Reports of the Academy of Sciences of the USSR
1957, Volume 112, No. 3

MATHEMATICS

G. GUSSI, V. POENARU, and K. FOIAȘ

A DIRECT METHOD IN THE CAUCHY PROBLEM FOR A QUASILINEAR HYPERBOLIC EQUATION IN THE CASE OF TWO INDEPENDENT VARIABLES

(Presented by Academician S. L. Sobolev on 29 VIII 1956)

The present note contains some results concerning the Cauchy problem for the equation

[
\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}
=
f\left(x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}\right).
\tag{1}
]

The differential operator (\partial^2 u/\partial x^2-\partial^2 u/\partial y^2) is replaced by the hyperbolic operator introduced by Begele ((^1)):

[
\widetilde{\Box}u
=
\lim_{h,k\to 0}
\frac{
u(x+h+k,y+h-k)-u(x+h,y+h)-u(x+k,y-k)+u(x,y)
}{hk}
]

and the Cauchy problem for the equation

[
\widetilde{\Box}u
=
f\left(x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}\right)
\tag{2}
]

is considered.

The Cauchy problem is posed as follows: find a solution of equation (2) satisfying the conditions

[
u(x,0)=\varphi(x),\qquad
\left.
\frac{\partial}{\partial y}
\left(
u-\frac{\varphi(x+y)+\varphi(x-y)}{2}
\right)
\right|_{y=0}
=
\psi(x),
]

where (\varphi) and (\psi) are functions of class (C^0), if (f) does not explicitly depend on (\partial u/\partial x), (\partial u/\partial y), and (\varphi) is a function of class (C^1) in the contrary case.

The solution of the problem thus posed is a generalized solution of equation (1) in the sense of S. L. Sobolev. This follows from arguments analogous to those used by Montel ((^2)).

For equation (2) one obtains theorems on existence, uniqueness, correctness of the Cauchy problem, as well as theorems on continuation, uniform boundedness, and Lyapunov stability of solutions.

For the formulation of these theorems the following definitions are necessary: A positive monotone nondecreasing function (\omega(z)) is called

1) of type (O) (Osgood), if

[
\int_0^{\eta>0}\frac{dz}{\omega(z)}=\infty,
]

2) of type (W) (Wintner), if

[
\int_{\eta>0}^{\infty}\frac{dz}{\omega(z)}=\infty.
]

Using normal families of functions and an analogue of Peano’s method (the method applied by Montel in the Goursat problem), we obtain the following theorem:

Theorem 1. If (f) is continuous and

[
\left|
f(x,y,u,v,w)-f(x,y,u,v',w')
\right|
\le
K(y)\omega\bigl(|v-v'|+|w-w'|\bigr),
]

where (K(y)) is a summable function and (\omega(z)) is of type (O), then the solution of the Cauchy problem for equation (2) exists. (Obviously, if (f) does not depend explicitly on (\partial u/\partial x), (\partial u/\partial y), then for existence it is sufficient that (f) be continuous.)

The uniqueness theorem is analogous to the theorem of Osgood—Tonelli ({}^{3}).

Theorem 2. If
[
\left| f(x,y,u,v,w)-f(x,y,u',v',w')\right|\leq
K(y)\,\omega\left(|u-u'|+|v-v'|+|w-w'|\right),
]
where (K(y)) is a summable function and (\omega(z)) is of type (O), then the Cauchy problem has at most one solution. (If (f) does not depend explicitly on (\partial u/\partial x), (\partial u/\partial y), then (K) may also depend on (x), with (K(x,y)) a summable function of two variables.)

Theorem 3. If the hypotheses of Theorems 1 and 2 are satisfied, then the Cauchy problem is well posed.

The following theorem is an analogue of Wintner’s theorems ({}^{4}) on the continuability of solutions.

Theorem 4. If the hypotheses of Theorems 1 and 2 are satisfied and, in addition, the initial data are defined for all (x), and
[
\left| f(x,y,u,v,w)\right|\leq K_1(y)\,\omega_1\left(|u|+|v+|w|\right),
]
where (K_1(y)) is summable on every finite interval and (\omega_1(z)) is of type (W), then the solution is continuable over the entire plane.

Theorem 5. Under the hypotheses of Theorem 4, if (K_1) is summable on the entire axis and the initial data are uniformly bounded, with
[
\int_{-\infty}^{+\infty} |\psi(\xi)|\,d\xi<+\infty,
]
then the solution is bounded over the entire plane.

Theorem 6. Under the hypotheses of Theorem 3, if (K(y)) is summable on the entire axis and if the condition of Theorem 4 is satisfied, then the solution is stable in the sense of Lyapunov.

Stability is understood in the following sense: for every (\varepsilon>0) one can indicate an (\eta>0) such that, if
[
\sup_{-\infty