Abstract
In this article, we consider the damped wave equation in the scale-invariant case with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: ( E ) u t t − t 2 m Δ u + μ t u t + ν 2 t 2 u = | u t | p , in R N × [ 1 , ∞ ) , that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, ν and μ > 0, respectively, we prove that blow-up region and the lifespan bound of the solution of ( E ) remain the same as the ones obtained for the case without mass, i.e. ( E ) with ν = 0 which constitutes itself a shift of the dimension N by μ 1 + m compared to the problem without damping and mass. Finally, we think that the new bound for p is a serious candidate to the critical exponent which characterizes the threshold between the blow-up and the global existence regions.