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We consider an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity. Under certain conditions on the operators in the equation, we prove its local (in time) solubility and give sufficient conditions for finite-time blow-up of solutions of the corresponding abstract Cauchy problem. The proof uses a modification of a method of Levine. We give examples of Cauchy problems and initial-boundary value problems for concrete non-linear equations of mathematical physics.

We obtain the asymptotic behaviour of solutions of the Cauchy problem for a fractional non-linear equation. We show that the remainder term in the asymptotic formula is also the remainder in the far field, that is, as the spatial and temporal coordinates tend to infinity simultaneously. We also consider the case in which the Cauchy data are not small.

We describe classes of continuous functions for which one has pointwise and uniform convergence of certain Lagrange-type operators (constructed from solutions of a Cauchy problem) and the Lagrange-Jacobi interpolation polynomials . We also obtain sufficient conditions for the equiconvergence of these interpolation processes.

We investigate the asymptotic behaviour as of the solution of the Cauchy problem for a Sobolev-type equation with quadratic non-linearity and develop ideas used by I. A. Shishmarev and other authors in the study of classical and Sobolev-type equations. Conditions are found under which it is possible to consider the case of an arbitrary dimension of the spatial variable.

This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries (GCSs). Complete group classification results are presented, and some examples are given to show the main reduction procedure.

The Cauchy problem for an elliptic partial differential equation is ill-posed. In this paper, we study a numerical method for solving the Cauchy problem. The numerical method is based on a reformulation of the Cauchy problem through an optimal control approach coupled with a regularization term which is included to treat the severe ill-conditioning of the corresponding discretized formulation. We prove convergence of the numerical method and present theoretical results for the limiting behaviors of the numerical solution as the regularization parameter approaches zero. Results from some numerical examples are reported.

In this paper, we propose an approximate optimal control formulation of the Cauchy problem, for an elliptic equation, equivalent to the original one under some regularity assumptions on the data. We prove the existence of a solution and show the convergence of a subsequence of approximate solutions, obtained via finite element approximation, to a solution of the continuous problem. Finally, we give some numerical results showing the efficiency of our approach and confirming the convergence result.

A problem of reconstruction of the radiation field in a domain from experimental data given on a part of boundary is considered. For the model problem described by a Cauchy problem for the Helmholtz equation, an approximate method based on regularization in the frequency space is analysed. Convergence and stability are proved under a suitable choice of regularization parameter. Numerical implementation of the method is discussed.

We study the Cauchy problem for the (2+1)-dimensional relativistic Abelian Chern-Simons-Higgs model. We prove the global existence and uniqueness of the solutions for the finite energy data.

In this paper, the highly ill posed Cauchy problem for the Laplace equation is transformed to a classical moment problem whose numerical approximation can be achieved. Proofs on its convergence and stability estimates are given based on the Backus-Gilbert algorithm. For numerical verification, several examples which include random noise in the initial Cauchy data are presented.

In this paper we consider the non-characteristic Cauchy problem where with appropriate coefficient functions a, b and c. Assuming that the Cauchy data are given inexactly by a function satisfying ||-||Hr for some r0 and that f(y,t): = u(l,y,t) exists and belongs to Hs(n-1 × ) for some s, it is desired to calculate f from the improper data . This problem is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. In this paper the following mollification method is suggested for this problem: if the Cauchy data are given inexactly then we mollify them by projection on elements of Meyers multiresolution approximation {Vj}j. Within every space Vj the solution of the above problem depends continuously on the data, and we can find a mollification parameter J depending on the noise level in the Cauchy data such that the error estimation between the exact solution and the mollified solution is of Holder type. At the end of this paper numerical examples for our method will be given.

The Cauchy problem for the sinh-Gordon equation on the whole axis with non-zero boundary values at infinity is studied. The main part of the paper is devoted to investigation of the direct and inverse problems for the associated Lax operator. The characteristic properties of the corresponding scattering data are obtained as well as their evolution in time. It allows us to prove the solvability of the Cauchy problem and to construct a multisoliton regular solution.

The solution of the Cauchy problem for the Toda lattice with step-like initial data is shown to split into an infinite series of solitons when time tends to infinity. An exact asymptotic formula for such solutions is obtained.

Measure-valued solutions of the Cauchy problem are considered for a first-order quasilinear equation with only continuous flow functions. A measure-valued analogue of the maximum principle (in Lebesgue spaces) is proved. Conditions are found under which a measure-valued solution is an ordinary function. Uniqueness questions are studied. The class of strong measure-valued solutions is distinguished and the existence and uniqueness (under natural restrictions) of a strong measure-valued solution is proved. Questions of the convergence of sequences of measure-valued solutions are studied.

In this paper the non-characteristic Cauchy problem ut- alpha (x)uxx-b(x)ux-c(x)u=0, x in (0,l), t in R; u(0,t)= phi (t), t in R; ux(0,t)=0, t in R; is considered. The problem is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. In this paper the following mollification method is suggested for this problem: if the Cauchy data are given inexactly then we mollify them by elements of well-posedness classes of the problem, namely by elements of an appropriate co-regular multiresolution approximation {Vj}j in Z of L2(R) which is generated by the father wavelet of Meyer (1992). Within VJ the problem is well posed, and we can find a mollification parameter J depending on the noise level epsilon in the Cauchy data such that the error estimation between the exact solution and the mollified solution is of Holder type. The method can be numerically implemented using fundamental results by Beylkin, Coifman and Rokhlin (1991) on representing (pseudo)differential operators in wavelet bases. A stable marching difference scheme based on this method is suggested. Several numerical examples are given.

For the functional ( satisfies the Caratheodory condition, and , , ) it is proved that: 1) is lower semicontinuous on a fixed pair of function with respect to convergence of to in and weak convergence of to in if an only if for a.e. the function is convex at the point ; 2) strong convergence of to in , weak convergence of to in , and convergence of the values of the functional to imply strong convergence of to if and only if for a.e. the function is strictly convex at the point . Analogous results are obtained for problems with restrictions on the ranges of the functions and in the gradient scalar case: , , .

An apparatus is developed for proving theorems on the existence of a solution of the Cauchy problem and on continuous dependence of solutions on the initial conditions and parameters, applicable to situations not covered by known results on this subject.Bibliography: 24 titles.

Results on uniqueness of solution of the Cauchy problem in the class of functions of power growth and on well-posedness of this problem in such a space are presented.Bibliography: 24 titles.

The non-characteristic Cauchy problem for the diffusion equation in plane geometry is re-examined and an analytic solution is derived. There is a well known series solution to this classic problem which involves derivatives of all orders of the Cauchy data. In this work, we present a new series solution to the problem which does not require higher derivatives of the Cauchy data. The solution is expressed in a series of Laguerre polynomials in the time variable with coefficients involving new specialized functions of the space variable. A complete analysis of these specialized functions is presented and their properties exploited to great effect in a number of numerical examples. All computations are performed using an efficient numerical algorithm based on a truncated form of the series solution.

Conditions are found under which the set of solutions of an evolution parabolic inequality is nonempty, compact, and connected. Included in the study is the Cauchy problem , with a multivalued and monotone operator , where is a nonreflexive -space. Questions connected with well-posedness of the Cauchy problem and convergence of Faedo-Galerkin approximations are investigated.Bibliography: 20 titles.

Numerical relativity can be performed within the context of the Cauchy problem or the characteristic initial value problem (CIVP). These two approaches are reviewed in the case of axial symmetry without rotation. It is shown how to compactify CIVP coordinates so that a finite grid extends to future null infinity. Many physical situations have matter in a central region and vacuum outside. It is shown how to interface Cauchy algorithms in the central region with CIVP algorithms in the external vacuum.

A special version of the Newton-Kantorovich method is applied to the three-dimensional potential inverse scattering problem in the time domain. The related hyperbolic Cauchy problem with data on the side of the time cylinder is solved by the quasi-reversibility method, and a new stability theorem is established by Carleman-type estimates. The geometrical convergence of the Newton-Kantorovich method, used here, is also established.

This paper is concerned with the Cauchy problem connected with the Laplace equation. We investigate the denseness of harmonic polynomials and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with a suitable choice of the regularization method. Numerical experiments are also presented to show the effectiveness of our method.

We consider the Cauchy problem for harmonic functions outside some disc in the plane with the Cauchy data on an interval. We obtain simple formulae for singular values of the operator solving this Cauchy problem and explicit bounds on the difference between the exact and truncated operators. For a typical particular geometry we compute numerically these singular values and analyse their dependence on the size of the interval, on its distance to the disc, etc. As a consequence, we can tell how many parameters of the harmonic function (or of a source producing this function) can be found from the Cauchy data.

We obtain an asymptotic expansion as for the solution of the Cauchy problem with initial functions of compact support for the equation where and satisfies the following condition for some : Bibliography: 4 titles.

CONTENTSIntroduction § 1. Survey of results on the asymptotic behaviour of solutions of first order non-linear equations § 2. Representation and asymptotic behaviour of the solution of the Cauchy problem for a quasilinear conservation law in the case of a monotonic initial function § 3. The stabilization to a constant of the solution of the Cauchy problem for a quasilinear conservation law References

The Cauchy problem is studied for systems of pseudodifferential hypoelliptic equations. An existence and uniqueness theorem is established for the solution, the asymptotic behavior of the solution as is examined, and various representations of the solution in the form of T-products of operators are studied. Bibliography: 18 titles.

It is shown that, arbitrarily close to the Cauchy data that a 'conical' singularity induces on a surface, there exist Cauchy data which are the limit of a family of regular Cauchy data, the limit of whose developments is a spacetime with a curvature singularity (i.e. not just a quasi-regular singularity). This result is used to formulate a notion of stability for quasi-regular singularities. It is shown that certain 'cone-like' singularities are unstable in this sense.

In this article we study the -well-posedness of the non-characteristic Cauchy problem for hyperbolic equations with characteristic roots of variable multiplicity. We obtain a necessary condition for the Cauchy problem with arbitrary lower order terms to be well-posed, and also a necessary condition for the smoothness of the solution to be independent of the lower order terms. For equations with characteristic roots of an arbitrary variable multiplicity we obtain necessary conditions on the lower order terms for the Cauchy problem to be well-posed. All the proofs are based on a single method: the construction of asymptotic solutions.

The Cauchy problem for any nonstrictly hyperbolic operator with analytic coefficients is correct in some Gevrey class if the domain is a lens of spatial type.Bibliography: 21 items.

For the equation where is an increasing function with , consider the Cauchy problem in different formulations determined by specifying the initial data in various forms on the curve . It is proved that the problems considered are uniquely solvable.Bibliography: 12 titles.

The classical solvability of the singular Cauchy problem for the Euler-Poisson-Darboux equation in a homogeneous, globally symmetric space of rank 1 is studied. Starting out from the mean value theorem for spaces of the indicated type, the Darboux and the Euler-Poisson-Darboux equations are introduced. For the Cauchy problem with specific singularity conditions, analogs of Kirchhoff's formulas are derived, i.e. a representation of the solution in terms of spherical means of the initial data is given. The representations so obtained permitted the establishment of necessary and sufficient conditions for the problems under consideration to satisfy Huygens' principle. In particular, Kirchhoff's formulas for the wave equation have been obtained.Bibliography: 27 titles.

In this paper we examine a problem connected with Cauchy's problem for linear elliptic equations.Let be a bounded region of , and let be its boundary. In we consider the elliptic equation (1)where is a regular elliptic expression with complex coefficients. Let , be a piece of the surface . The coefficients of the expression , the surface , and the boundary are assumed to be infinitely smooth. We are concerned with Cauchy's problem on with the initial conditions , , where designates the direction normal to . In this paper we prove that under our assumptions the set of Cauchy initial data for solutions of (1) in is dense in for any integer if Cauchy's problem is unique for the formal conjugate operator , as is the case, for example, when has no multiple complex characteristics.In addition, in this paper we give conditions under which the analogous assertion holds for certain elliptic systems.Bibliography: 4 items.

A maximal uniqueness class of solutions is determined for the Cauchy problem for linear partial differential equations whose coefficients depend on a space variable.