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 structure of low regularity solutions to the semilinear generalized Tricomi equation ∂ 2 t u − t m ∆u = f (t, x, u) with typical discontinuous initial data (u(0, x), ∂ t u(0, x)) = (0, ϕ(x)); here m ∈ N, x = (x 1 , ..., x n ), n ≥ 2, and f (t, x, u) is C ∞ smooth in its arguments. When the initial data ϕ(x) is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane {t = x 1 = 0}, it is shown that the local solutionand is C ∞ away from the forward cuspidal cone Γ 0 = (t, x): t > 0, |x| 2 = 4t m + 2 , respectively. On the other hand, for n = 2 and piecewise smooth initial data ϕ(x) singular along the two straight lines {t = x 1 = 0} and {t = x 2 = 0}, we establish the local existence of a solution2 )) in general due to the degenerate character of the equation under study; here Γ. This is an essential difference to the well-known result for

On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains

“…Remark 1.6. With respect to the existence, singularity structures, and singularity propagation theories of classical solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions/mixed type regions or to the higher order degenerate hyperbolic equations, so far there have been many interesting results (one can see [1][2][3][4][5]7,13,20,21,26,27] and the references therein). For the linear Tricomi equation in the mixed type region, when a closed boundary value is given, the authors in [15] establish the existence and uniqueness of weak solutions.…”

“…For more general nonlinear degenerate hyperbolic equations with discontinuous initial data, the authors in [20,21] obtained the local existence of low regularity solutions. In the present paper, we focus on the low regularity solution problem for the semilinear generalized Tricomi equation with an initial data u(0, x) in the mixed type region R × R n .…”

“…From this, one can get another initial data ∂ t u(0, x), which is necessary to solve (1.1) in the degenerate hyperbolic region {t ≥ 0}. Finally by using some techniques in degenerate hyperbolic equations (see [20,21,26,27] and the references therein), we can establish the solvability and regularity of the solution u of (1.1) in the domain {t ≥ 0}. Then a local solution u in the mixed type domain R × R n could be obtained by patching the two solutions got in the degenerate elliptic domain and the degenerate hyperbolic domain separately.…”

“…In [20,21], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we have a further study on the existence and regularities of solutions to the following n-dimensional semilinear generalized Tricomi equation in the mixed type domain R × R n where C T > 0 is a constant depending only on T , and the fixed constant exponent μ ≥ 0 fulfills…”

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In [20,21], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation f (t, x, u) is C 1 smooth in its arguments and has compact support with respect to the variable x.

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