A BLOW-UP RESULT FOR THE WAVE EQUATION: THE SCALE-INVARIANT DAMPING AND MASS TERM WITH COMBINED NONLINEARITIES

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) \quad u_{tt}-\Delta u+\frac{\mu}{1+t}u_t+\frac{\nu^2}{(1+t)^2}u=|u_t|^p+|u|^q, \quad \mbox{in}\ {{\mathbb{R}}}^N\times[0,\infty), $

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 $ |u|^q $. Indeed, the blow-up region is extended to $ p \in (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 $ |u|^q $).