M. Menteshashvili

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

ON SOME NON-LOCAL PROBLEMS FOR QUASI-LINEAR

HYPERBOLIC EQUATIONS WITH RECTILINEAR CHARACTERISTICS

On the x, y plane the non-local characteristic problem is considered for the quasi-linear hyperbolic second order equation

u_{xx} + (1 + u_{x} + u_{y})u_{xy} + (u_{x} + u_{y})u_{yy} = 0  (1)

with admissible parabolic degeneracy, depending on the behavior of the first order derivatives of an unknown solution. In terms of characteristic invariants ξ = y − x, η = y − (u_{x} + u_{y})x equation (1) has the explicit representation of solutions

u(ξ, η) = f(ξ) − g(η) + (ξ − η)[1 − g′(η) + e^{g′(η)}]  (2)

with arbitrary functions f, g.

For equation (1) the following non-local characteristic problem is posed and investigated: define the regular solution u(x, y) of equation (1) simultaneously with its domain of definition under the conditions

(u_{x} + u_{y})|_{y=0} = α(x), u(x, 0) + γ(x)u(\frac{α(x)x}{α(x) − 1}, \frac{α(x)x}{α(x) − 1}) = β(x), (3)

where α ∈ C^{1}[0, b], β, γ ∈ C^{2}[0, a], a = \frac{α(b)b}{α(b)−1} are the given functions. The following theorem is proved.

Theorem. If

α(x) > 1, x ∈ [0, b], α(x) + (log α(x))′(y − x) ≠ 1, x, y ∈ [0, a], (4)

then problem (1), (3) has the unique solution defined in the triangle, bounded by supports of the problem data and passing through its endpoints characteristic y = α(b)(x − b). In the case under consideration the parabolic degeneracy is excluded by conditions (4).