一类带有记忆项的双阻尼σ-发展方程解的整体存在性和爆破

本文研究一类带有记忆项的双阻尼σ-发展方程的柯西问题.借助方程线性问题的衰减估计,利用压缩映像原理证得小初值问题解的整体存在性.同时考虑初值积分为正的情形,利用检验函数方法得到解的爆破以及生命跨度上界的估计.推广了带有双阻尼项的σ-发展方程的有关结论.     

竞争型反应扩散方程的正解与时间最优控制

本文研究了一类竞争型反应扩散方程的时间最优控制问题的存在性问题.利用上下解的方法,得到了反应扩散方程正解的存在性.利用Kakutani不动点定理,证明了反应扩散方程关于正轨迹的局部精确能控性,并据此证明了时间最优控制的存在性.     

谱问题及发展方程族的等价性

本文讨论了四对谱问题、相应保谱发展方程族以及它们 Lax表示的等价性 ,并利用这种等价关系导出了一些方程的 Lax表示 .     

一类具有排斥型奇性的中立型Li\'{e}nard方程周期正解的存在性

利用Mawhin重合度拓展定理, 研究一类具有排斥型奇性的中立型Li\''{e}nard方程周期问题. 在强奇性条件下, 获得周期正解存在性的新结果. 本文允许方程在无穷远点具有不定型奇性, 改进了已有文献中的相关结论.     

具有幂函数反应项的分数阶多孔介质方程解的爆破性

本文研究了一类具有幂函数反应项的分数阶多孔介质方程Dirichlet边值问题解的爆破性.首先,由于分数阶Laplace算子的非局部性,利用Caffareli-Silvestre扩展方法将非局部的原问题等价地转化为具有动力边界条件的局部椭圆型方程定解问题.然后,在此基础上,通过凹函数法得到局部解的爆破性;最后,利用全局解的一致有界性,得到方程全局解的长时间渐近性态.     

具有未知源的某些非线性伪抛物型方程的反问题

考 虑具有未 知源项的 某些非 线性伪 抛物 型方程 的反演 问题． 首先 将伪抛 物型 方程初 边值问 题化为非线 性发展方 程 Ｃｏｕｃｈ ｙ 问题,然 后,利用半 群理论,论 证发展 方程反问 题解的存 在唯一 性,最后, 利用不 动点方法得到 伪抛物型方程反 问题的可解性     

一类GBBM方程的柯西问题

研究一类非线性拟抛物方程的柯西问题.首先利用与原方程等价的积分方程以及压缩映像原理得到了问题的局部W~(k,p)解的存在性,而后在某些假设下得到了问题W~(k,p)解的整体存在性,最后利用凸性方法得到了解的有限时间内爆破.     

R2中含临界位势的非线性椭圆问题

考虑R²中的含临界位势的非线性椭圆方程齐次Dirichlet问题. 通过建立一常数为最佳的含权不等式, 确定了临界位势, 并讨论了含临界位势的Laplace方程特征值问题. 通过建立含一个奇点的解的Pohozaev型恒等式并结合Cauchy-Kovalevskaya定理得到了含临界位势非线性椭圆型方程有奇点的解的不存在性结果. 此外, 利用山路定理和特征值的性质得到了这一问题多重解的存在性的一系列结果.     

非线性发展方程混合问题解的爆破性质

讨论了具有第三类非线性边界条件的非线性发展方程的混合问题,并在已知函数满足某些假设的条件下,利用抛物型方程的最大值原理和凸性方法,证明了该问题的解在有限时间内爆破.     

非自治Caputo分数阶发展方程弱解的适定性与不变集

该文分析一族含有依赖时间参数t线性算子的时间分数阶非自治发展方程,利用Lions表示定理,得到了弱解适定性的充分条件;基于正交投影,建立了时间分数阶发展方程弱解的不变性准则.该文所研究方程中的算子是依赖时间的.     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

An inverse coefficient problem with nonlinear parabolic equation

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.     

Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhge?m–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.     

Controllability for a Nonlinear Abstract Evolution Equation

We prove a theorem on the local controllability of a system described by a nonlinear evolution equation in Banach space when the control is a multiplier on the right-hand side. We obtain sufficient conditions on the size of the neighborhood from which we can take the function from the overdetermination condition so that the inverse problem is uniquely solvable.     

Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.     

On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum

We consider the sum of the Sturm-Liouville operator and a convolution operator. We study the inverse problem of reconstructing the convolution operator from the spectrum. This problem is reduced to a nonlinear integral equation with a singularity. We prove the global solvability of this nonlinear equation, which permits one to show that the asymptotics of the spectrum is a necessary and sufficient condition for the solvability of the inverse problem. The proof is constructive.     

Generating operators for integrable nonlinear evolution equations

For linear problems which are associated with known, exactly integrable nonlinear evolution equations, one gives the corresponding integrodifferential Λ-operators. Relative to the expansions with respect to the elgenfunctions of Λ-operators, the method of the inverse scattering problem can be considered as the analog of the Fourier transform of linear problems, while the Λ-operators are the analogues of the differentiation operator. One considers the equations: Koteweg-de Vries, the nonlinear Schrödinger equations, the nonlinear Schrödinger equations with a derivative, the system of three waves, the matricial analog of the KdV equation, the Toda chain equation.     

On Nonlinear Kelvin-Helmholtz Waves Near Direct Resonance

The evolution equation for the nonlinear Kelvin-Helmholtz wave envelope with its carrier wavenumber near direct resonance is formulated directly by using the nonlinear dispersion relation. The stability of a wavetrain is examined, and the long-time evolution for an arbitrary initial condition is studied through inverse scattering transforms.     

A Nonlinear Difference Scheme and Inverse Scattering

A nonlinear partial difference equation is obtained and solved by the method of inverse scattering. In a certain continuum limit it is shown how this equation approximates the nonlinear Schrodinger equation and a related nonlinear differential-difference equation. At all times the solutions can be compared, and the scheme is shown to be convergent. These ideas apply to other nonlinear evolution equations as well.