Blow-up result for semilinear damped wave equations with data from negative order Sobolev spaces: the critical case

In this paper, we address an open question posed in (Chen and Reissig, { J. Evol. Equ.} 23, 13, (2023)). Specifically, we show that the weak solution to the damped wave equation with power type nonlinearity |u|ᵖ, given initial data from the Sobolev spaces of negative order Ḣ-γ(Rⁿ), γ ∈ (0, n/2), will blow-up in the critical case p = pCrit(n, γ):=1+4/n+2γ. We also investigated a similar question in the context of the damped wave equations associated with sub-Laplacian on the Heisenberg group Hⁿ, filling the gap of (Dasgupta et. al, { J. Evol. Equ.} 24, 51, (2024)) by proving that the critical exponent p = pCrit(Q, γ)=1+4/Q+2γ belongs to the blow-up case, where $Q:=2n+2$ is the homogeneous dimension of Hⁿ. Our proof hinges upon suitable test function methods. Finally, we end this paper by exploring the diffusion phenomenon of the damped wave equation on the Heisenberg group with initial data belonging additionally to Sobolev spaces of negative order.