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<∞,thenlim supt↗Tmax‖u(⋅,t)‖L∞(Ω)=∞.(⋆)This is used to, secondly, make sure that if additionally f is convex and grows superlinearly in the sense that f′′≥0onR,f(ξ)ξ→+∞asξ→+∞and∫ξ0∞dξf(ξ)<∞for someξ0>0,then for some initial data the above solution must undergo some finite-time L∞ blow-up in the style described in (⋆).