Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture

Abstract

We study the small data Cauchy problem for semilinear generalized Tricomi equations with nonlinear term of derivative type:Blow-up result and lifespan estimate from above are established for 1 < p ≤ 1 + 2 (m+1)(n−1)−m . If m = 0, our result coincides with that of the semilinear wave equation. The novelty is that we construct a new test function by using cutoff functions, the modified Bessel function and a harmonic function. We also find a interesting phenomenon: if n = 2, the blow-up power is independent of m.

A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime

“…The upper bound p Gla (1 − k)n + 2k in Theorem 1.2 is consistent with the upper bound for the semilinear generalized Tricomi with nonlinearity of derivative type (when the power in the speed of propagation is positive and the Cauchy data are assumed at the initial time t = 0) see e.g. [15,10].…”

“…Proof. We are going to prove the representation formula in (15) by means of a suitable change of variables that transforms (13) in a linear wave equation with scale-invariant damping and mass terms and allows us to employ a result from [22]. More specifically, we perform the transformation…”

“…where ℓ > 0, p > 1, ε > 0, has been attracting a lot of attention and different approaches have been applied to establish several blow-up results for p in suitable ranges depending on n, ℓ. Among these approaches we find, on the one hand, comparison arguments based either on the fundamental solution for the operator ∂ 2 t − t 2ℓ ∆ in [16] or on the employment of a special positive solution of the corresponding homogeneous equation involving modified Bessel functions of second kind in [10], and, on the other hand, a modified test function method in [15]. In particular, the result that we are going to show for (5) is consistent with those for (6) in [15,10], although we shall use a different approach to prove this result.…”

“…Among these approaches we find, on the one hand, comparison arguments based either on the fundamental solution for the operator ∂ 2 t − t 2ℓ ∆ in [16] or on the employment of a special positive solution of the corresponding homogeneous equation involving modified Bessel functions of second kind in [10], and, on the other hand, a modified test function method in [15]. In particular, the result that we are going to show for (5) is consistent with those for (6) in [15,10], although we shall use a different approach to prove this result. Moreover, we will see that the upper bound for the exponent p in the blow-up result for (1) is a shift of the corresponding upper bound for (5).…”

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

“…We recall that for us the fact that p Str (n, ℓ) is the critical exponent for (2) with f (u, ∂ t u) = |u| p means the following: for any 1 < p < p Str (n, ℓ) local in time solutions blow up in finite time under suitable sign assumptions for the Cauchy data and regardless of their size, while for p > p Str (n, ℓ) (technical upper bounds for p may appear, depending on the space for the solutions) a global in time existence result for small data solutions holds. Very recently, even the cases with derivative type nonlinearity f (u, ∂ t u) = |∂ t u| p and with mixed nonlinearity f (u, ∂ t u) = |u| q + |∂ t u| p have been studied from the point of view of blow-up dynamics in [24,2,21,12]. In particular, in [24] a blow-up result when f (u,…”