Solutions for the Kirchhoff type equations with fractional Laplacian

Due to the singularity and nonlocality of the fractional Laplacian, the classical tools such as Sturm comparison, Wronskians, Picard--Lindel\"{o}f iteration, and shooting arguments (which are all purely local concepts) are not{\ applicable} when analyzing solutions in the setting of the nonlocal operator $\left( -\Delta \right) ^{s}$. Furthermore, the nonlocal term of the Kirchhoff type equations will also cause some mathematical difficulties. The present work is motivated by the method of semi-classical problems which show that the existence of solutions of the Kirchhoff type equations are equivalent to the corresponding associated fractional differential and algebraic system. In such case, the existence of the fractional Kirchhoff equation can be obtained by using the corresponding fractional elliptic equation. Therefore some qualitative properties of solutions for the associated problems can be inherited. In particular, the classical uniqueness results can be applied to this equation.     

Existence results for some Kirchhoff-Carrier problems

Analyzing the viscoelastic problem for small vibrations of elastic strings, Kirchhoff and Carrier proposed two different models of nonlinear partial differential equations. By combining these two models, we deal here with some nonlocal hyperbolic problems that cover a large class of Kirchhoff and Carrier type problems. The existence of local solutions of degenerate problems as well as local and nonlocal solutions of nondegenerate problems is established. The proofs are based on the combination of the Schauder fixed point theorem with some asymptotic method.     

KIRCHHOFF TYPE EQUATIONS DEPENDING ON A SMALL PARAMETER

ＫＩＲＣＨＨＯＦＦＴＹＰＥＥＱＵＡＴＩＯＮＳＤＥＰＥＮＤＩＮＧＯＮＡＳＭＡＬＬＰＡＲＡＭＥＴＥＲ￥Ｐ．Ｄ＇ＡＮＣＯＮＡＳ．ＳＰＡＧＮＯＬＯ（ＤｅｐａｒｔｍｅｌｌｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＰｉｓａＵｎｉｖｅｒｓｉｔｙ！Ｐｉｓａ，Ｉｔａｌｙ）Ａｂｓｔ...     

Spectrum comparison for a conserved reaction-diffusion system with a variational property

We are dealing with a two-component system of reaction-diffusion equations with conservation of a mass in a bounded domain subject to the Neumann or the periodic boundary conditions. We consider the case that the conserved system is transformed into a phase-field type system. Then the stationary problem is reduced to that of a scalar reaction-diffusion equation with a nonlocal term. We study the linearized eigenvalue problem of an equilibrium solution to the system, and compare the eigenvalues with ones of the linearized problem arising from the scalar nonlocal equation in terms of the Rayleigh quotient. The main theorem tells that the number of negative eigenvalues of those problems coincide. Hence, a stability result for nonconstant solutions of the scalar nonlocal reaction-diffusion equation is applicable to the system.     

Dynamics of the logistic equation with two delays

We study the logistic equation with two delays. When studying its nonlocal dynamics, we obtain a condition for the existence and the asymptotics of a relaxation cycle. When studying the local dynamics, we show that the behavior of solutions of the original equation is determined by the structure of solutions of special families of nonlinear boundary value problems of parabolic and degenerate-parabolic type.     

一类分数阶Kirchhoff型方程非平凡解的存在性

利用临界点理论中的山路引理,研究一类分数阶Kirchhoff型方程在次临界增长条件下非平凡解的存在性,进一步统一和丰富了已有文献的相关结果.     

Prohibitions caused by nonlocality for nonlocal Boussinesq‐KdV type systems

It is found that two different celebrate models, the Korteweg de‐Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The nonlocal KdV equation can be derived in two ways, via the so‐called consistent correlated bang companied by the parity and time reversal from the local KdV equation and via the parity and time reversal symmetry reduction from a coupled local KdV system which is a two‐layer fluid model. The same model can be called as the nonlocal Boussinesq system if the nonlocality is changed as only one of parity and time reversal. The nonlocal Boussinesq equation can be derived via the parity or time reversal symmetry reduction from the local Boussinesq equation. For the nonlocal Boussinesq equation, with help of the bilinear approach and recasting the multisoliton solutions of the usual Boussinesq equation to an equivalent novel form, the multisoliton solutions with even numbers and the head on interactions are obtained. However, the multisoliton solutions with odd numbers and the multisoliton solutions with even numbers but with pursuant interactions are prohibited. For the nonlocal KdV equation, the multisoliton solutions exhibit many more structures because an arbitrary odd function of can be introduced as background waves of the usual KdV equation.     

Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities.     

Decay estimates for evolution equations with classical and fractional time-derivatives

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviours, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation.     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

Stabilization by memory effects: Kirchhoff plate versus Euler–Bernoulli plate

In this paper we study the asymptotic behavior of solutions of a dissipative coupled system where we have interactions between a Kirchhoff plate and an Euler–Bernoulli plate. The dissipative mechanism is given by memory terms that act either collaboratively (in both equations) or unilaterally (in only one equation). We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We found explicit decay rates that depend on the fractional exponents of the memory in each of the following cases: when the memory only acts in the Kirchhoff equation, or only in the Euler–Bernoulli equation, or in both. We also show that all decay rates found are the best. The results obtained are surprising for the following facts: in the collaborative case, the best decay rates of the system are given by the worst decay rates of the uncoupled equations, and in the unilateral case, we conclude that the memory effects in the Euler–Bernoulli equation dissipate the system more slowly than memory effects in the Kirchhoff equation.     

Signed and sign-changing solutions for a Kirchhoff-type problem involving the fractional p-Laplacian with critical Hardy nonlinearity

In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on mountain pass theorem and constrained minimization in Nehari sets.     

Positive and negative solutions of a class of nonlocal problems

The existence of nontrivial solutions of Kirchhoff type equations is an important nonlocal quasilinear problem. In this paper, still by using the invariant sets of descent flow, we obtain positive and negative solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows:

     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

一类非局部非线性扩散方程解的全局爆破

主要研究在Dirichlet边界条件或Neumann边界条件下的一类非局部非线性的扩散方程问题.在适当的假设下,证明解的存在性、唯一性、比较原则、以及解对初边值条件的连续依赖性,并就给定的初边值条件,证明解在有限时刻全局爆破.     

Nonexistence and existence of positive solutions for the Kirchhoff type equation

In this paper, we study the nonexistence and existence of positive solutions for the Kirchhoff type equation with nonlinearity having prescribed asymptotic behavior. Because of the structure of our equation, our nonexistence results have some special properties, and our conditions for positive solutions are also different from the existing results.     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

Ground state solutions for an indefinite Kirchhoff type problem with steep potential well

In this paper we study an indefinite Kirchhoff type equation with steep potential well. Under some suitable conditions, the existence and the non-existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.     

Existence and multiplicity of solutions for Kirchhoff type equations

In this paper, the existence and multiplicity of weak solutions are obtained for a class of Kirchhoff type problems with Dirichlet boundary value conditions by using the mountain pass theorem, the local linking theorem, the fountain theorem and the symmetric mountain pass lemma in critical point theory.     

On a reaction diffusion equation with nonlinear time‐nonlocal source term

In this article, we prove the local existence of a unique solution to a nonlocal in time and space evolution equation with a time nonlocal nonlinearity of exponential growth. Moreover, under some suitable conditions on the initial data, it is shown that local solutions experience blow‐up. The time profile of the blowing‐up solutions is also presented. Copyright © 2015 John Wiley & Sons, Ltd.