Critical Exponent for a Class of Semilinear Damped Wave Equations with Decaying in Time Propagation Speed

Critical exponent of Fujita type for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

“…Then, taking into account the definition of the zone Z 2 , we use again (18) and both the asymptotic expansions ( 23) and ( 24), for large (1 + t) 1−𝓁 |𝜉| and small (1 + s) 1−𝓁 |𝜉| arguments, respectively. Finally, in the zone Z 3 , we have to consider separately the representations (19) and ( 20) according to the cases in which 𝛾 is integer or not. If 𝛾 ∉ Z, we use the asymptotic expansion (25) of the Bessel functions of the first kind for small arguments, whereas if 𝛾 ∈ Z we use (25) and (26).…”

“…If the assumption that the initial data $$$ {u}_1\in {L}^1\left({R}^n\right) $$$ is removed and we only assume that $$$ {u}_1\in {L}^2\left({R}^n\right) $$$, that is, $$$ {u}_1\notin {L}^{2-\delta}\left({R}^n\right) $$$ for all $$$ \delta \in \left(0,1\right] $$$, then the critical exponent to () is modified into $$$ 1+\frac{4}{n\left(1-\ell \right)} $$$ for a lower thresholds required for $$$ \beta $$$ (see Ebert and Marques 19 ). For the classical damped wave equation, this phenomenon has been investigated in Ikehata and Ohta 20 …”

“…If the assumption that the initial data u 1 ∈ L 1 (R n ) is removed and we only assume that u 1 ∈ L 2 (R n ), that is, u 1 ∉ L 2−𝛿 (R n ) for all 𝛿 ∈ (0, 1], then the critical exponent to (1) is modified into 1 + 4 n(1−𝓁) for a lower thresholds required for 𝛽 (see Ebert and Marques 19 ). For the classical damped wave equation, this phenomenon has been investigated in Ikehata and Ohta.…”