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 $u_{tt}+(-\Delta)^{\sigma}u_t -\Delta u =f(u)$, where $\sigma \in (0,\frac{1}{2})$ and $f(u) \sim |u|^p$ or $u |u|^{p-1}$ with $p>1 + {2}/(n - 2 \sigma)$. We first show the global existence for initial data small in some weighted Sobolev spaces on $\mathcal R^n$ ($n \ge 2$). 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 $L^1$ spaces.

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 [13,20,21] ). On the other hand, in the case p > p 0 , small data global existence is widely studied, (see [14,15,20,19,6,7,16,10,17,8], for example, and the references therein). The asymptotic profiles of the solutions are obtained as follows. Galley and Raugel (n = 1), Hosono and Ogawa [7] (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 [12] for higher order asymptotic expansion in the case n ≤ 3.) For general space dimensions, Hayashi, Kaikina and Naumkin [6] 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 [6]. 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 by a different method employing Lorentz spaces (Lemma 1). For the estimate of nonlinear term, we use the method of [6,8].
This paper is organized as follows.
• In section 2, we list some notations and state main results.
• In section 3, we list known preliminary lemmas.
• In section 4, we estimate kernels.
• In section 6, we estimate a nonlinear term.
• In section 7, we estimate a convolution term.
• In section 8, we prove Proposition 1 and Theorems 2 and 3. That is, we prove the global existence of the solution of semilinear wave equation with structural damping, and give the asymptotic profile of the solutions.

Main Results
Before stating our results, we list some notations.
Notation 3. For every q ∈ [1, ∞], we abbreviate R n in L q (R n ), and L q norm is denoted by · q .
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.
Remark 4. We note that L q j = L q j ,q j ⊂ L q j ,2 by Lemma A given later, since q j ≤ 2.
We prove Theorem 2 by using the following proposition.
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).
Theorem 3 (Asymptotic profile). Assume the assumption of Proposition 1 with r = 1. Let ε be a positive constant given by Proposition 1 for r = 1, and assume that initial data satisfy (2.17) and (2.18). Let ν be an arbitrary number satisfying Assume moreover that Then there is a constant C depending on such that the solution u ∈ C([0, ∞); Hs ∩ H 0,δ ) ∩ C 1 ((0, ∞); Hs −1 ) of (1.1), which is given by Proposition 1, satisfies the following:

Preliminary lemmas
We list some properties for weak L p and Lorentz spaces which are used in this paper (see [1, section 1.3], for example).
The next corollary immediately follows from Lemma B.
then the following hold.
then H s (R n ) ⊂ L q,2 (R n ).

Decay estimate for the kernels
In this section, we estimate the kernel of the following linear wave equation with structural damping (1.3).
We put Here we note that Estimate of the kernels for low frequency part. In this subsection, we consider low frequency region: Then for every q j ∈ [1, 2) (j = 0, 1) and ϑ ∈ [0, n 2 + α) satisfying 1 the following holds.
where · q denote Before proving Lemma 1, we state two corollaries: Let υ and λ be smooth functions on some interval (0, a) such that Then the conclusion of Lemma 1 holds.
In this paper, we show weighted L 2 estimates of K(t, x) * ϕ by using Fourier analysis.
Now we prove Lemma 1.
then the following hold provided the right-hand sides are finite: where · q is defined by (4.12).
(4.53) Then the following hold provided the right-hand sides are finite: where · q is defined by (4.12).
Proof. We first prove (4.54). Let λ ± be the functions defined by (4.5). By the same reason as in the proof of Lemma 2, we may assume that s 2 = 0.

66)
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 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 estimatê 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.
Since the solution u of (1.3) is written as the conclusion of Theorem 1 follows from the following lemma.

Estimates of a convolution term
Throughout this section, we suppose the assumption (1.2). 7.1. Decay estimates. Throughout this subsection, we suppose the assumption of Proposition 1.
where ζ r,ϑ is the number defined by (6.3).
The assumption (1.2) implies and we can prove (7.3) in the same way.

Proof of Theorems 2 and 3.
Proof of Theorems 2. We prove Theorem 2 by reducing it to Proposition 1.