Cauchy characteristic problem

The problem of finding a solution of a partial differential equation (or a system of partial differential equations) which assumes prescribed values on a characteristic manifold.

There is a large class of equations of hyperbolic and parabolic types for which a non-closed $n$-dimensional surface $S$, oriented in a certain way, may serve as the initial surface in the space $E_{n+1}$ of independent variables $x_1,\dots,x_n,t$. For example, if $S$ is a space-like surface then the Cauchy problem (with initial data on $S$) is always well-posed. In a characteristic Cauchy problem the initial surface is always a characteristic manifold (or a well-defined part thereof). In this case the Cauchy problem may have no solution at all; and if it has a solution it need not be unique.

For example, the characteristic Cauchy problem for the equation ($n=1$, $x_1=x$)

$$u_{xt}=0$$

with data on the characteristic $t=0$:

$$u(x,0)=\tau(x),\quad u_t(x,0)=\nu(x),$$

is not well-posed. If the characteristic Cauchy problem has a solution, then the equation and the second initial condition imply a necessary condition for its solvability: $\nu'(x)=0$, i.e. the characteristic Cauchy problem may be solvable only if $\nu(x)=\mathrm{const}=\alpha$. In that case, if $\tau(x)\in C^2$, $t\geq0$, a solution indeed exists and is given by

$$u(x,t)=\tau(x)+\alpha t+\rho(t),$$

where $\rho(t)$ is any function of class $C^2$, $t\geq0$, satisfying the conditions $\rho(0)=\rho'(0)=0$.

A necessary condition for the existence of a solution to a characteristic Cauchy problem for a linear system of hyperbolic equations is that the rank of the augmented matrix of the system equals the rank of the singular matrix along the characteristic surface $S$.

There is a wide class of hyperbolic equations and systems for which one may take a characteristic surface as an initial surface. For example, for the equation

\begin{equation}\sum_{i=1}^nu_{x_ix_i}-u_{tt}=0,\label{1}\end{equation}

when the characteristic surface $S$ is the cone

\begin{equation}\sum_{i=1}^n(x_i-x_i^0)^2-(t-t_0)^2=0,\label{2}\end{equation}

the characteristic Cauchy problem may be phrased as follows: Find a solution $u(x,t)$ of equation \eqref{1} which is regular within the cone \eqref{2} and takes prescribed values on the cone \eqref{2}.

In the case of a space-like variable ($n=1$, $x_1=x$), the cone \eqref{2} is a pair of straight lines $(x-x_0)^2=(t-t_0)^2$ passing through the point $(x_0,t_0)$. These straight lines divide the plane $E_2$ of the variables $x,t$ into four angles. Let the domain $\Omega$ be one of these angles. Then the characteristic problem is customarily known as the Goursat problem: Determine a solution $u(x,t)$ of the equation

$$u_{xx}-u_{tt}=0$$

which is regular in $\Omega$ and satisfies the conditions

$$u=\phi\quad\text{if }x-x_0=t-t_0,$$

$$u=\psi\quad\text{if }x-x_0=t_0-t,$$

$$\phi(x_0,t_0)=\psi(x_0,t_0).$$

If the characteristic surface $S$ is at the same time a surface of degenerate type or order, the characteristic Cauchy problem may prove to be well-posed.

For the equation

\begin{equation}y^mu_{yy}-u_{xx}+au_x+bu_y+cu=f,\label{3}\end{equation}

which is hyperbolic for $y>0$, the curve of degeneracy $y=0$ is a characteristic. If $0<m<1$, the Cauchy problem

\begin{equation}u(x,0)=\tau(x),\quad u_y(x,0)=\nu(x)\label{4}\end{equation}

for equation \eqref{3} is well-posed, but if $m\geq1$ it becomes ill-posed. In that case it is natural to investigate the problem either with modified initial data:

$$\lim_{y\to0}\alpha(x,y)u(x,y)=\tau(x),$$

$$\lim_{y\to0}\beta(x,y)u_y(x,y)=\nu(x),$$

$$\lim_{y\to0}\alpha(x,y)=0,\quad\lim_{y\to0}\beta(x,y)=0,$$

or with incomplete initial data, i.e. dropping one of the conditions \eqref{4}.

References

[1]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[2]	S.K. Godunov, "The equations of mathematical physics" , Moscow (1971) (In Russian)
[3]	F.G. Tricomi, "Equazioni a derivate parziale" , Cremonese (1957)
[4]	L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)
[5]	A.V. Bitsadze, D.F. Kalinichenko, "A collection of problems on the equations of mathematical physics" , Moscow (1977) (In Russian)
[6]	A.V. Bitsadze, "Linear partial differential equations of mixed type" , Proc. 3-rd All-Union Math. Congress , 3 , Moscow (1958) (In Russian)

Comments

The more common English term for this problem is characteristic Cauchy problem. A general discussion is given in [a1], Sect. 12.8.

References

[a1]	L.V. Hörmander, "The analysis of linear partial differential operators" , 2 , Springer (1983)
[a2]	A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian)
[a3]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)