Abstract
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 \({{\mathbb {R}}}^n\) (\(n \ge 2\)). 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.
Similar content being viewed by others
1 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 \(\sigma \in (0,\frac{1}{2})\):
where \(f \in C^{[\bar{s}],1 }({{\mathbb {R}}})\) (\(1 \le \bar{s},\; [\bar{s}] < p\)) satisfies
for a positive constant C. Here, \([\bar{s}]\) denotes the integer part of \(\bar{s}\).
For linear wave equations with structural damping:
with \(\sigma \in (0,\frac{1}{2})\), Narazaki and Reissig [20] gave some \(L^p-L^q\) (\(1 \le p \le q \le \infty \)) 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
with initial data \((-\Delta )^{\sigma }u_0 + u_1\). 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 \((u_0, u_1) \in ( L^1 \cap H^1) \times (L^1 \cap L^2)\) (see Remark 3).
For semilinear structural damped wave equation (1.1) with \(\sigma \in (0, \frac{1}{2})\), 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 \(1 \le n \le 4\) and \(p \in [2,n/[n-2]_+]\) such that
They showed the results by using \((L^1 \cap L^2)-L^2\) estimates of solutions of the linear wave equation with structural damping (1.3). In [5], they considered also for \(\sigma \in [\frac{1}{2},1]\) and showed that \(p_\sigma \) is critical in a particular case \(u_{tt}+ 2(-\Delta )^{\sigma }u_t -\Delta u = 0\). Using the \(L^p-L^q\) decay estimate (\(1 \le p \le q \le \infty \)) 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:
or
In [4], they also treated the case where \(-\Delta u\) is replaced by \((- \Delta )^\delta u\) with \(\delta > 0\).
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 \(\sigma \in (0, \frac{1}{2})\).
The second purpose is to give the asymptotic profile of the solutions of (1.1) as \(t \rightarrow \infty \), 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 \(\sigma \in (0,\frac{1}{2})\).
In the case \(\sigma = 0\), 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 \(1 < p \le p_0\) where
then the solution of the semilinear damped wave equation blows up when \(f(u) = |u|^p\) and the integrals of initial data on \({{\mathbb {R}}}^n\) 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 \(n \le 3\).) For general space dimensions, Hayashi, Kaikina and Naumkin [7] proved the unique existence of global solution \(u \in C([0,\infty );H^{\bar{s}} \cap H^{0,\delta })\) 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 (\(\sigma = 0\)), 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.
-
In Sect. 2, we list some notations and state main results.
-
In Sect. 3, we list known preliminary lemmas.
-
In Sect. 4, we estimate kernels.
-
In Sect. 5, we prove Theorem 1.
-
In Sect. 6, we estimate a nonlinear term.
-
In Sect. 7, we estimate a convolution term.
-
In Sect. 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.
2 Main results
Before stating our results, we list some notations.
Notation 1
We write \(\varphi (x) \lesssim \psi (x)\) on I if there exists a positive constant C such that
We write \(\varphi (x) \sim \psi (x)\) on I, if \(\varphi (x) \lesssim \psi (x)\) and \(\psi (x) \lesssim \varphi (x)\) on I.
Notation 2
For \(a \in {{\mathbb {R}}}\), \([a]_+: = \max \{a, 0\}\).
Notation 3
For every \(q \in [1,\infty ]\), we abbreviate \({{\mathbb {R}}}^n\) in \(L^q({{\mathbb {R}}}^n)\), and \(L^q\) norm is denoted by \(\Vert \cdot \Vert _q\).
Notation 4
Let \( H^{s,\delta } = H^{s,\delta }({{\mathbb {R}}}^n)\) denote the weighted Sobolev space equipped with the norm
\(H^{s,0}\) equals \(H^s\). Let \( \dot{H}^{s} = \dot{H}^{s}({{\mathbb {R}}}^n)\) denote the homogeneous Sobolev space equipped with the norm
Notation 5
(see [1, section 1.3], for example) Let \(q \in (1,\infty )\) and \(r \in [1,\infty ]\). Let \(\mu \) be the Lebesgue measure on \({{\mathbb {R}}}^n\). The distribution function \( m(\tau , \varphi )\) is defined by
The Lorentz space \(L_{q,r}\) consists of all locally integrable function \(\varphi \) on \({{\mathbb {R}}}^n\) such that
where \( \varphi ^*(t) = \inf \{\tau ; m(\tau ,\varphi ) \le t \} \) (the rearrangement of \(\varphi \)).
Notation 6
For \(\kappa \in (0,n)\), Riesz potential is the operator
First we give the asymptotic profile of the solutions to linear wave equation with structural damping.
Theorem 1
Let \(n \ge 1\). Let \((u_0, u_1) \in ( L^1 \cap L^2) \times (L^1 \cap H^{-2 \sigma })\) such that \(|\cdot |^{\theta _j} u_j \in L^1\) with \(\theta _j \in [0,1]\) \((j = 0,1)\). Let \(u \in C([0,\infty ); H^1) \cap C^1((0,\infty );L^2)\) be a unique global solution of (1.3). Then the following holds.
where
Remark 1
Remark 2
The function \(H_\sigma (t,x)\) is the fundamental solution of the parabolic equation (1.4). We easily see that
(see (5.8) and (5.16)). Putting \(\theta _0 = [\theta - 2 \sigma ]_+ \) and \(\theta _1 = \theta \) (\(\theta \in (0,1]\)) in (2.1), and taking (2.4) and the assumption \(\sigma \in (0,\frac{1}{2})\) into consideration, we obtain
Thus, the decay order of \(\Vert u(t,\cdot ) - \vartheta _1 G_{\sigma }(t, \cdot )\Vert _2 \) is larger than that of \(\Vert G_{\sigma }(t,\cdot )\Vert _2\) itself, and therefore, \(\vartheta _1 G_{\sigma }(t,x)\) gives the asymptotic profile of the solution if \(\vartheta _1 \ne 0\).
If \(u_1 = 0\), then (2.1) implies
Thus, the decay order of \(\Vert u(t,\cdot ) - \vartheta _0 H_{\sigma }(t, \cdot )\Vert _2 \) is larger than that of \(\Vert H_{\sigma }(t,\cdot )\Vert _2\) itself if \(\theta _0 > 0\), and therefore, \(\vartheta _0 H_{\sigma }(t,x)\) gives the asymptotic profile if \(\vartheta _0 \ne 0\).
Remark 3
Ikehata and Takeda [11, Theorem 1.2] showed
as \(t \rightarrow \infty \) for initial data in \((u_0, u_1) \in ( L^1 \cap H^1) \times (L^1 \cap L^2)\).
If \(u_1 = 0\), Karch [13, Corollary 4.1] showed
as \(t \rightarrow \infty \) for \(u_0 \in L^1 \).
Theorem 2
(Global existence of the solution) Let \(n \ge 2\), and
Assume that \(\bar{s} \ge 1\) and \([\bar{s}] < p\). If \( 2 \bar{s} < n\), assume moreover that
Assume that \(f \in C^{[\bar{s}],1 }({{\mathbb {R}}})\) satisfies (1.2). Let \(q_j\; (j=0,1) \) be numbers such that
- (Case 1). :
-
In the case \(p_\sigma < p \le 1 + \frac{4}{n + 2 - 4\sigma }\), let \(\delta \) be a number satisfying
$$\begin{aligned} 2 \left( \frac{1}{p-1} + \sigma \right) - \frac{n}{2} - 1 < \delta \le \frac{2}{p-1} - \frac{n}{2}. \end{aligned}$$(2.9)Then there exists a positive number \(\varepsilon \) such that if initial data
$$\begin{aligned} u_0 \in H^{\bar{s}} \cap H^{0,\delta }, \langle \cdot \rangle ^\delta u_0 \in L^{q_0,2 }, \quad u_1 \in \dot{H}^{\bar{s}-1}, \langle \cdot \rangle ^\delta u_1 \in L^{{ q_1,2 }} \end{aligned}$$(2.10)satisfy
$$\begin{aligned} \begin{aligned}&\Vert \langle \cdot \rangle ^\delta u_0 \Vert _{q_0,2 } + \Vert \langle \cdot \rangle ^\delta u_0 \Vert _2 + \Vert u_0 \Vert _{{H^{\bar{s}}}} + \Vert \langle \cdot \rangle ^\delta u_1 \Vert _{{ q_1,2 }} + \Vert u_1 \Vert _{{\dot{H}^{s-1} }} \le \varepsilon , \end{aligned} \end{aligned}$$(2.11)then initial value problem (1.1) has a unique global solution \(u \in C([0,\infty ); H^{\bar{s}}\cap H^{0,\delta }) \cap C^1((0,\infty );H^{\bar{s} - 1}) \).
- (Case 2). :
-
In the case \(p >1 + \frac{4}{n + 2 - 4\sigma }\), there exists a positive number \(\varepsilon \) such that if initial data
$$\begin{aligned} u_0 \in H^{\bar{s}} \cap L^{q_0,2}, \quad u_1 \in H^{\bar{s}-1} \cap L^{q_1,2}, \end{aligned}$$satisfy
$$\begin{aligned}&\Vert u_0 \Vert _{q_0,2} + \Vert u_0 \Vert _{H^{\bar{s}}} + \Vert u_1 \Vert _{q_1,2} + \Vert u_1\Vert _{{ H^{\bar{s}-1}}} \le \varepsilon , \end{aligned}$$(2.12)then initial value problem (1.1) has a unique global solution \(u \in C([0,\infty ); H^{\bar{s}}) \cap C^1((0,\infty );H^{\bar{s} - 1}) \).
Remark 4
We note that \(L^{q_j} = L^{q_j, q_j} \subset L^{q_j, 2}\) by Lemma A given later, since \(q_j \le 2\).
Remark 5
If the space dimension \(n = 2\), then \( 1 + \frac{4}{n + 2 - 4\sigma } \le p_{\sigma }\), and therefore, (Case 1) does not occur.
We prove Theorem 2 by using the following proposition.
Proposition 1
(Global existence of the solution) Let \(n \ge 2\) and \(r \in [1,\frac{2n}{n + 4 \sigma })\). Let
Assume that \(\bar{s} \ge 1\) and \([\bar{s}] < p\). If \( 2 \bar{s} < n\), assume moreover (2.7). Assume that \(f \in C^{[\bar{s}],1 }({{\mathbb {R}}})\) satisfies (1.2). Let \(\delta \) be a non-negative constant satisfying
and
Let
Then there exists a positive number \(\varepsilon \) such that if initial data
satisfy
in the case \(r = 1\), and
in the case \(r \in (1,2]\), then initial value problem (1.1) has a unique global solution \(u \in C([0,\infty ); H^{\bar{s}} \cap H^{0,\delta })) \cap C^1((0,\infty );H^{\bar{s}-1}) \).
Furthermore, the solution satisfies estimate:
Remark 6
The assumption \(r < \frac{2n}{n + 4\sigma }\) implies that \(n(\frac{1}{r} - \frac{1}{2}) - 2\sigma > 0\). The inequality
is equivalent to
which holds by (2.13) since \( \sigma < 1\). Hence, we can take a non-negative number \(\delta \) satisfying assumptions (2.14) and (2.15).
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 \(\varepsilon \) be a positive constant given by Proposition 1 for \(r = 1\), and let \(\theta \in [0,1]\). Assume that initial data satisfy (2.17) and (2.18) and that \( \langle \cdot \rangle ^{[\theta - 2\sigma ]_+} u_0, \langle \cdot \rangle ^{\theta } u_1 \in L^1\). Let \(\nu \) be an arbitrary number satisfying
Assume moreover that
Then there is a constant C depending on
such that the solution \(u \in C([0,\infty ); H^{\bar{s}} \cap H^{0,\delta }) \cap C^1((0,\infty );H^{\bar{s} - 1}) \) of (1.1), which is given by Proposition 1, satisfies the following:
where \(G_{\sigma }\) is defined by (2.2) and
Remark 7
The right-hand sides of (2.21) and (2.22) are positive. In fact, assumtion (2.15) implies \(\frac{n}{2}(p-2) + p \delta > 0\), and (2.7) implies \( n - \frac{p}{2}(n - 2 \bar{s}) > 0\). Hence we can take \(\nu \) satisfying (2.21) and (2.22).
Remark 8
Since (2.4) holds, (2.23) implies that \(G_\sigma \) gives the asymptotic profile of the solution if \(\varTheta \ne 0 \).
Remark 9
In the case \(\sigma = 0\), Hayashi, Kaikina and Naumkin [7] showed the existence of global solution \(u \in C([0,\infty );H^{\bar{s}} \cap H^{0,\delta })\) of the semilinear damped wave (1.1) with \(\sigma = 0\) for small initial data \( u_0 \in H^{\bar{s}} \cap H^{0,\delta }, u_1 \in H^{\bar{s}-1} \cap H^{0,\delta } \) with \(\delta > \frac{n}{2}\), and showed
for \(2 \le q \le \frac{2n}{n - 2 \bar{s}}\), where \( {\tilde{\varTheta }} = \int \nolimits _{{{\mathbb {R}}}^n} (u_0(y) +u_1(y))dy + \int \nolimits _0^\infty \int \nolimits _{{{\mathbb {R}}}^n}f(u(\tau ,y))dy d\tau \), \(G_0\) is the heat kernel ((2.2) with \(\sigma = 0\)) and \(0< \nu < 1\).
3 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).
Lemma A
Let \(q \in (0,\infty )\). Then
Lemma B
Assume that \(\mu , \rho , \nu \in (1,\infty )\) and \({\tilde{\mu }}, \tilde{\rho }, {\tilde{\nu }} \in [1,\infty ]\) satisfy
Then
provided the right-hand side is finite.
The next corollary immediately follows from Lemma B.
Corollary A
Let \(\omega >0\), \(\mu , \nu \in (1,\infty )\) and \({\tilde{\mu }} \in [1,\infty ]\). If
then the following hold.
Lemma C
Let \(q \in (2,\infty )\), and let \(q^\prime \) be the dual exponent of q, that is, \(\frac{1}{q} + \frac{1}{q^\prime } = 1\). Let \(\nu \in [1,\infty ]\). Then
Lemma D
(Young’s inequality) Let \(q, \rho \in (1,2]\) such that \(\frac{1}{q} + \frac{1}{\rho } = \frac{3}{2}\). Let \(s,t \in [2,\infty )\) such that \(\frac{1}{s} + \frac{1}{t} = \frac{1}{2}\). Then
Lemma E
(sharp Sobolev embedding theorem) Let \( q \in [2, \infty )\) and \(s \ge 0\). If
then
4 Decay estimate for the kernels
In this section, we estimate the kernel of the following linear wave equation with structural damping (1.3).
By Fourier transform, the equation (1.3) is transformed to
Hence the solution u of (1.3) is expressed as
where
We divide \(K_0\) and \(K_1\) into
Let \(\chi _{low}(\xi ) \in C^\infty ({{\mathbb {R}}}^n)\) be a function such that \(\chi _{low}(\xi ) = 1\) for \(|\xi | \le 2^{{ - \frac{3}{1 - 2 \sigma }}}\) and \(\chi _{low}(\xi ) = 0\) for \(|\xi | \ge 2^{{ - \frac{2}{1 - 2 \sigma }}}\). Let \(\chi _{high}(\xi ) \in C^\infty ({{\mathbb {R}}}^n)\) be a function such that \(\chi _{high}(\xi ) = 1\) for \(|\xi | \ge 2\) and \(\chi _{high}(\xi ) = 0\) for \(|\xi | \le 1\).
We put
Here we note that
We put
for \(j = 0,1\). Dividing the kernel into
for \(j = 0,1\), we estimate each part.
4.1 Estimate of the kernels for low frequency part
In this subsection, we consider low frequency region: \(|\xi | \le 2^{{ - \frac{2}{1 - 2 \sigma }}}\).
Lemma 1
Let \(\alpha > -\frac{n}{2}\) and \(\beta > 0\). Let \(a > 2^{{ - \frac{2}{1 - 2 \sigma }}}\). Let \(g(t,\rho )\) be a smooth function on \([0,\infty ) \times (0,a)\) satisfying
on \([0,\infty ) \times (0,a)\) for every \(k = 0,1,\cdots \). Put
Then for every \(q_j \in [1,2)\) \((j = 0,1)\) and \(\vartheta \in [ 0,\frac{n}{2} + \alpha )\) satisfying
the following holds.
where \(\Vert \cdot \Vert ^\prime _q\) denote
Before proving Lemma 1, we state two corollaries:
Corollary 1
Let \(\alpha > -\frac{n}{2}\) and \(\beta > 0\). Let \(a > 2^{{ - \frac{2}{1 - 2 \sigma }}}\). Let \(\upsilon \) and \(\lambda \) be smooth functions on some interval (0, a) such that
on (0, a) for every \(j = 0,1,\cdots \). Put
Then the conclusion of Lemma 1 holds.
In fact, we easily see that
on (0, a) for every \(k = 0,1,\ldots \). Hence, \(g(t,\rho ) = \upsilon (\rho )e^{\lambda (\rho )t}\) satisfies the assumption (4.9) of Lemma 1, and thus the conclusion holds.
Corollary 2
Let \(\alpha , \beta , \gamma \) be numbers such that \(\alpha - \beta + \gamma > -\frac{n}{2}\), \(\beta > 0\) and \(\gamma > 0\). Let \(a > 2^{{ - \frac{2}{1 - 2 \sigma }}}\). Let \(\upsilon \) and \(\lambda \) be smooth functions on (0, a) such that
on (0, a) for every \(j = 1,2,\cdots \). Put
Then for every \(q_j \in [1,2)\,(j = 1,2)\) and \(\vartheta \in [ 0,\frac{n}{2} + \alpha - \beta + \gamma )\) satisfying
the following holds.
Remark 10
D’Abbicco and Ebert [2] considered the kernels:
where \(\upsilon \) and \(\lambda \) satisfy the assumptions (4.13) and (4.14) for \(\alpha > -1\) (see [2, Lemma 3.1]), and
where \(\upsilon \), \(\lambda \) and \(\mu \) satisfy (4.17), (4.18) and (4.19) for \(\alpha> -1, \beta> 0, \gamma > 0\) (see [2, Lemma 3.2]), and showed \(L^p-L^q\) estimates of \(\varphi \mapsto K(t,\cdot )*\varphi \) for \(1 \le p \le q \le \infty \) such that
-
(i)
\(p \ne q\) if \(\alpha = 0\) and \(\upsilon \) is not a constant,
-
(ii)
\(\frac{1}{p} - \frac{1}{q} \ge - \frac{\alpha }{n}\) if \(\alpha \in (-1,0)\),
by using the description of kernels by Bessel functions.
In this paper, we show weighted \(L^2\) estimates of \(K(t,\cdot )*\varphi \) in a way different from [2] by employing Lorentz spaces.
Proof of Corollary 2
By the Leibniz rule, we have
By assumtion (4.19), we have
if \(j \le k - 1\), and
with \(\theta \in (0,1)\) if \(j = k\). From (4.22), (4.16) with k replaced by j, (4.23) and (4.24), it follows that
on (0, a). Hence, \(g(t,\rho ) = \upsilon (\rho )e^{\lambda (\rho )t}(1 - e^{\mu (\rho )t})\) satisfies the assumption (4.9) with \(\alpha \) replaced by \(\alpha - \beta + \gamma \), and therefore, Lemma 1 implies the assertion.
\(\square \)
Now we prove Lemma 1.
Proof of Lemma 1
(Step 1) Let k be a non-negative integer and \(\nu \in (0,\infty )\). We show that
for every \(t \ge 0\) if \((-\alpha + k)\nu < n\), and
for every \(t \ge 0\) if \((-\alpha + k)\nu =n\).
First, we assume that \((-\alpha + k)\nu < n\). Using assumtion (4.9) and changing variables by \(t^{1/\beta }\rho = r\), we have
By (4.27), we have
for \(0 < t \le 1\), which together with (4.28) yields (4.25).
Next we assume that \((-\alpha + k)\nu = n\). By (4.9), we have
for every \(t \ge 0\). Hence,
if \(|\gamma | = k\), and therefore,
for every \(t \ge 0\), that is, (4.26) holds in the case \((-\alpha + k)\nu = n\).
(Step 2) Let \(\vartheta \in [0,\delta ]\) and \(\kappa \in (1,2]\). We prove that
for every \(t > 0\) if \(-\alpha +\vartheta < n(1 - \frac{1}{\kappa })\), and
for every \(t > 0\) if \(-\alpha +\vartheta = n(1 - \frac{1}{\kappa })\).
Let \(\omega \) be a non-negative number such that \(\vartheta + \omega \) becomes an integer and that
Since \(n \ge 2\), we can take \(\omega \) satisfying above conditions. Let \(\nu \) and its dual exponent \(\nu ^\prime \) be the numbers defined by
Since \(0 < 1/\nu ^\prime = 1/\kappa - \omega /n \le 1/2\) by assumtion (4.31), we have
Now we prove (4.29) under the assumption \(-\alpha +\vartheta < n(1 - \frac{1}{\kappa })\). By Corollary A and Lemmas A and C together with the relation (4.33), we have
Since the assumption \(-\alpha +\vartheta < n(1 - \frac{1}{\kappa })\) and (4.32) imply that \(- \alpha + \vartheta + \omega < \frac{n}{\nu }\), we can take \(k = \vartheta + \omega \) in (4.25). Substituting the inequality into (4.34), we obtain (4.29).
Next we prove (4.30) under the assumption \(-\alpha +\vartheta = n(1 - \frac{1}{\kappa })\). By Corollary A and Lemma C together with (4.33), we have
Since the assumption \(-\alpha +\vartheta = n(1 - \frac{1}{\kappa })\) implies \( - \alpha + \vartheta + \omega = \frac{n}{\nu }\), we can take \(k = \vartheta + \omega \) in (4.26). Substituting the inequality into (4.35), we obtain (4.30).
(Step 3) We define \(r_j \in (1,2]\) by \(\frac{1}{q_j} + \frac{1}{r_j} = \frac{3}{2}\) \((j = 1,2)\). We estimate each term of the right-hand side of
If \(q_1 \in (1,2)\), Lemma D yields
The assumption (4.10) implies \(-\alpha +\vartheta \le n (\frac{1}{q_1} - \frac{1}{2}) = n(1 - \frac{1}{r_1})\). Hence, noting that \({\left\| |\cdot |^{\vartheta } K(t,\cdot ) \right\| _{{r_1, \infty }}} \le {\left\| |\cdot |^{\vartheta } K(t,\cdot ) \right\| _{{r_1, 2}}}\) and substituting (4.29) or (4.30) with \((\vartheta , \kappa ) =(\vartheta , r_1)\) into (4.37), we obtain
In the case \(q_1 = 1\), Young’s inequality yields
The assumption \(\vartheta < \frac{n}{2} + \alpha \) implies \( - \alpha + \vartheta < \frac{n}{2} = n(1 - \frac{1}{2})\). Hence, substituting (4.29) with \((\vartheta , \kappa ) =(\vartheta , 2)\) into (4.39), we see that (4.38) holds also for \(q_1 = 1\).
We can estimate the second term of (4.36) in the same way: If \(q_2 \in (1,2)\), Lemma D yields
The assumption (4.10) implies \(-\alpha \le n (\frac{1}{q_2} - \frac{1}{2}) = n(1 - \frac{1}{r_2})\). Hence, substituting (4.29) or (4.30) with \((\vartheta , \kappa ) =(0, r_2)\) into (4.40), we obtain
In the case \(q_2 = 1\), Young’s inequality yields
Since \(-\alpha < \frac{n}{2} = n (1 - \frac{1}{2})\), we have (4.29) with \((\vartheta , \kappa ) =(0, 2)\), which together with (4.42) yields (4.41) with \(q_2 = 1\).
Hence, (4.38) and (4.41) hold for every case. Substituting (4.38) and (4.41) into (4.36), we obtain (4.11). \(\square \)
Lemma 2
Assume that \(0 \le s_2 \le s_1 \) and \(\vartheta \ge 0\) satisfy \(\vartheta - s_1 + s_2 < \frac{n}{2} - 2 \sigma \). If \(q_j \in [1,2)\) \((j = 1,2,3,4)\) satisfy
then the following hold provided the right-hand sides are finite:
where \(\Vert \cdot \Vert ^\prime _q\) is defined by (4.12).
Proof
Since
the conclusion reduces to the case \(s_2 = 0\) by taking \(s_1 - s_2\) and \((-\Delta ) ^{s_2} \varphi \) as \(s_1\) and \(\varphi \) respectively.
Let \(\lambda _\pm \) be the functions defined by (4.4). From (4.5), it follows that
on the support of \(\chi _{low}\).
We first prove (4.44). It is written that
where \(K_1^+\) is defined by (4.6). By definition, \(K(t,x) = {{\mathcal {F}}}^{-1}[|\xi |^{s_1} \widehat{K_1^+}(t,\cdot ) \chi _{low}]\) has the form (4.15) with
By using (4.50) and (4.52), we easily see that \(\upsilon \) and \(\lambda \) above satisfy the assumption of Corollary 1 with \(\alpha = s_1 - 2 \sigma \), \(\beta = 2(1 -\sigma )\). The assumption \(n \ge 2\) and \(0< 2 \sigma < 1\) implies \(\alpha > -\frac{n}{2}\) and \(\beta > 0\), that is, \(\alpha \) and \(\beta \) satisfy the assumption of Corollary 1. Definition of \(\alpha \) and (4.43) imply (4.10) (here we note that we assume \(s_2 = 0\)). Hence, applying Corollary 1, we obtain (4.44).
\( K(t,x) = {{\mathcal {F}}}^{-1}[|\xi |^{s_1} \widehat{K_1^-}(t,\cdot ) \chi _{low}] \,(K_1^-\) is defined by (4.6)) has the form (4.15) with
By using (4.51) and (4.52), we easily see that \(\upsilon \) and \(\lambda \) above satisfy the assumption of Corollary 1 for \(\alpha =s_1 - 2 \sigma (> -\frac{n}{2}), \beta = 2 \sigma (> 0 )\) and therefore, (4.45) holds in the same way as in the proof of (4.44).
Since \(\sigma < 1- \sigma \) by the assumption that \(\sigma \in (0,1/2)\), the estimate (4.46) follows from (4.44) and (4.45).
\( K(t,x) = {{\mathcal {F}}}^{-1}[|\xi |^{s_1} \widehat{K_0^+}(t,\cdot ) \chi _{low}] \,(K_0^+\) is defined by (4.7)) has the form (4.15) with
By using (4.50)–(4.52), we easily see that \(\upsilon \) and \(\lambda \) satisfy the assumption of Corollary 1 with \(\alpha = s_1, \beta = 2(1 -\sigma )\). Definition of \(\alpha \) and (4.43) imply assumtion (4.10) of \(q_1\) for \(q_1 = q_3\). Since \(q_4 < 2\), the assumption on \(q_2\) of (4.10) holds for \(q_2 = q_4\). Hence, we can apply Corollary 1 to obtain (4.47).
Kernel \(K = {{\mathcal {F}}}^{-1}[|\xi |^{s_1} \widehat{K_0^-}(t,\cdot ) \chi _{low}]\) \((K_0^-\) is defined by (4.7)) has the form (4.15) with
By using (4.50)–(4.52), we easily see that \(\upsilon \) and \(\lambda \) above satisfy the assumption of Corollary 1 with \(\alpha =s_1 + 2(1- 2 \sigma ) (> 0), \beta = 2 \sigma (> 0)\). Definition of \(\alpha \) and (4.43) imply the assumption on \(q_1\) of (4.10) for \(q_1 = q_3\). The assumption on \(q_2\) of (4.10) holds for \(q_2 = q_4\) in the same reason as above. Hence, (4.48) holds by (4.11).
Since \(\sigma \in (0,1/2)\), inequality (4.49) follows from (4.47) and (4.48). \(\square \)
Lemma 3
Let \(\vartheta \ge 0\) and \(0 \le s_2 \le s_1 \) such that \(\vartheta - s_1 + s_2 < \frac{n}{2} - 2 \sigma \). Assume that \(q_j \in [1,2)\) \((j = 1,2,3,4)\) satisfy
Then the following hold provided the right-hand sides are finite:
where \(\Vert \cdot \Vert ^\prime _q\) is defined by (4.12).
Proof
We first prove (4.54). Let \(\lambda _\pm \) be the functions defined by (4.5). By the same reason as in the proof of Lemma 2, we may assume that \(s_2 = 0\). It follows from the definition that
We easily see that
satisfy the assumption (4.17)–(4.19) of Corollary 2 with \(\alpha = s_1 - 2 \sigma , \beta = 2(1 -\sigma ), \gamma = 2(2 - 3 \sigma )\). Then the assumption \(n \ge 2\) and \(2 \sigma < 1\) implies \(\alpha - \beta + \gamma = s_1 + 2(1 - 3\sigma ) > -\frac{n}{2}\) and \(\beta > 0\), that is, \(\alpha \), \(\beta \) and \(\gamma \) satisfy the assumption of Corollary 2. The assumption (4.53) and the definition of \(\alpha , \beta , \gamma \) above imply
that is, (4.20) holds. Hence, we can apply Corollary 2 for the above choice to obtain
We also see that
satisfy the assumption of Corollary 1 with \(\alpha = s_1 + 2(1 - 3 \sigma ) (>- \frac{n}{2})\) and \(\beta = 2(1 -\sigma ) (> 0)\). The assumption (4.10) is satisfied by (4.53) together with the definition of \(\alpha \). Hence, we can apply Corollary 1 to obtain
Inequality (4.54) follows from (4.56), (4.57) and (4.58).
Next we prove (4.55). It follows from (4.5), (4.50) and (4.52) that
We easily see that
satisfy the assumption of Corollary 1 with \(\alpha = s_1 + 2(1 - 2 \sigma ) , \beta = 2(1 -\sigma )\). The assumption (4.53) and the definition of \(\alpha \) above yield
that is, (4.10) holds for \(q_1 = q_3\) and \(q_2 = q_4\). Hence, we can apply Corollary 1 to obtain
We also see that
satisfy the assumption of Corollary 2 with \(\alpha = s_1, \beta = 2(1 -\sigma ), \gamma = 2(2 - 3 \sigma )\). The assumption (4.53) and the definition of \(\alpha , \beta , \gamma \) above yield
that is, (4.20) holds for \(q_1 = q_3, q_2 = q_4\). Hence, we can apply Corollary 2 to obtain
Inequality (4.55) follows from (4.60), (4.61) and (4.62). \(\square \)
4.2 Estimate of the kernels for high frequency part\((|\xi | \ge 1)\)
In this subsection, we consider high frequency region: \(|\xi | \ge 1\).
Lemma 4
For every \(s \ge 0, \delta \ge 0\), the following hold.
provided the right-hand sides are finite.
Proof
We easily see that
on the support of \( \chi _{high}\). Hence,
that is, (4.63) holds.
In the proof of [9, p. 10] (see also [7, p. 643]), the following Leibniz rule is shown:
Since \( |\partial _j^k (\widehat{K_1}(t,\xi ) \chi _{high}(\xi ) \langle \xi \rangle )| \le C_{k} e^{-\frac{t}{2}}\) for every nonnegative integer k, we have
Taking \(\vartheta = 0\) and \(\delta \) in this inequality, we obtain (4.64).
We can prove (4.65) and (4.66) in the same way. \(\square \)
4.3 Estimate of the kernels for middle frequency part
In this subsection, we consider the region: \( |\xi | \in [2^{{-\frac{3}{1 - 2\sigma } }}, 2]\).
Lemma 5
There is a constant \(\varepsilon _{\sigma } \in (0, \frac{1}{2})\) such that the following hold for every \(s \ge 0\), \( \delta \ge 0\):
provided the right-hand sides are finite.
Proof
By definitions (4.3), (4.4) and (4.5), we have
for some \(\theta \in (0,1)\). Hence,
in the case \(2|\xi |^{1 - 2 \sigma } \ge 1\), where \(\varepsilon _{\sigma } = 2^{- \frac{6}{1 - 2 \sigma }- 1}\). Next, we consider the case \(2|\xi |^{1 - 2 \sigma } \le 1\). Then
and thus,
on \([2^{-\frac{3}{1-2\sigma }}, 2^{-\frac{1}{1-2\sigma }}]\), which together with (4.74) yields
Calculating in the same way as in the proof of (4.63) by using (4.75) instead of (4.67), and noting that \(-\Delta \) is bounded operator on the support \(\chi _{mid}\), we obtain (4.69).
In the same way as in the proof of (4.75), we see that
for every \(k \in {{\mathbb {N}}}\cup \{0\}\). 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 \(K_{0,mid}\). \(\square \)
5 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.
Lemma 6
Let \(u_j \in L^1 \cap L^2\) for \(j = 0,1\). Then the following hold.
Proof
First we prove (5.1). We have
By (4.54) and (4.45) for \(q_1 = q_2 = 1\) and \(\vartheta = s_1 = s_2 = 0\), we have
Since the support of \(\chi _{mh} \) is included in \([2^{-\frac{3}{1 - 2\sigma }}, \infty )\) and \(2^{{ -\frac{6(1 - \sigma )}{1 - 2\sigma } }} > \varepsilon _\sigma \) , we have
It is written that
Since \(\widehat{G_{\sigma }}(t,\xi ) = |\xi |^{-2\sigma }e^{-|\xi |^{2(1-\sigma )}t }\), we have by the transformation \(t^{\frac{1}{2(1-\sigma )}} r = \rho \) that
that is,
On the other hand, since \(\theta \in [0,1]\), we have
for every \(\xi \in {{\mathbb {R}}}^n\). From (5.7), (5.8) and (5.9), it follows that
Substituting (5.4)–(5.6) and (5.10) into (5.3), we obtain (5.1).
Next we prove (5.2). We have
By (4.55) with \(q_j =1, s_j = 0 \;(j = 1,2)\) and \(\vartheta = 0\), we have
Inequality (4.48) implies
and inequalities (4.65) and (4.71) imply
Since the support of \(\chi _{mh} \) is included in \([2^{-\frac{3}{1 - 2\sigma }}, \infty )\) and \(2^{{ -\frac{6(1 - \sigma )}{1 - 2\sigma } }} > \varepsilon _\sigma \) , we have
By (5.8) with \(\theta \) replaced by \(2 \sigma + \theta \), we have
Then, in the same way as in the proof of (5.10), by using (5.16) instead of (5.8), we have
Since \(\sigma < 1 - \sigma \), (5.2) follows from (5.11)–(5.15) and (5.17). \(\square \)
6 Estimate of the nonlinear term
Throughout this section, we suppose assumtion (1.2), and estimate nonlinear terms by using the argument of [7] and [9]. For \(r \in [1,2)\), \(\delta \in [ 0,\frac{n}{2} - 2 \sigma )\) and \(\bar{s} \ge 1\), we define
where
For \(\vartheta \in [0, \frac{n}{2}- 2 \sigma )\), we put
For \(s \ge 0\), we define
Lemma 7
Let \(r \in [1,2)\), \(\delta \in [ 0,\frac{n}{2} - 2 \sigma )\) and \(\bar{s} > 2 \sigma \). Let \(X = X_{r,\delta ,\bar{s}}\). Then the following holds for every \(\vartheta \in [0,\delta ]\), \(s \in [0,\bar{s}]\) and \(u \in X\):
-
(i)
We have
$$\begin{aligned}&{\left\| (- \Delta )^{\frac{s}{2}} u(t,\cdot )\right\| _{2}} \lesssim \langle t \rangle ^{{\frac{1}{1 - \sigma } \left( -\frac{n}{2}(\frac{1}{r} - \frac{1}{2}) + \sigma - \frac{s }{2} \right) }}{\left\| u\right\| _X}, \end{aligned}$$(6.5)$$\begin{aligned}&{\left\| |\cdot |^\vartheta u(t,\cdot )\right\| _{2}} \lesssim \langle t \rangle ^{{\frac{1}{1 - \sigma } \left( -\frac{n}{2}(\frac{1}{r} - \frac{1}{2}) + \frac{\vartheta }{2} + \sigma \right) }}{\left\| u\right\| _X}. \end{aligned}$$(6.6) -
(ii)
We have
$$\begin{aligned}&{\left\| u(t,\cdot )\right\| _{q,2}} \lesssim \langle t \rangle ^{{\frac{1}{1 - \sigma } \left( -\frac{n}{2}(\frac{1}{r} - \frac{1}{q}) + \sigma \right) }}\Vert {u}\Vert _{X}, \; \quad \text {if} \quad q = \frac{2n}{n + 2\delta }, \end{aligned}$$(6.7)$$\begin{aligned}&{\left\| u(t,\cdot )\right\| _{q}} \lesssim \langle t \rangle ^{{\frac{1}{1 - \sigma } \left( -\frac{n}{2}(\frac{1}{r} - \frac{1}{q}) + \sigma \right) }}{\left\| u\right\| _X} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad \text {if} \quad q \in {\left\{ \begin{array}{ll} (\frac{2n}{n + 2\delta }, \frac{2n}{n - 2 \bar{s}}] \quad &{}(2\bar{s} < n), \\ (\frac{2n}{n + 2\delta }, \infty ) \qquad \quad &{}(2\bar{s} \ge n). \end{array}\right. } \end{aligned}$$(6.8) -
(iii)
We have
$$\begin{aligned}&\Vert (-\Delta )^{\frac{[s ]}{2}} f(u(t,\cdot ))\Vert _{{{{\tilde{q}}_s }}} \lesssim \langle t \rangle ^ {{ \frac{1}{1- \sigma } \left( (-\frac{n}{2r} + \sigma )p + \frac{n}{4} + \frac{1}{2} - \frac{s }{2} \right) }} \Vert u \Vert _X^p, \end{aligned}$$(6.9)$$\begin{aligned}&{\left\| (- \Delta )^{\frac{s}{2}} (1 - \Delta )^{-\frac{1}{2}} f(u(t,\cdot ))\right\| _{2}} \lesssim \langle t \rangle ^ {{ \frac{1}{1 - \sigma } \left( (-\frac{n}{2r}+\sigma )p + \frac{n}{4} + \frac{1}{2} - \frac{s }{2} \right) }} \Vert {u} \Vert _{X}^p \end{aligned}$$(6.10)$$\begin{aligned}&{\left\| \langle \cdot \rangle ^\vartheta f(u(t,\cdot )) \right\| _{ {2n/(n+2)} } } \lesssim \langle t \rangle ^{\zeta _{r,\vartheta }} \Vert u\Vert _{X}^p. \end{aligned}$$(6.11)
Proof
Except use of the weak \(L^p\) estimate, we follow the argument of [9, Lemmas 2.3 and 2.5], which is originated in [7, Lemma 2.1, 2.3 and 2.5].
-
(i)
By Plancherel’s theroem and Hölder’s inequality, we have
$$\begin{aligned}&{\left\| (- \Delta )^{\frac{s}{2}} u(t,\cdot )\right\| _{2}} = \Vert {|\cdot |^{s} {\hat{u}}(t,\cdot )}\Vert _{{2}} \\&\quad \le \Vert {|\cdot |^{\bar{s}} {\hat{u}}(t,\cdot )}\Vert _{{2}}^{\frac{s }{\bar{s}}} \Vert {\hat{u}}(t,\cdot )\Vert _{{2}}^{1 -\frac{s }{\bar{s}}} = \Vert {(-\Delta )^{\frac{\bar{s}}{2}} u(t,\cdot )}\Vert _{{2}}^{\frac{s }{\bar{s}}} \Vert u(t,\cdot ) \Vert _{{2}}^{1 -\frac{s }{\bar{s}}}, \end{aligned}$$which together with the definition of \(\Vert \cdot \Vert _X\) implies (6.5). In the same way, we see that (6.6) holds by Hölder’s inequality.
-
(ii)
We first consider the case \(\frac{2n}{n + 2\delta } = q\), that is, \(n(\frac{1}{q} - \frac{1}{2}) = \delta \). Then
$$\begin{aligned} \Vert u(t,\cdot ) \Vert _{{q,2}} \lesssim \Vert |\cdot |^{-\delta } \Vert _{{\frac{n}{\delta }, \infty }} \Vert |\cdot |^{\delta }u(t,\cdot )\Vert _{{2}} \lesssim \Vert |\cdot |^{\delta }u \Vert _{{2}}, \end{aligned}$$(6.12)which together with the definition of \(\Vert \cdot \Vert _X\) implies (6.7). Next we consider the case \(\frac{2n}{n + 2\delta }< q < 2\), that is, \(0< n \left( \frac{1}{q} - \frac{1}{2}\right) < \delta \). In [9, (2.12)] (see also [7, (2.5)]), the following is shown
$$\begin{aligned} \Vert u \Vert _{{q}} \lesssim \Vert u\Vert _{{2}}^{1 - \frac{n}{\delta }\frac{2 - q}{2q}} \Vert |\cdot |^\delta u\Vert _{{2}}^{\frac{n}{\delta }\frac{2 - q}{2q}}, \end{aligned}$$(6.13)when \( 0< n \left( \frac{1}{q} - \frac{1}{2}\right) < \delta \). This together with the definition of \(\Vert \cdot \Vert _X\) implies (6.8). We consider the case \(2 \le q \le \frac{2n}{n - 2 \bar{s}}\) and \(\bar{s} < 2 n\). Let \({\tilde{s}} = \frac{n}{2} - \frac{n}{q} (\le \bar{s})\). Then Sobolev’s embedding theorem together with (6.5) implies
$$\begin{aligned} \Vert u \Vert _{{q}} \le \Vert u \Vert _{{\dot{H}^{{\tilde{s}}}}} \le \langle t \rangle ^{{\frac{1}{1 - \sigma } \left( -\frac{n}{2}(\frac{1}{r} - \frac{1}{2}) + \sigma - \frac{s}{2} \right) }}\Vert {u}\Vert _{X} = \langle t \rangle ^{{\frac{1}{1 - \sigma } \left( -\frac{n}{2}(\frac{1}{r} - \frac{1}{q}) + \sigma \right) }}\Vert {u}\Vert _{X}, \end{aligned}$$that is, (6.8) holds. In the same way, (6.8) holds also in the case \(2 \le q < \infty \) and \(\bar{s} \le 2 n\).
-
(iii)
We put
$$\begin{aligned} \kappa := \frac{n}{2} - \frac{1}{p-1}. \end{aligned}$$(6.14)By the Leibniz rule together with assumtion (1.2), we have
$$\begin{aligned} \begin{aligned} \Vert \nabla ^{[s]} f(u(t,\cdot )) \Vert _{{{{\tilde{q}}_s}}} \le \Big \Vert u(t,\cdot )^{p-[s]} \sum _{{\sum _{j=1}^{[s]}|\nu _j |= [s] }} \prod _{j=1}^{[s]} | D_x^{\nu _j} u(t,\cdot )| \Big \Vert _{{{\tilde{q}_s}}}, \end{aligned} \end{aligned}$$(6.15)where \(\nu _j\) is a multi index. Put \(k_j = |\nu _j|\). Then, as in the proof of [9, Lemma 2.5], we can choose \(s_j \in [0,k_j - \frac{1}{p-1})\) such that \(q_j\) \((j =1,\ldots ,[s])\) defined by
$$\begin{aligned} \frac{1}{q_0}&= \left( \frac{1}{2} - \frac{\kappa }{n} \right) (p - [s]), \end{aligned}$$(6.16)$$\begin{aligned} \frac{1}{q_j}&= \frac{1}{2} - \frac{\kappa + s_j - k_j}{n} \;\; (j = 1,\ldots ,[s]) \end{aligned}$$(6.17)satisfies
$$\begin{aligned}&\sum _{j = 0}^{[s]}\frac{1}{q_j} = \frac{1}{{\tilde{q}}_s}, \end{aligned}$$(6.18)$$\begin{aligned}&(p - [s])q_0 \in [2,\infty ) \; \text {and} \; q_j \in [2,\infty ) \; \text {for} \; j = 1,\ldots ,[s]. \end{aligned}$$(6.19)Since
$$\begin{aligned}&\frac{1}{\tilde{q_{s}}} - \frac{1}{q_0} = \frac{1}{2} + \frac{1}{n} + \frac{[s] - s}{n} - \left( \frac{1}{2} - \frac{\kappa }{n} \right) (p - [s]), \\&\quad \sum _{j=1}^{[s]}\frac{1}{q_j} = \sum _{j=1}^{[s]} \left( \frac{1}{2} - \frac{\kappa + s_j - k_j}{n} \right) = \Bigg (\frac{1}{2} - \frac{\kappa }{n}\Bigg )[s] - \frac{1}{n} \sum _{j=1}^{[s]} s_j + \frac{[s]}{n}, \end{aligned}$$the condition (6.18) is equivalent to
$$\begin{aligned} \sum _{j=1}^{[s]} s_j = s - \kappa , \end{aligned}$$(6.20)and thus, \(\kappa + s_j \le s\). Taking (6.16)–(6.19) into account, we apply Hölder’s inequality and Sobolev’s embedding theorem to (6.15). Then we obtain
$$\begin{aligned} \begin{aligned} \Vert \nabla ^{[s]} f(u(t,\cdot )) \Vert _{{{{\tilde{q}}_s}}}&\lesssim \Vert u^{p-[s]}\Vert _{{{q_0}}} \sum _{{k_j \ge 0, \sum _{j=1}^{[s]}k_j = [s] }} \prod _{j=1}^{[s]} \Vert |\nabla |^{k_j} u(t,\cdot ) \Vert _{{{q_j}}} \\&\lesssim \Vert \nabla ^{\kappa } u \Vert _{{2}}^{p - [s] } \sum _{{k_j \ge 0, \sum _{j=1}^{[s]}k_j = [s] }} \prod _{j=1}^{[s]} \Vert |\nabla |^{\kappa + s_j} u(t,\cdot ) \Vert _{{2}}, \end{aligned} \end{aligned}$$(6.21)where \(|\nabla | := (-\Delta )^{1/2}\). Then estimating the right-hand side of (6.21) by the definition of \(\Vert \cdot \Vert _X\), and using (6.20) and (6.14), we obtain
$$\begin{aligned} \begin{aligned} \Vert \nabla ^{[s]} f(u(t,\cdot )) \Vert _{{{{\tilde{q}}_s}}}&\lesssim \langle t \rangle ^ {{ \frac{1}{1- \sigma } \left( (-\frac{n}{2}(\frac{1}{r} - \frac{1}{2}) + \sigma )p - \frac{\kappa }{2}(p - [s]) - \frac{1}{2} \sum _{j=1}^{[s]} (\kappa + s_j) \right) }} \Vert u \Vert _X^p \\&= \langle t \rangle ^ {{ \frac{1}{1- \sigma } \left( (-\frac{n}{2}(\frac{1}{r} - \frac{1}{2}) + \sigma - \frac{\kappa }{2})p - \frac{1}{2}\sum _{j=1}^{[s]} s_j \right) }} \Vert u \Vert _X^p \\&= \langle t \rangle ^ {{ \frac{1}{1- \sigma } \left( (-\frac{n}{2r} + \sigma )p + \frac{n}{4} + \frac{1}{2} - \frac{s}{2} \right) }} \Vert u \Vert _X^p, \end{aligned} \end{aligned}$$that is, (6.9) holds. Sobolev’s embedding theorem together with inequality (6.9) implies (6.10). By Hölder’s inequality and assumtion (1.2), we have
$$\begin{aligned} \begin{aligned}&\Vert \langle \cdot \rangle ^\vartheta f(u(t,\cdot )) \Vert _{{2n/(n+2)} } \lesssim \Vert \langle \cdot \rangle ^\vartheta u(t,\cdot )\Vert _2 \Vert u(t,\cdot )\Vert ^{p-1}_{{{{n(p-1)}} }}. \end{aligned} \end{aligned}$$(6.22)The assumption (2.13) implies
$$\begin{aligned} n(p-1) \ge \frac{2rn}{n - 2r\sigma } > \frac{2n}{n+ 2 \delta }, \end{aligned}$$and (2.7) implies
$$\begin{aligned} n(p-1) \le \frac{2n}{n - 2 \bar{s}} \end{aligned}$$if \(2 \bar{s} < n\). Thus, we can apply (6.6) and (6.8) with \(q = n(p-1)\), which together with (6.22) yields (6.11). \(\square \)
7 Estimates of a convolution term
Throughout this section, we suppose assumtion (1.2).
7.1 Decay estimates
Throughout this subsection, we suppose the assumption of Proposition 1.
Lemma 8
Let \(\vartheta \in [0,\delta ]\). For every \(u \in X = X_{r,\delta ,\bar{s}}\), we have
where \(\zeta _{r,\vartheta }\) is the number defined by (6.3).
Proof
By (4.64) and (4.70), Sobolev’s embedding theorem and (6.11), we have
which yields (7.1). \(\square \)
Lemma 9
For every \(u,v \in X = X_{r,\delta ,\bar{s}}\), we have
Proof
First, we estimate the low frequency part. By (4.46) with \(q_1 = r\), \(q_2 = \frac{2n}{n+2}\) and \(\vartheta = \delta \), \(s_1 = s_2 = 0\), we have
where
Since \(\bar{s} \ge 1\), assumtion (2.7) implies that \(pr \le 2 p \le \frac{2n}{n - 2 \bar{s}}\) if \(2 s_o < n\). From (2.15), it follows that \(\frac{2n}{n + 2 \delta }< pr\) in the case \(r = 1\), and \(\frac{2n}{n + 2 \delta } \le pr\) in the case \(r > 1\). In the case \(\frac{2n}{n + 2 \delta } < pr\), we can apply (6.8) with \(q = pr\) to obtain
In the case \(pr = \frac{2n}{n + 2 \delta }\), making use of the equality \((|u|^p)^* = (u^*)^p\), we have
Hence, by (6.7) with \(q = pr\), we obtain
Substituting (7.5) or (7.6) into \(I_1\), we obtain
The following inequality is commonly used to estimate the nonlinear term.
The assumption that \(\delta \ge n(\frac{1}{r} - \frac{1}{2}) - 1\) implies
The assumption (2.13) is equivalent to \( p \left( \frac{n}{2r} - \sigma \right) > \frac{n}{2r} - \sigma + 1, \) which is equivalent to
Hence, by using (7.7), we obtain
Since \(\frac{1}{1 - \sigma } (- \frac{1}{2}+ \sigma ) > -1\), it follows from (6.11) and (7.7) that
The assumption that \(\delta \ge n(\frac{1}{r} - \frac{1}{2}) - 1\) implies \(- \frac{1}{2}+ \sigma \le -\frac{n}{2}(\frac{1}{r} - \frac{1}{2}) + \frac{\delta }{2} + \sigma \). Hence, we have
in the case \(\zeta _{r,\delta } < - 1\). By definition (6.3) and assumption (2.13), we have
Hence, (7.11) holds also in the case \(\zeta _{r,\delta } \ge -1\). Substituting (7.10) and (7.11) into (7.4), we obtain
which together with (7.1) yields (7.2).
The assumption (1.2) implies
and we can prove (7.3) in the same way. \(\square \)
Lemma 10
Let \(s \in [0,\bar{s}]\). For every \(u,v \in X = X_{r,\delta ,\bar{s}}\), we have
for every \(u,v \in X\),
Proof
We first prove (7.12). We divide the left-hand side of (7.12) into three parts:
Substituting (4.44) with \(\vartheta = 0\), \(q_1 = q_2 = r\), \(s_1 = s \) and \(s_2 = 0\) and (7.5) into \(J_1^+\), and using (7.9), we obtain
By (4.44) with \(\vartheta = 0\), \(s_1 = s, \; s_2 =[s ]\), \(q_1 = q_2 = \tilde{q}_s \) (defined by (6.4)), and (6.9), we have
Last we estimate \(J^+_3\). Combining (4.63), (4.69) and (6.10), we have
The assumption (2.13) implies \( - (p-1)(\frac{n}{2r} - \sigma ) + 1 < 0\). Thus, (7.12) follows from (7.16)–(7.19).
We divide the left-hand side of (7.13) into three parts:
Substituting (4.45) with \(\vartheta = 0\), \(q_1 = q_2 = r\), \(s_1 = s \) and \(s_2 = 0\) and (7.5) into \(J^-_1\), and using (7.9), we obtain
Since \(\sigma < 1-\sigma \), the right-hand side of (4.45) is dominated by that of (4.44). Hence, \(J_2^-\) and \(J_3^-\) are estimated by the right-hand sides of (7.18) and (7.19), respectively, and thus,
Substituting (7.21), (7.22) and (7.23) into (7.20), we obtain (7.13).
Inequality (7.14) follows from (7.12) and (7.13), since \(\sigma < 1 -\sigma \) and \(- (p-1)(\frac{n}{2r} - \sigma ) + 1 < 0\).
By using assumtion (1.2), we can prove (7.15) in the same way. \(\square \)
7.2 Diffusion estimate
Lemma 11
Let \(\delta \) and \(\nu \) be an arbitrary number satisfying the assumption of Theorem 3. Let \(X = X_{1,\delta ,\bar{s}}\), where \(X_{1,\delta ,\bar{s}}\) is defined by (6.1). Then we have
Proof
We have
where
First we estimate \(L_1\) by dividing the integrand as
Taking \(q_1 = q_2 = \frac{2n}{n + 4 \sigma }\) and \(s_1 = s_2 = \vartheta = 0\) in (4.46), we obtain
Since \(\bar{s} \ge 1\), assumtion (2.13) implies
and (2.7) implies
if \(2 \bar{s} < n\). Thus, we can apply (6.8) with \(r = 1\) and \(q = \frac{2np}{n + 4 \sigma }\) to obtain
From the inequality above and (7.26), it follows that
By (7.19) and (7.23) with \(r = 1\) and \(s = 0\), we have
which together with (7.27) yields
By (5.1) with \(u_1 =f(u(\tau ,\cdot ))\) and \(\theta = 2 \tilde{\nu }\), we have
Inequality (7.5) and (7.9) with \(r = 1\) yield
Thus
Since \(\bar{s} \ge 1\), (2.7) and (2.15) with \(r = 1\) imply \(2p > \frac{2n}{n + 2\delta }\), and moreover \(2p \le \frac{2n}{n - 2 \bar{s}}\) if \(2 \bar{s} < n\). Hence, we can use (6.8) with \(r = 1\) and \(q = 2p\) to obtain
Thus
We estimate \(L_{2,3}\). Let \(\tilde{\nu }\) be an arbitrary number satisfying
Assume moreover that
if \(\bar{s} < \frac{n}{2}\). By Hölder’s inequality, we have
where \(q = \frac{ p \delta }{ \delta - \tilde{\nu }}\) and \({\tilde{q}} = q(1 -\frac{2 {\tilde{\nu }}}{p\delta })\). The assumption (7.32) implies
In fact, the condition \( \tilde{\nu } < \frac{n}{4}(p-2) + \frac{p \delta }{2} \) is equivalent to \( {\tilde{q}} = \frac{p \delta - 2 \tilde{\nu }}{ \delta - \tilde{\nu }} > \frac{2n}{n + 2\delta } \). The condition (7.33) is equivalent to \( {\tilde{q}}= \frac{p \delta - 2 \tilde{\nu }}{ \delta - \tilde{\nu }} \le \frac{2n}{n - 2 \bar{s}} \) in the case \(n > 2 \bar{s}\). Hence, using (6.8) with taking q as \({\tilde{q}}\), and definition of \(\Vert \cdot \Vert _X\) with \(r = 1\) (see (6.2)) in the right-hand side of (7.34), we obtain
Thus,
which yields
Inequalities (7.30), (7.31) and (7.36) yield
We estimate \(L_3\). By the definition of \(G_\sigma \),
Inequality (5.8) with \(\theta = 2 - 2 \sigma \) implies
uniformly to \(\theta \in [0,1]\) and \(\tau \in [0,t/2]\). Then by using (7.5) with \(r = 1\) together, we have
which yields
Last we estimate \(L_4\). Since \(n \ge 2\), the assumptions (2.14) and (2.13) imply
By this inequality and (2.7), we can apply (6.8) with \(r = 1\) and \(q = p\) to obtain
This together with (7.9) yields
Taking the product of (2.4) and (7.41), we obtain
Substituting (7.28), (7.37), (7.40) and (7.42) into (7.25), we obtain (7.24). \(\square \)
8 Proof of Proposition and Theorems
8.1 Proof of Proposition 1
Let \(\varepsilon > 0\), and
where \(X_{r,\delta , \bar{s}}\) and \(\Vert \varphi \Vert _{{X_{r,\delta , \bar{s}}}}\) are defined by (6.1) and (6.2), respectively. We put \(X = X_{r,\delta , \bar{s}}\) and \(\Vert \cdot \Vert _X =\Vert \varphi \Vert _{{X_{r,\delta , \bar{s}}}}\), throughout this subsection.
If u is a solution of (1.1), then Duhamel’s principles implies
where \(K_0\) and \(K_1\) are defined by (4.2) and (4.3). Taking account of the formula above, we define the mapping \(\Phi \) on \(X(\varepsilon )\) by
We prove that \(\Phi \) is a contraction mapping on \(X(\varepsilon )\) provided \(\varepsilon \) and initial data are sufficiently small.
First we estimate \(K_1(t,\dot{)}*u_1\). By (2.14), we see that assumtion (4.43) of Lemma 2 is satisfied for \(\vartheta = 0\) and \(\delta \), \(s_1 = s_2 = 0\), \(q_1 = r\) and \(q_2 = \frac{nr}{n - r \vartheta }(\in [r,2))\) (that is, \(\frac{1}{q_2} = \frac{1}{r} - \frac{\vartheta }{n}\)). Then, (4.46) gives estimate of low frequency part. The high and middle frequency parts are given by (4.64) and (4.70). Then we have
Assumption (2.14) implies \( \frac{ n - r \delta }{r} - 1 \le \frac{n}{2} \), and therefore, sharp Sobolev’s embedding theorem (Lemma E) yields
Substituting this inequality into (8.2), we obtain
Inequality (4.46) with \(\vartheta = 0\), \(s_1 = \bar{s}\), \(s_2 = 0\), \(q_1 =q_2 = r\), and inequalities (4.63) and (4.69) with \(s = \bar{s}\) yield
Next we estimate \(K_0(t,\dot{)}*u_0\). By (2.14), we see that the assumption of Lemma 2 is satisfied for
Then, (4.49) gives estimate of low frequency part. Inequalities (4.66) and (4.72) give estimate of high and middle frequency parts. Then we obtain
By Corollary A, we have
Hence, we have
Inequality (4.49) with \(\vartheta = 0\), \(s_1 = \bar{s}\), \(s_2 = 0\) and \(q_3 = q_4 =\frac{nr}{n - 2 r \sigma }\) and inequalities (4.65) and (4.71) with \(s = \bar{s}\) imply
For the last inequality, we used (8.5).
By (8.1), (8.3), (8.4), (8.6), (8.7), (7.2) and (7.14) with \(s = 0,\bar{s}\), we have
(i) First we consider the case \(r = 1\). By (8.8), there is a positive constant \(C_1\) independent of initial data such that
Hence, taking \(\varepsilon _1 > 0\) such that \(C_1 \varepsilon _1^{p-1} \le \frac{1}{2}\), and assuming \(u_0\) and \(u_1\) satisfy
we see that \(\Phi \) is a mapping from \(X(\varepsilon _1)\) to \(X(\varepsilon _1)\).
By (7.3) and (7.15), there is a positive constant \(C_2\) independent of initial data such that
for every \(u, v \in X\). We take \(\varepsilon \in (0,\varepsilon _1)\) such that
Then we see that \(\Phi \) is a contraction mapping from \(X(\varepsilon )\) to \(X(\varepsilon )\), and therefore \(\Phi \) has the only one fixed point u, which is the unique solution.
(ii) Next we consider the case \(r \in (1,\frac{2n}{n + 2 \sigma }]\). By Corollary A, we have
Substituting this inequality into (8.8), we obtain
for a positive constant \(C_3\) independent of initial data. Hence, taking \(\varepsilon _2 > 0\) such that \(C_3 \varepsilon _2^{p-1} \le \frac{1}{2}\), and assuming \(u_0\) and \(u_1\) satisfy
we see that \(\Phi \) is a mapping from \(X(\varepsilon _2)\) to \(X(\varepsilon _2)\). In the same way as (8.9), we see that there is a positive constant \(C_4\) independent of initial data satisfying
for every \(u, v \in X\). We take \(\varepsilon \in (0,\varepsilon _2)\) such that \( C_4 (2 \varepsilon )^{p-1} < 1 \). Then \(\Phi \) becomes a contraction mapping \(X(\varepsilon )\) to \(X(\varepsilon )\), and therefore \(\Phi \) has the only one fixed point u, which is the unique solution.
8.2 Proof of Theorems 2 and 3
Proof of Theorems 2
We prove Theorem 2 by using Proposition 1.
(Case 1) In the case \(p_\sigma < p \le 1 + \frac{4}{n}\), we can \(\eta > 0\) sufficiently small such that r defined by
satisfies
Then \(\frac{1}{r} > \frac{2}{n} \left( \frac{1}{p-1} + \sigma \right) \) implies assumtion (2.13).
(Case 1-1) First we consider the case \(p_\sigma < p \le 1 + \frac{4}{n + 2 - 4 \sigma }\). We put
By definition (8.11), we have
Comparing (2.9) and (8.13), we see that
if \(\eta >0\) is sufficiently small. Hence, taking \(\eta > 0\) sufficiently small, we can assume that r and \(\delta ^\prime \) defined above satisfy (8.12) and (8.15). We check that the assumptions (2.14) and (2.15) of Proposition 1 are satisfied with \(\delta \) replaced \(\delta ^\prime \). Since \(2 \sigma < 1\), (2.14) is trivial by (8.14). Since \(n \ge 2\) and \(2 \sigma < 1\), we have
from which it follows that
From this and the definition of \(\delta ^\prime \) and r, it follows that
that is, (2.15) is satisfied with \(\delta \) replaced by \(\delta ^\prime \). Hence the assumption of Proposition 1 is satisfied. Let \({\hat{q}}_j\) \((j = 0,1)\) be the constants defined by (2.16) with \(\delta = \delta ^\prime \) and r defined above. Then
that is, \({\hat{q}}_j = q_j\) \((j = 0,1)\). Since \(\delta ^\prime \le \delta \), the assumptions (2.10) and (2.11) imply (2.17) and (2.19) with \(\delta \) replaced by \(\delta ^\prime \). Thus, Proposition 1 guarantees the existence of the solution \(u \in C^1([0,\infty ), H^{\bar{s}}) \cap C([0,\infty ), H^{\bar{s} - 1})\) if \(\varepsilon \) is sufficiently small. By the standard argument, the uniqueness holds in the class \(C^1([0,\infty ), H^{\bar{s}}) \cap C([0,\infty ), H^{\bar{s} - 1})\).
(Case 1-2) Next we consider the case
We show that \(\delta = 0\) satisfies the assumptions (2.14) and (2.15), that is,
if \(\eta > 0\) is sufficiently small.
The assumption (8.12) implies \(n(\frac{1}{r} - \frac{1}{2}) - 2\sigma > 0\). The assumption \(1~+~\frac{4}{n + 2 - 4\sigma } < p\) is equivalent to
Hence (8.19) holds if \(\eta \) is sufficiently small.
From the assumption that \(p \ge 1 + \frac{4}{n + 2 - 4 \sigma }\), it follows that
and thus assumtion (8.20) holds if \(\eta > 0\) is sufficiently small. Hence the assumption of Proposition 1 is satisfied with \(\delta = 0\). Let \({\hat{q}}_j\) \((j = 0,1)\) be the constants defined by (2.16) with \(\delta = 0\), and r be defined by (8.11) for sufficiently small \(\eta \) satisfying conditions described above. Then
Then by the assumption of \(q_j\) (\(j = 0,1\)), we have
if \(\eta > 0\) is sufficiently small. Since \(p \le 1 + \frac{4}{n}\),
Hence (2.11) implies (2.19). Thus the conclusion holds by Proposition 1 in the same way as above.
(Case 2) Last we consider the case \(p \ge 1 + \frac{4}{n}\). We define r by
Since \(2 \sigma < 1\), (2.14) holds for \(\delta = 0\) if \(\eta > 0\) is sufficiently small. The assumption
is equivalent to
which holds if \(\eta > 0\) is sufficiently small, since \(\sigma < 1\) and \(p \ge 1 + \frac{4}{n}\). Hence, defining r by (8.25) with sufficiently small \(\eta > 0\), we can take \(\delta = 0\) in Proposition 1. Let \({\hat{q}}_j\) \((j = 0,1)\) be the constants defined by (2.16) with \(\delta = 0\) and r defined above. Then, considering the asumption of \(q_j\) (\(j = 0,1\)), we see that
if \(\eta > 0\) is sufficiently small. This imply that \(q_j< {\hat{q}}_j < 2\) \((j = 0,1\)). Hence (2.12) implies (2.19) with \(\delta = 0\), and the conclusion holds by Proposition 1.
\(\square \)
Proof of Theorem 3
Since \(u =\Phi u\) (\(\Phi \) is defined by (8.1)), we can write
where \(\vartheta _1\) is defined by (2.3). Since \(K_0(t,\cdot )*u_0 + K_1(t,\cdot )*u_1\) is a solution of the linear equation (1.3), the first term of the right-hand side of (8.27) is estimated by Theorem 1. The second term is estimated in (7.24). Combining these estimates, we obtain the assertion. \(\square \)
References
Bergh, J., Löfström, J.: Interpolation Spaces, An Introduction. Springer, New York (1976)
D’Abbicco, M., Ebert, M.R.: Diffusion phenomena for the wave equation with structural damping in the \(L^p-L^q\) framework. J. Differ. Equ. 256, 2307–2336 (2014)
D’Abbicco, M., Ebert, M.R.: An application of \(L^p-L^q\) decay estimates to the semi-linear wave equation with parabolic-llike structural damping. Nonlinear Anal. 99, 16–34 (2014)
D’Abbicco, M., Ebert, M.R.: A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations. Nonlinear Anal. 149, 1–40 (2017)
D’Abbicco, M., Reissig, M.: Semilinear structural damped waves. Math. Methods Appl. Sci. 37, 1570–1592 (2014)
Galley, T.H., Raugel, G.: Scaling variables and asymptotic expansions in damped wave equations. J. Differ. Equ. 150, 42–97 (1998)
Hayashi, N., Kaikina, R.I., Naumkin, P.I.: Damped wave equation with super critical nonlinearities. Differ. Integr. Equ. 17, 637–652 (2004)
Hosono, T., Ogawa, T.: Large time behavior and \(L^q-L^p\) estimate of solutions of 2-dimensional nonlinear damped wave equations. J. Differ. Equ. 203, 82–118 (2004)
Ikeda, M., Inui, T., Wakasugi, Y.: The Cauchy problem for the nonlinear damped wave equation with slowly decaying data. Nonlinear Differ. Equ. Appl. 24(10), 1–53 (2017)
Ikehata, R., Ohta, M.: Critical exponents for semilinear dissipative wave equations in \({\mathbb{R}}^N\). J. Math. Anal. Appl. 269, 8797 (2002)
Ikehata, R., Takeda, H.: Asymptotic profiles of solutions for structural damped wave equations. J. Dyn. Differ. Equ. 31, 537–571 (2019)
Ikehata, R., Tanizawa, K.: Global existence of solutions for semilinear damped wave equationsn \({\mathbb{R}}^n\) with noncompactly supported initial data. Nonlinear Anal. 61, 1189–1208 (2005)
Karch, G.: Selfsimilar profiles in large time aysmptotics of solutions to damped wave equation. Stud. Math. 143(2), 175–197 (2000)
Kawakami, T., Takeda, H.: Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation. Nonlinear Differ. Equ. Appl. 23(54), 1–30 (2016)
Li, T.T., Zhou, Y.: Breakdown of solutions to \(\square u + u_t = |u|^{1+\alpha }\). Discrete Contin. Dyn. Syst. 1, 503–520 (1995)
Matsumura, A.: On the asymptotic behavior of solution of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12, 169–189 (1976)
Nakao, M., Ono, K.: Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations. Math. Z. 214, 325–342 (1993)
Narazaki, T.: \(L^p-L^q\) estimates for the damped wave equation and their applications to semi-linear problem. J. Math. Soc. Jpn. 56, 585–626 (2004)
Narazaki, T., Nishihara, K.: Asymptotic behavior of solutions for damped wave equation with slowly decaying data. J. Math. Anal. Appl. 338, 803–819 (2008)
Narazaki, T., Reissig, M.: \(L^1\) estimates for oscillating integrals related to structural damped wave models. In: Cicognani, M., Colombini, F., Del Santo, D. (eds.) Studies in Phase Space Analysis with Applications to PDEs. Nonlinear Differential Equations Applications, pp. 215–218. Birkhäuser, New York (2013)
Nishihara, K.: \(L^p-L^q\) estimates of solutions to the damped wave equation in 3-dimensional space and their application. Math. Z. 244, 631–649 (2003)
O’Neil, R.: Convolution operators and \(L(p, q)\) spaces. Duke Math. J. 30, 129–142 (1963)
Todorova, G., Yordanov, B.: Critical exponent for a nonnlinear wave equation with damping. J. Differ. Equ. 174, 464–489 (2001)
Zhang, Q.S.: A blow-up result for a nonlinear wave equation with damping: the critical case. C. R. Acad. Sci. Paris Série I(333), 109–114 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partly supported by Grant-in-Aid for Scientific Research (C) 17K05338 of JSPS.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Yamazaki, T. Asymptotic profile of solutions for semilinear wave equations with structural damping. Nonlinear Differ. Equ. Appl. 26, 16 (2019). https://doi.org/10.1007/s00030-019-0562-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-019-0562-x