J. Gvazava

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON THE NONLINEAR GOURSAT PROBLEM WITH A FREE CHARACTERISTIC

An alternative nonlinear version of the Goursat characteristic problem is considered for the quasilinear mixed type hyperbolic-parabolic equation

x{u_{t}u_{xx} − (u_{x} − u_{t})u_{xt} − u_{x}u_{tt}} + u_{x}(u_{x} + u_{t}) = 0  (1)

with order degeneracy on x = 0.

This problem consists in finding a solution of equation (1) by the characteristic condition

u|_{t=x} = φ(x), 0 ≤ x ≤ a,  (2)

simultaneously with the domain of its definition provided that this solution satisfies the condition

αu_{x} + βu_{t}|_{γ} = ν(x), a ≤ x ≤ b,  (3)

on the unknown characteristic curve γ.

Under the conditions φ ∈ C^{2}[0, a], ν ∈ C^{1}[a, b] an explicit representation of the desired characteristic curve γ by the relation

t = x + (α − β) ∫_{a}^{x} \{\frac{zν(z)}{αaφ′(a)} − 1\}^{−1}dz  (4)

is constructed.

Using representation (4), a nonlinear version of the Asgeirssonian mean value property and general representation of solutions for equation (1), we prove the unique solvability of problem (1)–(3).