Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L 2 Initial Data

In this paper we study solvability of the Cauchy problem of the Kawahara equation 偏导dtu ＋ au偏导dzu ＋ β偏导d^3xu ＋γ偏导d^5xu = 0 with L^2 initial data. By working on the Bourgain space X^r,s（R^2） associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H^r（R） and -1 〈 r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L^2（R）.     

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D _s), whereD _s is the differentiation operator, withF bounded linear andK andD _sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.     

The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

In this paper we consider the Cauchy problem for the equation , where the matrix {a _jk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs<q/(q−1).     

Local solvability of the cauchy problem of a fifth-order nonlinear dispersive equation

The solvability of the fifth-order nonlinear dispersive equation δtu＋au （δxu）^2＋βδx^3u＋γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5（R） and s〉1/4,then the Cauchy problem has a unique solution in C（[-T,T],H^5（R）） for some T〉0.     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

On existence and uniqueness of solution of stochastic differential equations with heredity

We consider the Cauchy problem for the stochastic differential equation with the heredity where x _t(s) = x(s)for s?(- ∞,t).Existence and uniqueness theorems for the problem (1),(2)are proved inthe case,when instead of the Lipschitz condition for the functions a(t,u) and b(t,u)on u someless restrictive conditions (Ousgood or Hölder type)are satisfied, and the operator(Fx)(t) = x(t)-f(t,x _t) is invertible.Similar questions were considered in[1-4]     

Integrable and nonintegrable cases of the Lax equations with a source

The Korteweg-de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that, depending on the choice of the basis of the eigenfunctions, we have the following three possibilities: (1) evolution equations for the scattering data are nonintegrable; (2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg-de Vries equation with a source at somet>t _o leaves the considered class of functions decreasing rapidly enough asx±; (3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg-de vries equation with a source exists at allt>t _o. All these possibilities are widespread and occur in other Lax equations with a source.Bogoliubov Theoretical Laboratory, Joint Institute for Nuclear Research 141980 Dubna, Moscow Region, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 471–477, June 1994.     

Non autonomous evolution operators of hyperbolic type

In this paper we consider a Cauchy problem in a Banach spaceE:u′(t)=A(t)u(t)+f(t), t∈[t ₀, T], u(t₀)=u₀, whereA(·) is a family of linear operators inE which satisfy all the requirements of Kato's semigroup approach to the non autonomous hyperbolic equations except for the density of the common domains ofA(t). An application is given to a hyperbolic partial differential equation with discontinuous coefficients.     

Asymptotic behavior relative to a large parameter of the solution of the Fok-Klein-Gordon equation in the case of a discontinuous initial condition

One considers the problem of the asymptotic behavior for K→+∞ of the solution of the Cauchy problem $$u_{tt} - u_{xx} + \kappa ^2 u = 0; u|_{t = 0} = \theta (x), u_t |_{t = 0} = 0 (t > 0 - fixed)$$ Hereθ(x) is the Heaviside function. In the neighborhood of the characteristics x=±t function u(x,t)_?2 oscillates exceptionally fast (the wavelength is of order k^?2). Near the t axis the asymptotics of u(x,t) contains the Fresnel integral.     

The self-similar solutions of the Tricomi-type equations

This paper analyses the properties of the family of self-similar solutions of the generalized Tricomi equation u_tt - t^2k Du = 0 (2k ? \mathbbN)u_{{tt}} - t^{{2k}} \Delta u = 0\,(2k \in {{\mathbb{N}}}) in the domain \mathbbR ₊ ^{1 + n}{{\mathbb{R}}}_{ + }^{{1 + n}} by considering initial conditions on the functions and their derivatives, posed as the Cauchy problem with homogeneous initial data. For specific values of the power k ( = 1/2 or = 3/2) and n = 1 this problem has applications in the aerodynamics of airfoils operating in transonic flows of perfect or dense gases, respectively. An integral transformation is suggested and used to represent the solutions of the Cauchy problem with homogeneous initial functions in terms of fundamental solutions of the classical wave equation (the case k = 0). Then the Cauchy problem with homogeneous initial functions for the wave equation in \mathbbR^{1 + n}{{\mathbb{R}}}^{{1 + n}} is solved. These results are used to derive estimates of the upper bound for solutions’ size and to obtain the asymptotics for self-similar solutions of the wave equation and of the Tricomi-type equation in the neighbourhood of their light cones.     

Stability of the solutions of the Cauchy problem for first-order quasilinear equations

In this paper we establish estimates of the stability of the solutions of the Cauchy problem for the multidimensional quasilinear equation \(u_t + (\varphi _i (u))_{x_i } = \psi (t, x, u)\) relative to variation of the function ?_i(u), ψ(t, x, u).     

Well-posedness of a class of operator-differential equations

We consider a linear first-order ordinary operator-differential equation A(t)u′(t) + B(t)u(t) = f(t) in a Banach space, where the operator A(t) is not invertible in general. Sufficient conditions for the existence, uniqueness, and well-posedness of the Cauchy problem for this equation are obtained.     

On the Cauchy problem for a doubly nonlinear degenerate parabolic equation with strongly nonlinear sources

In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||_f||| _h and 〈f〉_h, we give the sufficient and necessary conditions on the initial value to the existence of local solution of doubly nonlinear equation. Moreover some results on the global existence and nonexistence of solutions are considered. This work was supported by the National Natural Science Foundation of China (Grant No. 10531020)     

L 2-estimates of solutions of boundary value problems for equations with mixed elliptic-parabolic structure

For the linear equation L u=0 in which L is a product of a 2b₁-parabolic operator and a 2b₂r-elliptic operator (b₁, b₂, and r are integers), we obtain L₂-estimates for solutions of model boundary value problems, namely, for the Cauchy problem and the half-space problem. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 4–27, 1992. Translated by N. A. Karazeeva.     

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      16. A periodic problem for the inhomogeneous equation of string oscillations      Yu. A. Mitropol’skii  G. P. Khoma  P. V. Tsynaiko 《Ukrainian Mathematical Journal》1997,49(4):618-627 We study a periodic problem for the equation u tt−uxx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ2 and establish conditions of the existence and uniqueness of the classical solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 558–565, April, 1997.      17. Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem      N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(12):1917-1923 In three spaces, we find exact classical solutions of the boundary-value periodic problem utt - a2uxx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.      18. Second order incomplete expiring Cauchy problems      Ralph Delaubenfels 《Semigroup Forum》1989,39(1):75-84 We extend results of Balakrishnan and Dorroh, on 2nd-order incomplete Cauchy problems, from differentiable to stronglycontinuous semigroups of operators. We show that the Cauchy problem (*) $$\begin{gathered} u''(t) = A(u(t)), t \geqslant 0, u(0) = x, \hfill \\ \mathop {lim}\limits_{t \to \infty } \left\| {u^{(k)} (t)} \right\| = 0, k = 0, 1, 2, \hfill \\ \end{gathered} $$ where A is a linear operator with nonempty resolvent on a Banach space, is well-posed if and only if A has a squares root that generates a Co semigroup, {T(t)} t>0, that converges to zero, as t goes to infinity, in the strong operator topology. This extension leads to the following application. If A is a linear constant coefficient partial differential operator on L2(?n), then there exist orthogonal closed subspaces, H1, H2, such that Hl⊕H2=L2(?n), and (*), on H1, is well-posed, while the complete Cauchy problem u″(t)=A(u(t)), t??, u(O)=x, u′ (O)=y is well-posed on H2. We also apply our results to the dying wave equation, on Co[0, ∞) and Lp(?dv) (1≤p <∞), for a large class of measures v.      19. Uniqueness and stability of solutions for Cauchy problem of nonlinear diffusion equations      Junning Zhao  Peidong Lei 《中国科学A辑(英文版)》1997,40(9):917-925 The uniqueness of solutions for Cauchy problem of the form $$\frac{{\partial u}}{{\partial t}} = \Delta A(u) + \sum\limits_{i = 1}^N {\frac{{\partial b^i (u)}}{{\partial x_i }} + c(u)} $$ is studied. It is proved that ifu ∈BVx and A(u) is strictly increasing, the solution is unique.      20. Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices   总被引：1，自引：0，他引：1   Hua WANG Shang Bin CUI Dong Gao DENG 《数学学报(英文版)》2007,23(8):1435-1446 We first prove that the Cauchy problem of the Kawahara equation, δtu ＋ uδxu ＋βδx^3u＋γδx^5u = 0, is locally solvable if the initial data belong to H^r（R） and r〉 r≥-7/5, thus improving the known local well-posedness result of this equation. Next we use this local result and the method of ＂almost conservation law＂ to prove that global solutions exist if the initial data belong to H^r（R） and r〉-1/2.

A periodic problem for the inhomogeneous equation of string oscillations

We study a periodic problem for the equation u _tt−u_xx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ² and establish conditions of the existence and uniqueness of the classical solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 558–565, April, 1997.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

Second order incomplete expiring Cauchy problems

We extend results of Balakrishnan and Dorroh, on 2^nd-order incomplete Cauchy problems, from differentiable to stronglycontinuous semigroups of operators. We show that the Cauchy problem (*) $$\begin{gathered} u''(t) = A(u(t)), t \geqslant 0, u(0) = x, \hfill \\ \mathop {lim}\limits_{t \to \infty } \left\| {u^{(k)} (t)} \right\| = 0, k = 0, 1, 2, \hfill \\ \end{gathered} $$ where A is a linear operator with nonempty resolvent on a Banach space, is well-posed if and only if A has a squares root that generates a C_o semigroup, {T(t)} t>0, that converges to zero, as t goes to infinity, in the strong operator topology. This extension leads to the following application. If A is a linear constant coefficient partial differential operator on L²(?ⁿ), then there exist orthogonal closed subspaces, H₁, H₂, such that H_l⊕H₂=L²(?ⁿ), and (*), on H₁, is well-posed, while the complete Cauchy problem u″(t)=A(u(t)), t??, u(O)=x, u′ (O)=y is well-posed on H₂. We also apply our results to the dying wave equation, on C_o[0, ∞) and L^p(?dv) (1≤p <∞), for a large class of measures v.     

Uniqueness and stability of solutions for Cauchy problem of nonlinear diffusion equations

The uniqueness of solutions for Cauchy problem of the form $$\frac{{\partial u}}{{\partial t}} = \Delta A(u) + \sum\limits_{i = 1}^N {\frac{{\partial b^i (u)}}{{\partial x_i }} + c(u)} $$ is studied. It is proved that ifu ∈BVx and A(u) is strictly increasing, the solution is unique.     

Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices

We first prove that the Cauchy problem of the Kawahara equation, δtu ＋ uδxu ＋βδx^3u＋γδx^5u = 0, is locally solvable if the initial data belong to H^r（R） and r〉 r≥-7/5, thus improving the known local well-posedness result of this equation. Next we use this local result and the method of ＂almost conservation law＂ to prove that global solutions exist if the initial data belong to H^r（R） and r〉-1/2.