On the solutions of the Cauchy problem for a class of partial differential equations with double characteristic at a point

We give an explicit representation of the solutions of the Cauchy problem, in terms of series of hypergeometric functions, for the following class of partial differential equations with double characteristic at the origin:(xk∂t+a∂x)(xk∂t+b∂x)u+cxk−1∂tu=0,(xk∂t+a∂x)(xk∂t+b∂x)u+cxk−1∂tu=0,u(0,x)=u0(x),u(0,x)=u0(x),∂tu(0,x)=u1(x).∂tu(0,x)=u1(x). We show that the solutions are holomorphic, ramified around the characteristic surfaces K=K1∪K2∪K3K=K1∪K2∪K3 withK1:a(k+1)t−xk+1=0,K2:b(k+1)t−xk+1=0,K3:x=0.K3:x=0.