Second-order hyperbolic equations with strong characteristic degeneracy at the initial hypersurface

Equations of the following form are considered: psi(2) (t, x) u(tt) + phi(t, x )u(t) - Sigma/i,j (a(ij)(t, x)u(xi))(xj) +Sigma/i b(i)(t, x)u(xi) + c(t, x)u = f(t, x), (1) where (t, x) epsilon H = (0, T] x R-n, psi(t, x) greater than or equal to 0, phi(t, x) greater than or equal to 0; Sigma/i,j a(ij) (t, x)xi(i)xi(j) greater than or equal to 0 for all (t, x) is an element of H and xi = (xi 1,...,xi n) is an element of R-n. In place of the Cauchy problem for (1), a problem without initial data but with constraints on the admissible growth of the solution as t --> 0 and as \x\ --> infinity is discussed. The unique solubility of (1) in certain Sobolev-type weighted spaces is proved. The smoothness properties of generalized solutions are studied. Bibliography: 25 titles.