General decay rate for a Moore–Gibson–Thompson equation with infinite history

Well‐posedness and exponential decay for the Moore–Gibson–Thompson equation with time‐dependent memory kernel

“…Different from the energy functional used in Dell'Oro et al [22] and Liu and Chen [25], since we are deal with the time-dependent memory kernel, the energy functional E(t) that we use here is in a simpler form for the subcritical case (𝛾 > 0). However, since the energy derivative d dt E(t) does not contain the term ∫ ∞ 0 𝜇 ′ t (s)||𝜂 t (s)|| 2 1 ds, we cannot deal with the redundant terms brought by the estimation of the new perturbation functional as in Dell'Oro et al [22] and Liu and Chen [25] for the critical case (𝛾 = 0), and thus, we cannot currently obtain the desired decay estimate for the critical case. We left this as an open question for interested readers.…”

“…(1.4) in a history space setting by using the linear semigroup theory. Liu and Chen [25] established general decay results of energy in both subcritical and critical cases.…”

On the Moore-Gibson-Thompson equation with memory with nonconvex kernels

“…for some δ > 0 (see [1,19,20,22,23]), implying in particular the exponential decay of the kernel g(s) ≤ g(0)e −δs . The differential inequality (2.2) is to some extent unsatisfactory, for it imposes severe restrictions on the shape of the kernel, whereas the exponential decay of E(t) should be reasonably expected if g is exponentially decaying solely, with no further information.…”

“…Nonetheless, our techniques can be applied to obtain also decay types other than exponential (see e.g. [19,22]), without assuming a differential inequality on g, but only requiring a proper control from above, along the line of the recent work [5].…”

New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory

“…We refer to [19] for the non-equal wave speeds case. And, Liu et al [22,23] also concerned with the similar question for third-order MGT equations with memory term.…”