show the lack of uniform decay in general domains. Further, we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain. Finally, in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface. We also analyze the difficulty there is to extend this result to the more natural transmission conditions.     

Decay Rates for the Three‐Dimensional Linear System of Thermoelasticity

. We consider the two and three‐dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as . First of all, by a decoupling method, we reduce the problem to an observability inequality for the Lamé system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component. We show that when the domain is convex, the decay rate is never uniform. In fact, the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary. We also show that, in three space dimensions, the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement. In two space dimensions we prove a sufficient (and almost necessary) condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays, under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents. We also give an example, due to D. Hulin, in which these geometric properties hold. In three space dimensions we indicate (without proof) how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold. In two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite‐dimensional subspace of solutions except when the domain is a ball or an annulus. Finally we discuss some closely related controllability and spectral issues. (Accepted May 14, 1998)