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 consider
(Tr) u(tt) - t(2m) Delta u = vertical bar u(t)vertical bar(p) + vertical bar u vertical bar(q), in R-N x [0, infinity),
with small initial data, where m >= 0.
For the problem (Tr) 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 (vertical bar u(t)vertical bar(p) or vertical bar u vertical bar(q)). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation (Tr) 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 (Tr) 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 vertical bar u(t)vertical bar(p) in the right-hand side of (Tr).