On the global solution problem for semilinear generalized Tricomi equations, I

Abstract

In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equationWe show that there exists a critical exponent p crit (m, n) > 1 such that the solution u, in general, blows up in finite time when 1 < p < p crit (m, n). We further show that there exists a conformal exponent p conf (m, n) > p crit (m, n) such that the solution u exists globally when p > p conf (m, n) provided that the initial data is small enough. In case p crit (m, n) < p ≤ p conf (m, n), we will establish global existence of small data solutions u in a subsequent paper [13].

Finite lifespan of solutions of the semilinear wave equation in the Einstein–de Sitter spacetime

“…Thus, for the self-interacting waves propagating in the radiation dominated universe with k = 1/2 and n = 3 there is small data blowing up solution if 1 < p < 7 3 . For the semilinear generalized Tricomi equation ∂ 2 t u − t m ∆u = |u| p with increasing coefficient, that is with m ∈ N, the critical exponent p crit (m, n) and conformal exponent p conf (m, n) are suggested in [13]. Then, there are interesting articles on the non-linear higher-order degenerate hyperbolic equations [14], the low regularity solution problem for the semilinear mixed type equation [16], and the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with discontinuous initial data [15].…”

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

“…In the authors treated a generalized Tricomi equation which after a change of variables can be reduced to the semi‐linear time‐dependent scale‐invariant wave equation with damping term and with the power non‐linearity . For this special power non‐linearity with and the paper proposes that the critical exponent is given by the positive root of the equation In this paper we consider besides the case the case , too (see also ).…”

Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case

“…it is still open to determine the critical exponent. we also remark that for β = 1, the case of µ ≈ 1 is related to the semilinear generalized Tricomi equation, which comes from the gas dynamics, see [7,8].…”