On the strauss index of semilinear tricomi equation

Abstract

In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution u of the multi-dimensional semilinear Tricomi equation ∂ 2 t u − t ∆u = |u| p , u(0, •), ∂tu(0, •) = (u 0 , u 1), where t > 0, x ∈ R n , n ≥ 2, p > 1, and u i ∈ C ∞ 0 (R n) (i = 0, 1). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of u(t, •) L ∞ x (R) is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of u in t, then the crucial weighted Strichartz estimates are established and the global existence of solution u is proved when p > 5.

On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity

“…Moreover, it is quite reasonable to conjecture that the exponent p c (n, ℓ, µ, ν 2 ) is critical even for higher dimensions. Indeed, for µ = ν 2 = 0 we have that p c (n, ℓ, 0, 0) = p Str (n, ℓ) for n 2 and p c (n, ℓ, 0, 0) = p Fuj (ℓ) for n = 1 (see [18,Remark 1.6] for the one-dimensional case) according to the results for (2) with power nonlinearity that we recalled in the introduction. In particular, when ℓ = 0 too we find that p c (n, 0, 0, 0) is the solution to the quadratic equation (n − 1)p 2 − (n + 1)p − 2 = 0, namely, the celebrated exponent named after the author of [31] which is the critical exponent for the semilinear wave equation.…”

“…Plugging this last estimate from below for the space integral of the nonlinear term in (18), for t T 1 we get…”

“…On the other hand, in [38,39] the fundamental solution for the generalized Tricomi operator (sometimes called also Gellerstedt operator in the literature) is employed to derive an integral representation formula, that is used in turn to prove (under suitable assumptions on p) the local existence of solutions and the global existence of small data solutions, respectively. Afterwards, in the series of papers [15,16,17,37,18,32] it was established that the critical exponent for (2) with f (u, ∂ t u) = |u| p and n 2 is the Strauss-type exponent (cf. [31] and the literature citing this renowned paper), that we denote p Str (n, ℓ) in the present paper, given by the biggest root of the quadratic equation…”