Abstract
We consider in this article the damped wave equation in the scale-invariant case with combined two nonlinearities as source term, namely vertical bar u(t)vertical bar(p) + vertical bar u vertical bar(q), and with small initial data. Owing to a better understanding of the influence of the damping term (mu/1+tu(t)) in the global dynamics of the solution, we obtain a new interval for mu that we conjecture to be closer to optimality, or probably optimal, and, thus, characterizes the threshold between the blow-up and the global existence regions. Moreover, taking advantage of the techniques employed in the problem with mixed nonlinearities, we prove for the damped wave equation with only one nonlinearity term vertical bar u(t)vertical bar(p) that the blow-up region is now given by p is an element of (1, p(G)(N + mu)] where p(G) (N) is the Glassey exponent. We think that this new interval for mu has better chances to characterize the threshold in this case. (C) 2020 Elsevier Ltd. All rights reserved.