Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations - Aldashev, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 1994)
Forthe equation$$\sum^{m}_{i=t}t^{k_i}U_{x_i,x_i} - U_n + \sum^{m}_{i=t} a_i(x,t)U_{x_i} + b(x, t)u_t + c(x,t)u = 0,$$$$k_i = \text{const} ≥ 0,\; i = l ..... m, x = (x_1,..., x_m),\; m_>2,$$we find a many-dimensional analog of the well-known "Gellerstedt condition"$$ a_i(x,t) = O(1)t^{α},\; i = 1,..., m,\, α >\frac{k_1}{2} - 2.$$We prove that if this condition is satisfied, then the Darboux problems are uniquely solvable.