A note on initial value problem for the generalized Tricomi equation in a mixed-type domain

In this paper, we study the well-posedness of initial value problem for n-dimensional generalized Tricomi equation in the mixed-type domain {(t, x): t . [.1,+.), x . .n} with the initial data given on the line t = .1 in Hadamard.s sense. By taking partial Fourier transformation, we obtain the explicit expression of the solution in terms of two integral operators and further establish the global estimate of such a solution for a class of initial data and source term. Finally, we establish the global solution in time direction for a semilinear problem used the estimate.