共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper we analyse the Gevrey well-posedness of the Cauchy problem for weakly hyperbolic equations of general form with time-dependent coefficients. The results involve the order of lower order terms and the number of multiple roots. We also derive the corresponding well-posedness results in the space of Gevrey Beurling ultradistributions. 相似文献
2.
In this paper, we prove that for non-effectively hyperbolic operators with smooth double characteristics exhibiting a Jordan
block of size 4 on the double manifold, the Cauchy problem is well-posed in the Gevrey 5 class, beyond the generic Gevrey
class 2 (see, e.g., [5]). Moreover, we show that this value is optimal, due to certain geometric constraints on the Hamiltonian
flow of the principal symbol. These results, together with results already proved, give a complete picture of the well-posedness
of the Cauchy problem around hyperbolic double characteristics. 相似文献
3.
We prove well-posedness results for the initial value problem of the periodic KdV equation as well as Kam type results in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential decay of Fourier coefficients, and ‘almost well posed’ in spaces with exponential decay of Fourier coefficients. 相似文献
4.
The aim of this note is to present the almost global well-posedness result for the Cauchy problem for the Kirchhoff equation with large data in Gevrey spaces. We also briefly discuss the corresponding results in bounded and in exterior domains. 相似文献
5.
Alessia Ascanelli 《Journal of Differential Equations》2005,217(2):305-340
The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C∞. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable. 相似文献
6.
本文研究了非线性柯西问题的适定性问题.利用经典的能量法和抽象柯西-柯瓦列夫斯卡娅定理,得到非线性柯西问题在Gevrey空间中是适定的.推广了已有文献在非线性柯西问题适定性方面的研究. 相似文献
7.
Ferruccio Colombini Tamotu Kinoshita 《Journal of Mathematical Analysis and Applications》2003,282(1):410-420
We prove that the Cauchy problem for a class of weakly hyperbolic equations satisfying a condition of finite order degeneration and having non-Lipschitz-continuous coefficients is well-posed in Gevrey spaces. 相似文献
8.
A. M. Biryukov 《Differential Equations》2017,53(10):1303-1317
We study the Cauchy problem for a system of complex linear differential equations in scales of spaces of functions of exponential type with an integral metric. Conditions under which this problem is well posed are obtained. These sufficient conditions are shown to be also necessary for the well-posedness of the Cauchy problem in the case of systems of ordinary differential equations with a parameter. 相似文献
9.
S. A. Aldashev 《Ukrainian Mathematical Journal》2000,52(5):673-679
We prove the well-posedness of the Cauchy, Goursat, and Darboux problems for multidimensional in-tegro-differential equations
of the hyperbolic type encountered in biology. 相似文献
10.
In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$ -regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the $C^\infty $ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation (which have been untreatable until now in these problems) as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms. 相似文献
11.
V. A. Litovchenko 《Differential Equations》2008,44(6):835-854
For a class of evolution systems of the parabolic type with unbounded coefficients, we study the properties of the fundamental solution matrices and establish the well-posed solvability of the Cauchy problem for these systems in spaces of distributions similar to Gevrey ultradistributions. For a subclass of such systems, we describe the maximal classes of well-posed solvability of the Cauchy problem. 相似文献
12.
Keisuke Uchikoshi 《Mathematische Nachrichten》2005,278(3):318-343
We consider local solvability of semilinear hyperbolic Cauchy problems for Gevrey functions. To obtain a general result, we define the notion of irregularities, and we give a criterion for the local solvability. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
13.
数学和物理中许多重要问题均可归结为算子在某些函数空间中的有界性质.奇异积分算子有界性质的研究是调和分析理论的核心课题之一,由此发展起来的各种方法和技巧已广泛应用于偏微分方程的研究.借助奇异积分算子在Lebesgue空间或Morrey型空间中建立的时空估计和半群理论,可以得到非线性色散方程在低阶Sobolev空间中Cauchy问题的适定性.本文首次定义一类单边振荡奇异积分算子并研究该类算子的经典加权有界性质.受经典交换子刻画理论的启发,本文首次引入Morrey空间的交换子刻画理论.利用不同于常规极大函数的方法得到两类象征函数在Morrey空间中的交换子刻画.以上结果为偏微分方程的研究提供了新的工具.最后,结合能量方法和数论知识,本文解决几类KdV型色散方程的适定性问题. 相似文献
14.
Summary. Edge Sobolev spaces are proposed as a main new tool for the investigation of weakly hyperbolic equations. The well-posedness
of the linear and semilinear Cauchy problem in the class of these edge Sobolev spaces is proved. An application to the propagation
of singularities for solutions to the semilinear problem is considered.
Received: October 3, 2000 Published online: December 19, 2001 相似文献
15.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》2001,332(4):311-314
The well-posedness of a Cauchy problem issue from an hyperbolic linear system is linked to the spectral properties of a real matrix pencil. It is known that such a problem is well posed in L2 if and only if the imaginary exponential of the pencil is bounded. We give a condition to have a bounded exponential when the eigenvalues don't have the same multiplicities. For pencils spanned by two 3×3 matrices, we prove that the exponential is bounded if and only if the pencil is analytically diagonable. 相似文献
16.
In this article we study the well-posedness of the initial value problem for quasi-linear weakly hyperbolic equations of second order. We obtain a sufficient condition for the Cauchy problem to be locally solvable in the class of smooth function. 相似文献
17.
We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed. 相似文献
18.
We shall consider the Cauchy problem for weakly hyperbolic equations of higher order with coefficients depending only on time. The regularities of the distinct characteristic roots and the multiple characteristic roots independently influence Gevrey well posedness of the Cauchy problem. 相似文献
19.
This paper is devoted to the well-posedness of abstract Cauchyproblems for quasi-linear evolution equations. The notion ofHadamard well-posedness is considered, and a new type of stabilitycondition is introduced from the viewpoint of the theory offinite difference approximations. The result obtained here generalizesnot only some results on abstract Cauchy problems closely relatedwith the theory of integrated semigroups or regularized semigroupsbut also the Kato theorem on quasi-linear evolution equations.An application to some quasi-linear partial differential equationof weakly hyperbolic type is also given. 2000 Mathematics SubjectClassification 34G20, 47J25 (primary), 47D60, 47D62 (secondary). 相似文献
20.
Tatsuo Nishitani 《偏微分方程通讯》2013,38(9):1289-1319
We study the Cauchy problem for second order hyperbolic equations with non negative characteristic form of two independent variables. We show that for such equations in divergence-free form, the Cauchy problem is well posed in the Gevrey class of order less than 5/2. 相似文献