Differential equation, partial, data on characteristics

A problem which consists in solving a partial differential equation or a system of partial differential equations with given conditions on the characteristic manifolds (cf. Characteristic manifold). The principal problems of this type are the characteristic Cauchy problem (cf. Cauchy characteristic problem) and the Goursat problem.

In the former case, when the initial manifold is characteristic at every point, the initial data cannot be set arbitrarily. They must satisfy certain conditions, determined by the differential equation. Accordingly, the characteristic Cauchy problem will usually be not well-posed unless supplementary conditions (especially so along a manifold which is not tangent to the initial manifold) are imposed on the class of solutions sought and on the given functions. For instance, for the thermal-conductance equation (heat equation)

$$u_t=u_{xx},$$

the characteristic Cauchy problem

$$\left.u\right|_{t=0}=\phi(x)$$

is well-posed in the class of functions which grow at infinity not faster than $\exp(cx^2)$; however, if the exponent 2 of $x$ is replaced by $2+\epsilon$, uniqueness is no longer guaranteed. There exists a wide class of equations for which the characteristic Cauchy problem is well-posed.

If the initial manifold $S$ is at the same time a manifold of degeneration of type or order of the equation, the characteristic problem may turn out to be well-posed. E.g., for the equation

$$u_{xx}-y^mu_{yy}=0,\quad y>0,\quad0<m=\mathrm{const}<1,$$

with sufficiently smooth initial data on any interval $S$ of the characteristic $y=0$, the problem is solvable and the solution is unique.

Problems with data on characteristics include problems with incomplete and modified initial data which arise in the theory of degenerate hyperbolic and parabolic equations and systems of equations. For equations of the type

$$u_{xx}-y^mu_{yy}+au_x+bu_y+cu=0,\quad y>0,\quad m>0,$$

these problems are posed as follows. One has to find the solution $u(x,y)$ of the equation which corresponds to the modified initial data

$$\lim\phi(x,y)u=\tau(x),\quad\lim\psi(x,y)u_y=\nu(x),$$

where $\alpha<x<\beta$, $\phi$, $\tau$, $\psi$, and $\nu$ are given functions, or to incomplete initial data, i.e. to one of these conditions.

References

 [1] A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian) [2] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) [3] A.N. Tikhonov, Mat. Sb. , 42 (1935) pp. 199–216 [4] L. Hörmander, "Linear partial differential operators" , Springer (1976) [5] L. Hörmander, "Hypoelliptic second order differential equations" Acta. Math. , 119 (1967) pp. 147–171