L1 − L1 estimates for the strongly damped plate equation

Some $L^1$-$L^1$ estimates for solutions to visco-elastic damped $σ$-evolution models

“…Remark 1.1. Here we want to underline that at the first glance the decay estimates for solutions produced from the results of [3] or [1] are somehow better than those of Theorem 1.1 when we choose σ " 1 formally or σ " 2, respectively. The fact is that this comes from the very special structure of solutions to (1) in the cases of σ " 1 and σ " 2 coupled with some used techniques under the suitable restriction to space dimensions.…”

“…At prsent, there has been very little work on the question of getting L 1 ´L1 estimates for solutions to (1). Back in 2000, let us first recall the pioneering paper of Shibata [3] devoting to the study of one of the most well-known equations of (1) in the case σ " 1, the so-called strongly damped wave equation.…”

“…odd, by taking into considerations the connection to Fourier multipliers appearing for wave models. Quite recently, regarding a different interesting model of (1), namely that with σ " 2, D'Abbicco and his collaborators [1] have employed a different technique in comparison with that used in [3] to derive L 1 ´L1 estimates for solutions to the strongly damped plate equation as follows:…”

The critical exponent for nonlinear damped $σ$-evolution equations

“…It is important to remark that, even if the noneffective damping does not influence the critical exponent, the damping has a great influence on the regularity of the solution to our problems. It produces a smoothing effect which smooths out oscillations at high frequencies, in particular, allowing us to derive L 1 − L 1 estimates, for instance, as done in [13]. Such a property is typical of noneffective damping; in the effective case, the regularity of the solution is not influenced by the damping action.…”

A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

“…A study on asymptotic profile and L p -L q estimates to the equation (1.6) has been done in the papers [6], [10], [12], [13], [30], [33], and [27], and the corresponding critical exponent problems (mainly) to the equation (1.7) are treated in the papers [11], [17,18], [29], and [34]. As for the related motivated topics concerning the asymptotic profiles for the higher order PDEs and the other types of linear equations one can also cite [7,8], [36], [39], [38], and the references therein.…”