Damped wave equation with a critical nonlinearity

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Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption (2006, background): “Note that Hayashi et al. [13] have recently obtained (1.15) for p c (N) − ε < p < p c (N) ( ε is a small positive constant) with N = 1 and the small data with suitable positivity (see Hayashi et al. [12] in the critical case).”
Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption (2006, result): “For small data refer to Hayashi, Kaikina and Naumkin [9–11].”
$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data (2017, background): “Based on the linear estimates (3), many mathematicians studied the global existence and the blow-up of solutions (see [9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 24, 27, 30, 34, 36, 37] and the references therein).”