On the Moore–Gibson–Thompson Equation and Its Relation to Linear Viscoelasticity

We discuss the parallel between the third-order Moore–Gibson–Thompson equation ∂ t t t u + α ∂ t t u - β Δ ∂ t u - γ Δ u = 0 depending on the parameters α , β , γ > 0 , and the equation of linear viscoelasticity ∂ t t u ( t ) - κ ( 0 ) Δ u ( t ) - ∫ 0 ∞ κ ′ ( s ) Δ u ( t - s ) d s = 0 for the particular choice of the exponential kernel κ ( s ) = a e - b s + c with a , b , c > 0 . In particular, the latter model is shown to exhibit a preservation of regularity for a certain class of initial data, which is unexpected in presence of a general memory kernel κ .