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One of the fundamental problems in the theory of (ordinary and partial) differential equations: To find a solution (an integral) of a differential equation satisfying what are known as initial conditions (initial data). The Cauchy problem usually appears in the analysis of processes defined by a differential law and an initial state, formulated mathematically in terms of a differential equation and an initial condition (hence the terminology and the choice of notation: The initial data are specified for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209501.png" /> and the solution is required for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209502.png" />). Cauchy problems differ from boundary value problems in that the domain in which the desired solution must be defined is not specified in advance. Nevertheless, Cauchy problems, like boundary value problems, are defined by the imposition of limiting conditions for the solution on (part of) the boundary of the domain of definition.

The main questions connected with Cauchy problems are as follows:

1) Does there exist (albeit only locally) a solution?

2) If the solution exists, to what space does it belong? In particular, what is its domain of existence?

3) Is the solution unique?

4) If the solution is unique, is the problem well-posed, i.e. is the solution in some sense a continuous function of the initial data?

The simplest Cauchy problem is to find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209503.png" /> defined on the half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209504.png" />, satisfying a first-order ordinary differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209505.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209506.png" /> is a given function) and taking a specified value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209507.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209508.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c0209509.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

In geometrical terms this means that, considering the family of integral curves of equation (1) in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095010.png" />-plane, one wishes to find the curve passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095011.png" />.

The first proposition concerning the existence of such a function (on the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095012.png" /> is continuous for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095013.png" /> and continuously differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095014.png" />) was proved by A.L. Cauchy (1820–1830) and generalized by E. Picard (1891–1896) (who replaced differentiability by a Lipschitz condition with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095015.png" />). It turns out that under those conditions the Cauchy problem has a unique solution which, moreover, depends continuously on the initial data. Modern concepts of the Cauchy problem are essentially a far-reaching generalization of this problem.

The fact that questions 1) to 4) touch profoundly on the very heart of the matter — i.e. to answer them satisfactorily requires the imposition of certain conditions — is already illustrated in the theory of ordinary differential equations. Thus, a solution of the Cauchy problem for equation (1) with the condition (2), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095016.png" /> is given on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095017.png" /> and is only continuous, exists on some interval depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095019.png" /> (see [[Peano theorem|Peano theorem]]), but it need not be unique. The solution need not exist at all points in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095020.png" />.

Repeating the above account almost word for word, one formulates the Cauchy problem for systems of ordinary differential equations, i.e. for an ordinary differential equation of type (1) with initial condition (2), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095021.png" /> is a function with values in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095024.png" /> is a function defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095025.png" />. Here, again, the Picard conditions are sufficient for the existence and uniqueness of the solution and for the problem to be well-posed.

For ordinary differential equations of a higher order,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095026.png" /></td> </tr></table>

the Cauchy problem the initial data of which involve, besides the function itself, the derivatives

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095027.png" /></td> </tr></table>

can be reduced by the standard device to a corresponding problem of type (1), (2).

In the case of first-order ordinary differential equations which cannot be expressed directly in terms of the derivative of the unknown function (as in equation (1)), the formulation of the Cauchy problem is similar, except that it relies to a high degree on the geometrical interpretation; however, the actual investigation of the equation may be complicated by the impossibility of (even locally) reducing the equation to the normal form (1).

While neither the formulation nor the investigation of the Cauchy problem for an ordinary differential equation involve essential difficulties, the situation is considerably more complicated in the case of partial differential equations (this applies, in particular, to answering questions 1) to 4)). This is true even if the functions involved are sufficiently regular (smooth). A major source of the difficulty is the fact that the space of independent variables is higher-dimensional, resulting in problems of (algebraic) solvability. E.g., consider the Cauchy problem for a system of equations in total differentials,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095028.png" /></td> </tr></table>

such equations being in a sense intermediate between  "ordinary"  and  "partial"  differential equations. The problem here is to determine an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095029.png" />-dimensional integral surface passing through a given point. Then the solvability condition is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095030.png" /></td> </tr></table>

(in a neighbourhood of the given point; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095031.png" /> are the symbols for the exterior differential and the exterior product, respectively) (see [[Frobenius theorem|Frobenius theorem]]).

For linear partial differential equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

the Cauchy problem may be formulated as follows. In a certain region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095033.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095034.png" /> it is required to find a solution satisfying initial conditions, i.e. taking specified values, together with its derivatives of order up to and including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095035.png" />, on some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095036.png" />-dimensional hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095038.png" />. This hypersurface is known as the carrier of the initial conditions (or the initial surface). The initial conditions may be given in the form of derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095039.png" /> with respect to the direction of the unit normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095040.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095041.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095044.png" />, are known functions (Cauchy data).

The formulation of the Cauchy problem for non-linear differential equations is similar.

A concept related to the Cauchy problem is that of a non-characteristic surface. If a non-singular coordinate transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095045.png" />  "straightens out"  the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095046.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095047.png" />, i.e. it transforms it into a part of the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095048.png" />, then the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095049.png" /> in the transformed equation (3) is proportional to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095050.png" /></td> </tr></table>

The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095051.png" /> is said to be non-characteristic at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095052.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

In that case equation (3) may be written in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095054.png" /> in the so-called normal form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

Cauchy problems are usually studied when the carrier of the initial data is a non-characteristic surface, i.e. when condition (5) holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095056.png" />.

The [[Cauchy–Kovalevskaya theorem|Cauchy–Kovalevskaya theorem]] occupies an important position in the theory of Cauchy problems; it runs as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095057.png" /> is an analytic surface in a neighbourhood of one of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095058.png" />, if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095062.png" />, are analytic in the same neighbourhood, and if moreover condition (5) is satisfied, then the Cauchy problem (3), (4) has an analytic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095063.png" /> in a neighbourhood of the point; this solution is unique in the class of analytic functions. With the analyticity assumption, this theorem is also valid for general non-linear equations if the latter can be reduced to the normal form (6), and also for systems of such equations. The theorem is universal in nature, since it is applicable to analytic equations regardless of their type (elliptic, hyperbolic, etc.) and yields the local existence of a solution. The solution is unique in the class of non-analytic functions.

The Cauchy problem for partial differential equations of order exceeding 1 may turn out to be ill-posed if one drops the analyticity assumption for the equation or for the Cauchy data in the Cauchy–Kovalevskaya theorem. An illustration is Hadamard's example: The Cauchy problem for the Laplace equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095064.png" /></td> </tr></table>

with initial conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095065.png" /></td> </tr></table>

has no solution if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095066.png" /> is not an analytic function.

The hyperbolic equations constitute a broad class of equations for which the Cauchy problem is well-posed. In this case the Cauchy problem is global in nature, but the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095067.png" /> be non-characteristic is no longer sufficient. It is necessary that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095068.png" /> is a space-like surface. A typical hyperbolic equation is the wave equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

considered in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095070.png" />-dimensional region, with variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095071.png" />. The Cauchy problem for this equation with data

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095072.png" /></td> </tr></table>

on the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095073.png" /> is uniquely solvable for any sufficiently smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095074.png" />, and the solution depends continuously (in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095075.png" /> metric) on these functions. For the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095077.png" />, an explicit form of the solution is given by the formulas of d'Alembert, Poisson and Kirchhoff, respectively:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095078.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095079.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095080.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095082.png" />;

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095083.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095084.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095086.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095087.png" /> is the surface element on the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095088.png" />.

The set of points in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095089.png" /> for which the Cauchy data completely determine the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095090.png" /> of the solution of the wave equation (7) at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095091.png" /> is called the domain of dependence of the latter point. The domains of dependence of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095092.png" /> in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095094.png" /> are, respectively, the closed interval, disc and ball defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095095.png" /> (in the appropriate space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095096.png" />). If the carrier of the Cauchy data is some region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095097.png" /> on the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095098.png" />, then the Cauchy data in that region affect the solution at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c02095099.png" /> of the set such that the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950100.png" /> is not empty; this set is known as the domain of influence.

The set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950101.png" /> at which the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950102.png" /> is completely determined by the Cauchy data on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950103.png" /> is called the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950104.png" /> with initial data on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950105.png" />. In cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950106.png" /> and 3 the domain of definition consists of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950107.png" /> for which the closed interval, disc or ball, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950108.png" /> (as the case may be), lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950109.png" />.

These results carry over to the more general case in which the carrier of the Cauchy data is a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950110.png" /> of spatial type, i.e. a surface for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950111.png" /> (see (5)) remains positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950112.png" />.

There are other problems besides the Cauchy problem which prove to be well-posed for hyperbolic equations; examples are the [[Cauchy characteristic problem|Cauchy characteristic problem]] and mixed initial-boundary value problems. In the latter type of problem, a solution exists in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950113.png" />-dimensional cylinder with generatrix parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950114.png" />-axis and a base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950115.png" /> which is some region in the space of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950116.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950117.png" />. The carrier of the initial conditions is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950118.png" />, while the value of the function, its normal derivative (in the case of second-order equations), or more general boundary value conditions, are given on the lateral surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950119.png" /> of the cylinder.

In the case of degenerate equations the formulation of the Cauchy problem also has to be modified. For example, if the equation is of hyperbolic type and the carrier of the Cauchy data is a surface on which the equation becomes parabolically degenerate, then, depending on the nature of degeneracy, the initial value conditions may involve the use of some weight function.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kovalevskaya,   "Scientific works" , Moscow  (1948)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hadamard,   "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint  (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Bers,   F. John,   M. Schechter,   "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S. Mizohata,   "The theory of partial differential equations" , Cambridge Univ. Press  (1973)  (Translated from Japanese)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L. Hörmander,   "Linear partial differential operators" , Springer  (1964)</TD></TR></table>



====Comments====
The name  "Cauchy problem"  is mostly used in relation with hyperbolic initial value problems. Hörmander [[#References|[8]]] has been followed by 4 volumes. Two of them [[#References|[a2]]] deal with Cauchy problems.

It is clear from the Kirchhoff solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950120.png" /> of (7) that the domain of dependence of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950121.png" /> is, in fact, the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020950/c020950122.png" />. The relevance of this remark for the general definition of domain of dependence for hyperbolic partial differential equations is discussed in [[#References|[5]]], Sect. VI.7.

It is modern practice to define hyperbolicity of a partial differential operator as a necessary condition for the well-posedness of the Cauchy problem, see [[#References|[a2]]], Vol. 2, Sect. 12.3.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1–2''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Treves,   "Basic partial differential equations" , Acad. Press  (1975)</TD></TR></table>