On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

Abstract

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation [Formula: see text] with typical discontinuous initial data (u(0, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x1 = 0}, it is shown that the local solution u(t, x) ∈ L∞([0, T] × ℝn) exists and is C∞ away from the forward cuspidal conic surface [Formula: see text] or the cuspidal wedge-shaped surfaces [Formula: see text] respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x1 = 0} and {t = x2 = 0}, we establish the local existence of a solution [Formula: see text] and further show that [Formula: see text] in general due to the degenerate character of the equation under study, where [Formula: see text]. This is an essential difference to the well-known result for solution [Formula: see text] to the two-dimensional semilinear wave equation [Formula: see text] with (v(0, x), ∂tv(0, x)) = (0, φ(x)), where Σ0 = {t = |x|}, [Formula: see text] and [Formula: see text].