Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities

Abstract

We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we considerwith small initial data, where m ≥ 0.For the problem (T r) with m = 0, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity (|u t | p or |u| q ). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation (T r) with m ≥ 0, and we derive an estimate of the lifespan in terms of the Tricomi parameter m. As an application of the method developed for the study of the equation (T r) we obtain with a different approach the same blow-up result as in [18] when we consider only one time-derivative nonlinearity, namely we keep only |u t | p in the right-hand side of (T r).

Blow-up result for a weakly coupled system of wave equations with a scale-invariant damping, mass term and time derivative nonlinearity

“…In the context of combined nonlinearities, similar results are obtained in [3,12]. Now, in a more general context, namely m, µ, ν > 0, the damped wave equation with Tricomi and mass terms is considered in [1] which somehow constitutes an improvement of [13,14] for −1 < m < 0 and without the mass term.…”

“…Hence, in the same spirit of use as e.g. in [10,12,13,14], we define the positive test function ψ η i (x, t) as the solution of the conjugate equation associated with the linear problem of (1.1), namely (3.1) ) x + e −η (1+m) x for N = 1.…”

Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques

“…for which the possibly optimal blow-up results in the case λ > 1 and λ = 1 are given in [22] and [9] respectively (in the latter reference a more general combined nonlinearity is considered, see also [27,19,4,8] for related problems). For the model with λ = 1 the critical exponent seems to be p G (n + µ), in the sense that we have blow-up for any 1 < p ≤ p G (n + µ) and global existence for p > p G (n + µ).…”