Abstract
We are interested in this article in studying the damped wave equation in the scale invariant case with mass term and two combined nonlinearities. More precisely, we consider the following equation:
(E) u(tt) - Delta u + mu/1+tu(t) + nu(2)/(1+t)(2)u = vertical bar ut vertical bar(p) + vertical bar u vertical bar(q), in R-N x [0, infinity),
with small initial data. Under some assumptions on the mass and damping coefficients, nu and mu > 0, respectively, we show that blow-up region and the lifespan bound of the solution of (E) remain the same as the ones obtained in [12] in the case of a mass-free wave equation, i. e. (E) with nu = 0. Furthermore, using in part the computations done for (E), we enhance the result in [30] on the Glassey conjecture for the solution of (E) with omitting the nonlinear term vertical bar u vertical bar(q) Indeed, the blow-up region is extended to p is an element of (1, p(G) (N + mu)] yielding, hence, a better estimate of the lifespan when (mu, - 1)(2) - 4 nu(2) < 1. Otherwise, the two results coincide. Finally, we may conclude that the mass term has no influence on the dynamics of (E) (resp. (E) without the nonlinear term vertical bar u vertical bar(q)).