Asymptotic profile and optimal decay of solutions of some wave equations with logarithmic damping
- PDF / 469,607 Bytes
- 26 Pages / 547.087 x 737.008 pts Page_size
- 21 Downloads / 185 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Asymptotic profile and optimal decay of solutions of some wave equations with logarithmic damping Ruy Coimbra Char˜ ao and Ryo Ikehata Abstract. We introduce a new model of the nonlocal wave equation with a logarithmic damping mechanism, which is rather weak as compared with frequently studied fractional damping cases. We consider the Cauchy problem for the new model in Rn and study the asymptotic profile and optimal decay rates of solutions as t → ∞ in L2 -sense. The damping terms considered in this paper is not studied so far, and in the low-frequency parameters, the damping is rather weakly effective than that of well-studied power type one such as (−Δ)θ ut with θ ∈ (0, 1). When getting the optimal rate of decay, we meet the so-called hypergeometric functions with special parameters, so the analysis seems to be more difficult and attractive. Mathematics Subject Classification. Primary 35L05; Secondary 35B40, 35C20, 35S05. Keywords. Wave equation, Logarithmic damping, L2 -decay, Asymptotic profile, Optimal decay.
1. Introduction We present and consider a new type of wave equation with a logarithmic damping term: utt + Au + Lut = 0,
(t, x) ∈ (0, ∞) × Rn ,
u(0, x) = u0 (x), ut (0, x) = u1 (x),
n
x∈R ,
where (u0 , u1 ) are initial data chosen as u0 ∈ H 1 (Rn ),
u1 ∈ L2 (Rn ),
and the operator Au := −Δu for u ∈ H 2 (Rn ), and a new operator L : D(L) ⊂ L2 (Rn ) → L2 (Rn ) is defined as follows:
⎫ ⎧ ⎬ ⎨ (log(1 + |ξ|2 ))2 |fˆ(ξ)|2 dξ < +∞ , D(L) := f ∈ L2 (Rn ) ⎭ ⎩ Rn
for f ∈ D(L),
−1 log(1 + |ξ|2 )fˆ(ξ) (x), (Lf )(x) := Fξ→x
and symbolically writing, one can see L = log(I + A). Here, we denote the Fourier transform Fx→ξ (f )(ξ) of f (x) by Fx→ξ (f )(ξ) = fˆ(ξ) := e−ix·ξ f (x)dx Rn
0123456789().: V,-vol
(1.1) (1.2)
148
Page 2 of 26
R. C. Char˜ ao and R. Ikehata
ZAMP
√ −1 as usual with i := −1, and Fξ→x expresses its inverse Fourier transform. Since the new operator L is constructed by a nonnegative-valued multiplication one, it is nonnegative and self-adjoint in L2 (Rn ). Then, by a similar argument to [18, Proposition 2.1] based on Lumer–Phillips theorem one can find that problem (1.1)–(1.2) has a unique mild solution u ∈ C([0, ∞); H 1 (Rn )) ∩ C 1 ([0, ∞); L2 (Rn )) satisfying the energy inequality Eu (t) ≤ Eu (0),
(1.3)
where
1 ut (t, ·) 2L2 + ∇u(t, ·) 2L2 . 2 (1.3) implies the decreasing property of the total energy because of the existence of some kind of dissipative term Lut . For details, see Appendix in this paper. A main topic of this paper is to find an asymptotic profile of solutions in the L2 topology as t → ∞ to problem (1.1)–(1.2) and to apply it to get the optimal rate of decay of solutions in terms of the L2 -norm. Now let us recall several previous works related to linear damped wave equations with constant coefficients. We mention them from the viewpoint of the Fourier transformed equations. (1) In the case of weak damping for Eq. (1.1) with L = I: Eu (t) :=
ˆ+u ˆt = 0, u ˆtt + |ξ|2 u
(1.4)
as is usually observed, for s
Data Loading...