Solutions of the autonomous Kirchhoff type equations in RN

Recently, the existence, the multiplicity and the concentration of solutions for the Kirchhoff type equations or systems have been extensively established by a number of authors. Such problems contain a nonlocal term which implies that the equation or system is no longer a pointwise identity. Thus, some researchers think that the nonlocal phenomenon will cause some mathematical difficulties. In this note, however, we will give a simple transformation so that the solutions of the autonomous Kirchhoff type equation or system are easily obtained by using the known solutions of the corresponding local equation or system. In particular, some qualitative properties of solutions for the local problems are also inherited.     

Bagley-Torvik型分数阶微分方程和包含非局部积分边值问题

In this article,we consider the Bagley-Torvik type fractional differential equation ~cD~(ν1) l(t)-a~cD~(ν2) l(t) = g(t, l(t)) and differential inclusion ~cD~(ν1) l(t)-a~cD~(ν2) l(t) ∈ G(t, l(t)),t ∈(0, 1) subjecting to l(0) = l_0,and■ ds where 1 ν_1 ≤ 2, 1 ≤ν_2 ν_1,0 ω≤ 1, χ = ν_1-ν_2 0, a, λ′are given constants. By using Leray-Schauder degree theory and fixed point theorems, we prove the existence of solutions. Our results extend the existence theorems for the classical Bagley-Torvik equation and some related models.     

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[ \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.     

Ground state and nodal solutions for a class of biharmonic equations with singular potentials

In this paper, we are concerned with a class of fourth order elliptic equations of Kirchhoff type with singular potentials in $\mathbb{R}^{N}.$ The existence of ground state and nodal solutions are obtained by using variational methods and properties of Hessian matric. Furthermore, the "energy doubling" property of nodal solutions is also explored.     

On the Lie Algebras,Generalized Symmetries and Darboux Transformations of the Fifth-Order Evolution Equations in Shallow Water

By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave(OWW) equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions{Xiβ}i=1,2,···,nβ =1,2,···,N depending on a finite number of partial derivatives of the nonlocal variables vβand a restriction ∑iα,β(? ξi/?vβ)2+(?ηα/?vβ)2≠0, i.e.,∑i,α,β(?Gα/?vβ)2≠0. Furthermore,the authors investigate some structures associated with the Olver water wave(AOWW)equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, It and Caudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, the symmetries are applied to investigate the initial value problems and Darboux transformations.     

Convergence results for a class of nonlinear fractional heat equations

In this article we study various convergence results for a class of nonlinear fractional heat equations of the form $\left\{ \begin{gathered} u_t (t,x) - \mathcal{I}[u(t, \cdot )](x) = f(t,x),(t,x) \in (0,T) \times \mathbb{R}^n , \hfill \\ u(0,x) = u_0 (x),x \in \mathbb{R}^n , \hfill \\ \end{gathered} \right.$ where I is a nonlocal nonlinear operator of Isaacs type. Our aim is to study the convergence of solutions when the order of the operator changes in various ways. In particular, we consider zero order operators approaching fractional operators through scaling and fractional operators of decreasing order approaching zero order operators. We further give rate of convergence in cases when the solution of the limiting equation has appropriate regularity assumptions.     

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.     

On the asymptotic behavior of solutions of certain integro-differential equations

The authors present conditions under which every positive solution $x(t)$ of the integro--differential equation $x^{\prime \prime }(t)=a(t)+\int_{c}^{t}(t-s)^{\alpha-1}[e(s)+k(t,s)f(s,x(s))]ds, \quad c>1, \ \alpha >0,$ satisfies $x(t)=O(tA(t))\textrm{ as }t\rightarrow \infty,$ i.e, $\limsup_{t\rightarrow \infty }\frac{x(t)}{tA(t)}<\infty, \textrm{where} \ A(t)=\int_{c}^{t}a(s)ds.$ From the results obtained, they derive a technique that can be applied to some related integro--differential equations that are equivalent to certain fractional differential equations of Caputo type of any order.     

A general study on random integro-differential equations of arbitrary order

Here the broad study is depending on random integro-differential equations (RIDE) of arbitrary order. The fractional order is in terms of $\psi$-Hilfer fractional operator. This work reveals the dynamical behaviour such as existence, uniqueness and stability solutions for RIDE involving fractional order. Thus initial value problem (IVP), boundary value problem (BVP), impulsive effect and nonlocal conditions are taken in account to prove the results.     

Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.     

一类耗散型电流体动力学方程组自相似解的渐近稳定性

主要考虑一类来源于电流体动力学中的由非线性非局部方程组耦合而成的耗散型系统的初值问题.利用Lorentz空间中广义L~p-L~q热半群估计和广义Hardy-Littlewood-Sobolev不等式,首先证明了该系统在Lorentz空间中自相似解的整体存在性和唯一性,然后建立了自相似解当时间趋于无穷时的渐近稳定性.因为Lorentz空间包含了具有奇性的齐次函数,因次上述结果保证了具有奇性的初值所对应的自相似解的整体存在性和渐近稳定性.     

An Inverse Coefficient Problem for an Integro-differential Equation

In this article we consider the inverse coefficient problem of recovering the function { ( x ) system of partial differential equations that can be reduced to a second order integro-differential equation $ -u_{xx} + c(x)u_{x} + d\phi (x)u-\gamma d\phi (x)\int _{0}^{t}e^{-\gamma (t-\tau )}u(x,\tau )\, d\tau = 0 $ with boundary conditions. We prove the existence and uniqueness of solutions to the inverse problem and develop a numerical algorithm for solving this problem. Computational results for some examples are presented.     

On ground state of fractional $p$-Kirchhoff equation involving subcritical and critical exponential growth

In this paper, we concern the existence of nontrivial ground state solutions of fractional $p$-Kirchhoff equation $$\left\{\begin{array}{ll} m\left(\|u\|^p\right) [(-\Delta)_p^su+V(x)|u|^{p-2}u] =f(x,u) \quad\text{in}\, \mathbb{R}^N, \vspace{0.2 cm}\\ \|u\|=\left(\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy +\int_{\mathbb{R}^N}V(x)|u|^pdx\right)^{\frac{1}{p}}, \end{array}\right.$$ where $m:[0,+\infty)\rightarrow [0,+\infty)$ is a continuous function, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator with $0相似文献   

On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order

In this paper, we prove the existence of solutions of a nonlocal boundary value problem for nonlinear integro-differential equations of fractional order given by $$ \begin{array}{ll} ^cD^qx(t) = f(t,x(t),(\phi x)(t),(\psi x)(t)), \quad 0 < t < 1,\\x(0) = \beta x(\eta), x'(0) =0, x''(0) =0, \ldots, x^{(m-2)}(0) =0, x(1)= \alpha x(\eta), \end{array}$$ where $${q \in (m-1, m], m \in \mathbb{N}, m \ge 2}$, $0< \eta <1$$ , and ${\phi x}$ and ${\psi x}$ are integral operators. The existence results are established by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented.     

Operational Calculus Approach to Nonlocal Cauchy Problems

Let Φ be a linear functional of the space ${\mathcal{C} =\mathcal{C}(\Delta)}$ of continuous functions on an interval Δ. The nonlocal boundary problem for an arbitrary linear differential equation $$ P\left(\frac{d}{d t}\right)y = F(t) $$ with constant coefficients and boundary value conditions of the form $$ \Phi\{\,y^{(k)}\} =\alpha_k,\,\,\,k = 0,\,1,\,2,\, \ldots,\,{\rm deg} P-1 $$ is said to be a nonlocal Cauchy boundary value problem. For solution of such problems an operational calculus of Mikusiński’s type, based on the convolution $$ (f*g)(t) = \Phi_\tau\, \left\{{\int\limits_\tau^t} f(t+\tau - \sigma)\,g(\sigma)\, d \sigma\, \right\}, $$ is developed. In the frames of this operational calculus the classical Heaviside algorithm is extended to nonlocal Cauchy problems. The obtaining of periodic, antiperiodic and mean-periodic solutions of linear ordinary differential equations with constant coefficients both in the non-resonance and in the resonance cases reduces to such problems. Here only the non-resonance case is considered. Extensions of the Duhamel principle are proposed.     

KIRCHHOFF TYPE EQUATIONS DEPENDING ON A SMALL PARAMETER

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

Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form(P_(a,b)){D~αu(x) + f(x, u(x)) = 0, x ∈(0, 1),u(0) = u(1) = 0, D~(α-3)u(0) = a, u(1) =-b,where 3 α≤ 4, Dαis the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) =-p(x)t~σ, with σ∈(-1, 1)and p being a nonnegative continuous function that may be singular at x = 0 or x = 1and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch¨auder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem(P_(0,0)).Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in(0, 1)×[0, ∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem(P_(a,b)), which behaves like the unique solution of the homogeneous problem corresponding to(P_(a,b)). Some examples are given to illustrate the existence results.     

Approximation Technique for Fractional Evolution Equations with Nonlocal Integral Conditions

We consider a class of fractional evolution equations with nonlocal integral conditions in Banach spaces. New existence of mild solutions to such a problem are established using Schauder fixed-point theorem, diagonal argument and approximation techniques under the hypotheses that the nonlinear term is Carathéodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional evolution equations with general nonlocal integral conditions. Finally, as a sample of application, the results are applied to a fractional parabolic partial differential equation with nonlocal integral condition. The results obtained in this paper essentially extend some existing results in this area.     

Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $\left\{\begin{array}{l}i\varepsilon_{t}+\varepsilon_{xx}=n\varepsilon+\alpha|\varepsilon|^{2}\varepsilon,\\n_{t}=(|\varepsilon|^{2})_{x}, x\in R.\end{array} \right.$ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $L$ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $L$. As the modulus of the Jacobian elliptic function $k\rightarrow 1$, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $\alpha=0$, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.