L∞ blow-up in the Jordan–Moore–Gibson–Thompson equation

The Jordan–Moore–Gibson–Thompson equation τuttt+αutt=βΔut+γΔu+(f(u))ttis considered in a smoothly bounded domain Ω⊂Rn with n≤3, where τ>0,β>0,γ>0, and α∈R. Firstly, it is seen that under the assumption that f∈C3(R) is such that f(0)=0, gradient blow-up phenomena cannot occur in the sense that for any appropriately regular initial data, within a suitable framework of strong solvability, an associated Dirichlet type initial–boundary value problem admits a unique solution u on a maximal time interval (0,Tmax) which is such that ifTmax