Asymptotic profile of solutions for semilinear wave equations with structural damping

This paper is concerned with the initial value problem for semilinear wave equation with structural damping utt+(-Δ)σut-Δu=f(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{tt}+(-\Delta )^{\sigma }u_t -\Delta u =f(u)$$\end{document}, where σ∈(0,12)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,\frac{1}{2})$$\end{document} and f(u)∼|u|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(u) \sim |u|^p$$\end{document} or u|u|p-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u |u|^{p-1}$$\end{document} with p>1+2/(n-2σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p> 1 + {2}/(n - 2 \sigma )$$\end{document}. We first show the global existence for initial data small in some weighted Sobolev spaces on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^n$$\end{document} (n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}). Next, we show that the asymptotic profile of the solution above is given by a constant multiple of the fundamental solution of the corresponding parabolic equation, provided the initial data belong to weighted L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} spaces.


Introduction
In this paper, we consider the unique global existence of solutions and diffusion phenomina for the Cauchy problem of the semilinear wave equation with structural damping (damping term depends on the frequency) for σ ∈ (0, 1 2 ): u tt − Δu + (−Δ) σ u t = f (u), t≥ 0, x ∈ R n , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R n , (1.1) with σ ∈ (0, 1 2 ), Narazaki and Reissig [20] gave some L p − L q (1 ≤ p ≤ q ≤ ∞) estimates of the solutions. D'Abbicco and Ebert [2] showed the diffusion phenomena, by giving the L p − L q decay estimates of the difference between the low frequency part of the solution of (1.3) and that of the corresponding parabolic equation (1.4) with initial data (−Δ) σ u 0 +u 1 . Ikehata and Takeda [11] showed that a constant multiple of the fundamental solution of the parabolic equation (1.4) gives the asymptotic profile of the solutions of (1.3) with (u 0 , u 1 ) ∈ (L 1 ∩H 1 )×(L 1 ∩L 2 ) (see Remark 3). For semilinear structural damped wave equation (1.1) with σ ∈ (0, 1 2 ), D'Abbicco and Reissig [5] first showed global existence and decay estimates of the solution of (1.1) with small initial data for space dimension 1 ≤ n ≤ 4 and p ∈ [2, n/[n − 2] + ] such that p > p σ := 1 + 2 n − 2σ . (1.5) They showed the results by using (L 1 ∩ L 2 ) − L 2 estimates of solutions of the linear wave equation with structural damping (1.3). In [5], they considered also for σ ∈ [ 1 2 , 1] and showed that p σ is critical in a particular case u tt + 2(−Δ) σ u t − Δu = 0. Using the L p − L q decay estimate (1 ≤ p ≤ q ≤ ∞) of solutions of the linear wave equations with structural damping (1.3) by [2] for low frequency part, D'Abbicco and Ebert [4] (see also [3]) showed the unique existence of solutions of (1.1) for small initial data in some Sobolev spaces and gave the decay estimates of the solutions, in the following two cases: p σ < p, n < 1 + 2 max m ∈ N; m < 1 + 2σ 1 − 2σ , (1.6) or p σ < p < 1 + 2(1 + 2σ) [n − 2(1 + 2σ)] + , n 2 (1.7) In [4], they also treated the case where −Δu is replaced by (−Δ) δ u with δ > 0. The assumption (1.6) and (1.7) for p < 2 restrict the space dimension from above. The first purpose of this paper is to remove restriction of the space dimension n from above for every σ ∈ (0, 1 2 ). The second purpose is to give the asymptotic profile of the solutions of (1.1) as t → ∞, if small initial data belongs to some weighted L 1 spaces. We show that a constant multiple of the fundamental solution of the parabolic equation (1.4) gives the asymptotic profile of (1.1) (Theorem 3). As as far as the author knows, there seems to be no results on the asymptotic profile for semilinear wave equation with structural damping (1.1) for σ ∈ (0, 1 2 ). In the case σ = 0, the asymptotic profile for semilinear damped wave equation is investigated. Since we treat nonlinear term not necessarily absorbing, we only refer to the results for non-absorbing type nonlinear term. Then if 1 < p ≤ p 0 where p 0 := 1 + 2 n : Fujita Exponent, then the solution of the semilinear damped wave equation blows up when f (u) = |u| p and the integrals of initial data on R n are positive (see [10,15,23,24] ). On the other hand, in the case p > p 0 , small data global existence is widely studied, (see [7][8][9]12,[16][17][18][19]21,23], for example, and the references therein). The asymptotic profiles of the solutions are obtained as follows. Galley and Raugel [6] (n = 1), Hosono and Ogawa [8] (n = 2), showed that the asymptotic profile of the solutions is given by a constant multiple of the heat kernel G(t, x), provided the initial data belong to some Sobolev spaces. (See also Kawakami and Takeda [14] for higher order asymptotic expansion in the case n ≤ 3.) For general space dimensions, Hayashi, Kaikina and Naumkin [7] proved the unique existence of global solution u ∈ C([0, ∞); Hs ∩ H 0,δ ) for small initial data belonging to some weighted L 1 spaces, and showed that a constant multiple of the heat kernel gives the asymptotic profile of the solutions (see Remark 9). We consider the equation in weighted Sobolev spaces as in [7]. The high frequency part of the structural damped wave equation has a good regularizing property. However, unlike the damped wave equation (σ = 0), the Fourier transform of the kernel of the linear structural damped wave equation is singular at the origin. This fact causes the difficulty when we treat the equation in weighted Sobolev spaces. To get around this difficulty, we estimate the low frequency part in a new way employing Lorentz spaces (Lemma 1). For the estimate of nonlinear term, we use the method of [7,9]. This paper is organized as follows.
The Lorentz space L q,r consists of all locally integrable function ϕ on R n such that where ϕ * (t) = inf{τ ; m(τ, ϕ) ≤ t} (the rearrangement of ϕ).

Notation 6.
For κ ∈ (0, n), Riesz potential is the operator First we give the asymptotic profile of the solutions to linear wave equation with structural damping.
be a unique global solution of (1.3). Then the following holds.
If initial data belong to weighted L 1 space, the asymptotic profile of the solution is given by a constant multiple of the fundamental solution of the parabolic equation (1.4).

23)
where G σ is defined by (2.2) and

Preliminary lemmas
We list some properties for weak L p and Lorentz spaces which are used in this paper (see [1, section 1.3], [22], for example). Then provided the right-hand side is finite.
The next corollary immediately follows from Lemma B.
then the following hold.
We put Here we note that We put for j = 0, 1. Dividing the kernel into for j = 0, 1, we estimate each part.

Estimate of the kernels for low frequency part
In this subsection, we consider low frequency region:

Then for every
the following holds.
where · q denote (4.12) Before proving Lemma 1, we state two corollaries: Let υ and λ be smooth functions on some interval (0, a) such that on (0, a) for every j = 0, 1, · · · . Put Then the conclusion of Lemma 1 holds.
, by using the description of kernels by Bessel functions.
In this paper, we show weighted L 2 estimates of K(t, ·) * ϕ in a way different from [2] by employing Lorentz spaces.
Now we prove Lemma 1.

Lemma 3. Let ϑ
(4.53) Then the following hold provided the right-hand sides are finite:
provided the right-hand sides are finite.
Proof. We easily see that on the support of χ high . Hence, that is, (4.63) holds.
In the proof of [9, p. 10] (see also [7, p. 643]), the following Leibniz rule is shown: for every nonnegative integer k, we have Taking ϑ = 0 and δ in this inequality, we obtain (4.64). We can prove (4.65) and (4.66) in the same way.
In the same way as in the proof of (4.75), we see that for every k ∈ N ∪ {0}. Then by the same calculation as in the proof of (4.64), we obtain (4.70). We can estimate in the same way, and obtain the assertion for K 0,mid .

Asymptotic profile of the solutions of linear equation
In this section, we prove Theorem 1.

NoDEA
Asymptotic profile of solutions for semilinear Page 23 of 43 16 Lemma 6. Let u j ∈ L 1 ∩ L 2 for j = 0, 1. Then the following hold.
The assumption (1.2) implies and we can prove (7.3) in the same way.