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].
On the Proximity of Solutions of Unperturbed and Hyperbolized Heat Equations for Discontinuous Initial Data
