Affiliations: Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, Saudi Arabia
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)utt−t2mΔu+μtut+ν2t2u=|ut|p,in RN×[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 μ1+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.
