Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations

Abstract

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" $$ a_i(x,t) = O(1)t^{\alpha},\; i = 1,..., m,\, \alpha >\frac{k_1}{2} - 2.$$ We prove that if this condition is satisfied, then the Darboux problems are uniquely solvable.