Cauchy Problem For Equation Research Articles

Well-posedness of the Cauchy problem of Ostrovsky equation in anisotropic Sobolev spaces

We study the Cauchy problem of the Ostrovsky equation ∂ t u − β ∂ x 3 u − γ ∂ x −1 u + u ∂ x u = 0 , with β γ < 0 . By establishing a bilinear estimate on the anisotropic Bourgain space X s , ω , b , we prove that the Cauchy problem of this equation is locally well-posed in the anisotropic Sobolev space H ( s , ω ) ( R ) for any s > − 5 8 and some ω ∈ ( 0 , 1 2 ) . Using this result and conservation laws of this equation, we also prove that the Cauchy problem of this equation is globally well-posed in H ( s , ω ) ( R ) for s ⩾ 0 .

Solvability of the Cauchy problem of nonlinear beam equations in Besov spaces

In this paper we study solvability of the Cauchy problem of the nonlinear beam equation ∂ t 2 u + △ 2 u = ± u p with initial data in Besov spaces. We prove that, for any 1 ≤ q < ∞ , the Cauchy problem of this equation is locally well-posed in the Besov spaces B ̇ 2 , q s p ( R n ) and B 2 , q s ( R n ) , where s p = n 2 − 4 p − 1 and s > s p , and globally well-posed in these spaces if initial data are small. Moreover we obtain scattering results in B ̇ 2 , q s p ( R n ) .

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Blowing-up solutions to the Cauchy problem for the master equation

The master equation is a nonlinear integro-partial differential equation that plays a very important role in investigating various sociodynamic phenomena. The purpose of this paper is to fully study blowing-up solutions to the Cauchy problem for the equation.

The behavior of solutions to the Cauchy problem for the master equation

We deal with the master equation, which is a nonlinear integro-partial differential equation. The equation plays a very important role in quantitative sociodynamics. The purpose of this paper is to investigate how solutions to the Cauchy problem for the equation behave as the time variable increases.

The equicontinuity condition for sequences of solution spaces

The verification of an appropriately stated equicontinuity condition for a sequence of solution spaces is one of the two key points in the theory of the Cauchy problem for equations with singular right-hand sides. We obtain a related sufficient condition.

On the Regularity Properties for Solutions of the Cauchy Problem for the Porous Media Equation

We consider the Cauchy problem for the equation ${\partial _t}u = \Delta {u^m}$ in ${R^N} \times (0,T)$. We assume that $1 < m < 3N/(3N - 2)$ and the initial data ${u_0}$ is in $C_0^1({R^N})$ and ${u_0} \geq 0$ in ${R^N}$. Then we prove that the second derivatives of ${u^m}$ with respect to the space-variable are in ${L^2}({R^N} \times (0,T))$.

On the regularity properties for solutions of the Cauchy problem for the porous media equation

We consider the Cauchy problem for the equation ∂ t u = Δ u m {\partial _t}u = \Delta {u^m} in R N × ( 0 , T ) {R^N} \times (0,T) . We assume that 1 &gt; m &gt; 3 N / ( 3 N − 2 ) 1 &gt; m &gt; 3N/(3N - 2) and the initial data u 0 {u_0} is in C 0 1 ( R N ) C_0^1({R^N}) and u 0 ≥ 0 {u_0} \geq 0 in R N {R^N} . Then we prove that the second derivatives of u m {u^m} with respect to the space-variable are in L 2 ( R N × ( 0 , T ) ) {L^2}({R^N} \times (0,T)) .

Cauchy problem for equations with loadings on surfaces

Cauchy problem for equations with loadings on surfaces

Estimates of solutions of the Cauchy problem for equations of Schr�dinger type in the spaces L p ? and B pq ? . I

Estimates of solutions of the Cauchy problem for equations of Schr�dinger type in the spaces L p ? and B pq ? . I

Estimates of solutions of the cauchy problem for equations of Schr�dinger type in the spaces L p ? and B pq ? . II

Estimates of solutions of the cauchy problem for equations of Schr�dinger type in the spaces L p ? and B pq ? . II

Cauchy problem for equations with multiple characteristics

This paper contains a proof of γ n( χ) correctness of the noncharacteristic Cauchy problem for nonstrictly hyperbolic equations with analytic coefficients under the condition that its characteristic roots are smooth and under some additional assumptions on the lower-order terms. There are two extreme cases: (1) χ < r r − 1 . In this case condition (0.6) is “void,” and we do not require conditions on P s for s < m. For this case, see [3, 8]. (2) Case of constant multiplicity of characteristic roots and χ = +∞. In this case condition (0.6) implies conditions on P s , where s = m, m − 1,…, m − r + 1, i.e., up to the same order as the necessary condition for C ∞-correctness [2]. Recall that in the case of equations with characteristics of constant multiplicity condition (0.6) (Levi's condition in this case) for χ = ∞ is necessary [2, 4] and sufficient [1] for C ∞-correctness.

Linear Einstein equations. III

The Cauchy problem for equations of the theory of gravitation, linearized around the non-Galilean metric $$\mathop {g_{\alpha \beta } }\limits_{(0)} $$ , can have a nonunique solution.

Stability of relative motion of phases in two-phase flows

turbances (the problem of the correctness of the Cauchy problem for equations of two-phase media) is examined. We show that a consideration of the effect of particle (bubble) diffusion caused by relative motion of the phases is of fundamental importance. The pressure in the disperse phase is a subsidiary factor. The critical stability loss curve is obtained. The problem of stability of two-phase media has been examined in many papers [i-5]. Existing theories predict a short-wave instability of sedimenting suspensions, fluidized beds, and layers of liquid with bubbles. This instability should lead to the rapid appearance of inhomogeneities within the medium and to the practical unattainability of the homogeneous state. Contradictory to theory, however, manifestly stable states are obtained in experiments [4]. Stability of a liquid with bubbles has been obtained only in [6, 7]. In [6] stability was secured by the action of electrical forces. In the problem of thermocapillary motion in a gas--liquld mixture stability in the short-wave region is obtained by bubble diffusion [7], i. Equations and Method of Solution The equations for the change of momentum and conservation of mass of a two-phase medium have the form [i]

Remark on behavior of solutions of the Cauchy problem for equations with variable coefficients

Remark on behavior of solutions of the Cauchy problem for equations with variable coefficients

Cauchy problem for equations in radial differentials

Cauchy problem for equations in radial differentials

Asymptotic solution of the Cauchy problem for equations with complex characteristics

Differential equations with a small parameter in the derivative are considered. A method is developed for constructing formal asymptotic solutions for the case of complex characteristics. For this a new class of manifolds is introduced which is a natural generalization of real Lagrangian manifolds to the complex case. The theory of the canonical Maslov operator is constructed in this class of manifolds. Asymptotic solutions are expressed in terms of the canonical Maslov operator.

Uniqueness classes of solutions of the Cauchy problem for equations with variable coefficients

Uniqueness classes of solutions of the Cauchy problem for equations with variable coefficients

The cauchy problem for the Yang-Mills equations

The Yang-Mills equation in Minkowski space with given initial data is considered. The gauge group is formulated in terms of Sobolev-Banach-Lie group, and the Cauchy problem for the equations thereby reduced to the temporal (hamiltonian) gauge. Given data for which are square-integrable over have respectively one and two derivatives which are square-integrable over space, a strong solution exists throughout space in a nontrivial time interval. If the initial data are infinitely differentiable in L 2, the solution may be represented as a C ∞ function on space-time satisfying the equations in the elementary sense. Strong solutions which agree at one time and have square-integrable derivatives as earlier agree throughout their regions of definition. For arbitrary finite-energy Cauchy data, there exists a quasi-solution (weak limit of solutions of truncated equations) which is global on Minkowski space. The solutions may take values in an arbitrary separable Hilbert Lie algebra.

Generic properties of functional equations

A PROPERTY is said to be generic on a Baire space if the subset on which it is true is residual, that is its complement is a set of Baire first category. The study of generic properties of differential equations was started by W. Orlicz [l] who showed that the subset allf for which the Cauchy problem for the equation y’ = f(t, y) has not uniqueness is of first category in the space of all continuous and boundedf with values in R”, equipped with a natural metric. An analogous result for hyperbolic equations was proved by A. Alexiewicz and W. Orlicz [2]. More recently A. Lasota and J. Yorke [3] studied properties concerning existence, uniqueness and continuous dependence of solutions of the Cauchy problem for ordinary differential equations in a Banach space. Similar problems for functional differential equations y’ = f(t, y,) in a Banach space have been discussed in [4]. In [5] ([6]) it has been shown that the convergence of successive approximations for ordinary differential equations in a Banach space (for a class of hyperbolic partial differential equations) is a generic property in the corresponding spaces of continuous functions. (For the R” case see [7].) Generic properties of fured points for nonlinear nonexpansive mappings in a Banach space have been investigated in [8,9]. Other problems in the same spirit of the aforementioned ones have been treated in [l&12]. In this note we shall present some contributions to the theory of generic properties for functional equations and functional differential equations. Section 1 contains results concerning generic properties of existence uniqueness, continuous dependence of solutions and convergence of successive approximations for abstract functional equations. In Sections 2 and 3 we establish, by using similar arguments, generic properties of existence, uniqueness, continuous dependence of solutions and convergence of successive approximations for a class of functional integral equations. In Section 4 the results of Section 1 are used to obtain generic properties of nonexpansive mappings in a Banach space. The question how those maps, for which the generic properties under consideration fail to be true, are scattered is discussed in Section 5.