一类非线性Schrdinger方程的整体自相似解

本文应用调和分析的方法研究了一类非线性Schrodinger方程Cauchy问题整体自相似解的存在唯一性．     

一类非线性Schr(o)dinger方程的整体自相似解

本文应用调和分析的方法研究了一类非线性Sehrodinger方程Cauchy问题整体自相似解的存在唯—性．     

弱齐次空间中半线性抛物型方程的Cauchy问题及其在Naiver-Stokes方程中的应用

致力于研究弱齐次空间中半线性抛物型方程的Cauchy问题. 通过引入五元容许簇、相容空间并建立线性抛物型方程解的时空估计, 给出了构造局部温和解的一种方法. 借此证明了弱齐次空间中半线性抛物型方程的Cauchy问题的局部适定性, 与此同时, 获得了小初值情形下的整体适定性. 进而, 研究了半线性抛物型方程的Cauchy问题在C_σ,s,p中解的正则性. 作为应用, 获得了Naiver-Stokes方程的Cauchy 问题在弱齐次Sobolev 空间中的适定性.     

高阶非线性Schrdinger方程的自相似解

研究非线性项的形式为|u|~pu,p>0的2m阶非线性Schrdinger方程的自相似解.利用scaling和压缩映象原理证明了当初值满足一定条件时Cauchy问题解的整体存在性,据此给出了当初值的形式为U(x/(|x|))|x|~(-(2m)/p)时,自相似解的存在性.     

高阶非线性Schr(o)dinger方程的自相似解

研究非线性项的形式为|u|pu,p>0的2m阶非线性Schr(o)dinger方程的自相似解.利用scaling和压缩映象原理证明了当初值满足一定条件时Cauchy问题解的整体存在性,据此给出了当初值的形式为U(x/|x|)|x|-2m/p时,自相似解的存在性.     

一类非线性Schrodinger方程的整体解和自相似解

对于α的某一取值范围,应用广义Strichartz不等式和压缩映射原理研究了初值在弱Lp空间中足够小的条件下,非线性Schr(o)dinger方程Cauchy问题整体解和自相似解的存在性.     

空时估计方法及一类非线性抛物方程的解

在形如BC( [0 ,T) ;L^p)及L^q( 0 ,T ;L^p)中研究了非线性抛物型方程的Cauchy问题和初边值问题 .类同于波动方程及色散波方程 ,首先对线性抛物方程给出了空时估计 ,进而利用空时估计方法给出了一系列的非线性估计 .借助于Banach不动点定理及通常的迭代技术 ,当 φ(x)∈L^r 时 ,构造了非线性抛物方程在BC( [0 ,T) ;L^p)和Lq( 0 ,T ;L^p)的局部解的存在唯一性 ,这里 ( p ,q ,r)是容许三元簇 .进而 ,对临界增长情形 ,证明了当初值函数充分小时 ,解的整体存在性 .     

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。     

关于抽象泛函微分方程与乘积空间上的非线性半群

本文研究Banach空间x上的非线性自治泛函微分方程的Cauchy问题x(t)=F(x_t)a.e.t≥0,x_0=φ,x(0)=η,(*)这里初值(η,φ)∈X×L~1(-r,0;X),r>0假定F是L~1→X的Lipschitz连续的条件下,通过对乘积空间X×L~1上方程(*)相关的一个极大增生算子及其所生成的非线性半群的研究,得到了问题的解的存在唯一性及其表示。在非线性抽象泛函微分方程的研究中,本文首次将Cauehy问题的讨论推进到具不连续初值函数的情形。     

一类具阻尼非线性波动方程的Cauchy问题

本文研究一类具阻尼非线性波动方程Cauchy问题整体广义解和整体古典解的存在唯一性,并用凸性方法给出解爆破的充分条件.     

Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data n the homogeneous spaces.We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in C(σ,s,p)and L~q([O,T);H~(s,p)by introducing the concept of both admissiblequintuptet and compatible space and establishing estblishing time-space estimates for solutions to the linear parabolic typeequations For the small data,we prove that these results can be extended globally in time. We also study the     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Vortex Filament Solutions of the Navier-Stokes Equations

We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions that are locally approximately self-similar. © 2023 Wiley Periodicals LLC.     

Asymptotic behavior of solutions of a free boundary problem modeling multi-layer tumor growth in presence of inhibitor

In this paper we study well-posedness and asymptotic behavior of solution of a free boundary problem modeling the growth of multi-layer tumors under the action of an external inhibitor. We first prove that this problem is locally well-posed in little Hölder spaces. Next we investigate asymptotic behavior of the solution. By computing the spectrum of the linearized problem and using the linearized stability theorem, we give the rigorous analysis of stability and instability of all stationary flat solutions under the non-flat perturbations. The method used in proving these results is first to reduce the free boundary problem to a differential equation in a Banach space, and next use the abstract well-posedness and geometric theory for parabolic differential equations in Banach spaces to make the analysis.     

Self-similar Solutions of Nonlinear Elasticity and Poroelasticity Equations

Recent advances in nonlinear wave propagation in elastic and porous elastic (poro-elastic) material have presented new nonlinear evolutionary equations. The derivation of these equations in three-dimensional space is based on the semilinear Biot theory. The nonlinear elastodynamic equations are derived form the more general model of poro-elastodynamic using consistency arguments. For simplicity, we discuss and carry out the analysis for the nonlinear elastic model. It is found in this article that the methods of symmetry groups and self-similar solutions can furnish solutions to the nonlinear elastodynamic wave equation. It is also found that these models lead to shock wave development in finite time. Necessary conditions for the existence of the solution are given and well-posedness of the Cauchy problem is discussed.     

On a nonlinear heat equation with functional dependence

We reduce the Cauchy problem for a heat equation with the nonlinear right-hand side which depends on some functionals to an equivalent integral equation. Considering mainly Banach spaces of continuous, bounded and exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We give a counterexample which shows that the Lipschitz condition is, in general, insufficient for the Cauchy problem with unbounded data and with functional dependence to guarantee an existence result     

Regular and self-similar solutions of nonlinear Schrödinger equations

We study the Cauchy problem for the nonlinear Schrödinger equations with nonlinear term |u|^ou. For some admissible α we show the existence of global solutions and we calculate the regularity of those solutions. Also we give some necessary conditions and some sufficient conditions on initial data for the existence of self-similar solutions.     

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.