Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation

Abstract

In this paper we derive asymptotic-in-time linear estimates in Hardy spaces Hp(Rn) for the Cauchy problem for evolution operators with structural dissipation. The obtained estimates are a natural extension of the known Lp- Lq estimates, 1 ≤ p≤ q≤ ∞, for these models. Different, standard, tools to work in Hardy spaces, are used to derive optimal estimates

A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations

“…This interesting phenomenon only appears in the case of structural damping, i.e. if δ > 0 (since (10) reduces to an ordinary differential equation if δ = 0).…”

“…In particular, it follows that (22) does not hold for q = p 1 , so that the decay rate t −β p 1 −1 is the same obtained for I 2δ w t (t, •) L p 1 in (8). Indeed, p 1 is related to the scaling of (10).…”

“…Pohozaev [18]. In particular, higher order diffusion equations (see later, (10)) are considered in Section 2.8 in [18] (see also [17]).…”

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

“…However, this zero average condition P j = 0( j = 0,1) seems to be just a technical one (at least) in the case of , while in the case of , such a zero average condition P 1 = 0 seems to be essential in the low dimensional case n = 1,2 in order to get the L 2 ‐decay of solutions. It should be emphasized that the paper due to D'Abbicco‐Ebert‐Picon has also pointed out its importance of the zero average condition P j = 0( j = 0,1) in order to get the decay estimates of solutions to the equations in terms of real Hardy spaces.…”

Optimal energy decay rates for some wave equations with double damping terms

“…recently so many new results are intermittently announced. In particular, we can cite Karch [21], D'Abbicco-Ebert [4,5,6], D'Abbicco-Reissig [8], D'Abbicco-Ebert-Picon [7], Charaõ-da Luz-Ikehata [1], Ikehata-Natsume [14], Ikehata-Takeda [19], and all these papers have contributed to derive several decay estimates and asymptotic profiles of solutions to problem (4)-(2) with θ ∈ (0, 1] (for θ = 0, one can cite Matsumura [23], Nishihara [25], Racke [27], Sobajima-Wakasugi [29], Taylor [30], Todorova-Yordanov [31], and the references therein). In this connection, to the best of authors' knowledge, the paper due to Lu-Reissig [22] first presented a Cauchy problem of (4) with a more generalized time dependent structural damping b(t)(−∆) θ u t to study parabolic aspects of the equation from the viewpoint of energy estimates.…”