Weighted energy method for semilinear wave equations with time-dependent damping

Of concern is the energy decay property of solutions to wave equations with time-dependent damping. A reasonable class of damping coefficients for the framework of weighted energy methods is proposed, which contains not only the model of “effective” damping (1+t)-β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+t)^{-\beta }$$\end{document}(-1≤β<1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1\le \beta <1)$$\end{document}, but also non-differentiable functions with a suitable behavior at t→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \infty $$\end{document}. As an application of the weighted energy estimate, global existence for the corresponding semilinear wave equation is discussed.


Introduction
As is well-known, the damped wave equation ∂ 2 t u − u + ∂ t u = 0 has a diffusive structure close to that of the heat equation ∂ t v − v = 0. Actually, the long-time behavior of solutions to the damped wave equation is characterized as the ones to the heat equation. This phenomenon is so-called diffusion phenomenon and it is wellstudied in the literature. In the analysis of the damped wave equation, the above diffusive structure can be seen everywhere. Particularly, the validity of "weighted energy method" may be understood as one of the greatest features of the diffusive structure of the damped wave equation. It is natural to ask what kind of generalization preserves this kind of the diffusive structure. In this paper, we focus our attention to such a problem for the wave equation with time-dependent damping coefficients.
To fix the target problem, we introduce the following initial-boundary value problem of semilinear wave equations with the time-dependent damping coefficient b(t): with some constants 1 < p < ∞ and C f ≥ 0. The pair (u 0 , u 1 ) is assumed to belong to the energy class H = H 1 0 ( ) × L 2 ( ) with some additional assumption. The main interest of this paper is the validity of "weighted energy method" to the linear ( f ≡ 0) and semilinear ( f ≡ 0) problems (1.1). In particular, we aim to propose a reasonable class of damping coefficients, which is admissible to the framework of weighted energy method.
The Cauchy problem of the usual damped wave equation has been studied in the literature. In Matsumura [26], the fundamental properties of solutions are discussed and he found the estimates of linear solutions as follows: Nowadays, the above inequality is called "Matsumura estimate" and it has been investigated to clarify the structure of dissipation and applied to semilinear problems (see e.g., a recent research Ikeda-Taniguchi-Wakasugi [18] and the references therein). In Hsiao-Liu [12], they proved diffusion phenomena, that is, the solution u of (1.3) satisfies u(·, t) − v(·, t) L 2 = o v(·, t) L 2 , as t → ∞, (1.4) where v is the solution of the following Cauchy problem of the heat equation (see also Yan-Milani [47]). For the semilinear problem ( f ≡ 0), global existence and finite time blowup of solutions to (1.3) also have been considered in the previous works. In , the problem (1.3) with compactly supported initial data is discussed. It is proved by a framework of weighted energy methods that if 1 , then the problem (1.3) possesses a global-in-time solution. On the one hand, if f (ξ ) = |ξ | p and 1 < p < 1 + 2 N , then the problem has a finite blowup solution for any small initial data the blowup result for the threshold case p = 1 + 2 N is filled in Zhang [48]. Therefore the critical exponent (threshold of p for small data global existence and small data blowup) has been determined as p = 1 + 2 N ; note that the semilinear heat equation ∂ t u − u = u p has the Fujita exponent p = 1 + 2 N as the critical exponent for existence and nonexistence of positive solutions (see Fujita [9]). Later, in Ikehata-Tanizawa [23], the assumption for the compactness of supports of initial data has been removed, but still an exponential decay of initial data is required in the sense of the restriction The class of initial data including polynomially decaying functions in L r (1 ≤ r < 2) can be found in Nakao-Ono [27], Galley-Raugel [10], Ikehata-Ohta [22], Hayashi-Kaikina-Naumkin [11] (see also Ikeda-Inui-Okamoto-Wakasugi [15] and the references therein).
In the case of time-dependent damping, Wirth [43] studied the classification of the asymptotic profiles of the solutions to the linear problem via the representation of solutions via the Fourier analysis. Roughly speaking, the following asymptotic profiles for the case b(t) = (1 + t) −β are shown: (i) (Non-effective case) If β ∈ (1, ∞), then the solution of (1.6) behaves like the one of wave equation without damping. (ii) (Effective case) If β ∈ [−1, 1), then the solution of (1.6) behaves like the one of the parabolic equation of the form (iii) (Overdamping case) If β ∈ (−∞, −1), then the solution of (1.6) converges to a non-trivial state.
An abstract version of the effective case (ii) can be seen in Yamazaki [44,45] via the spectral analysis with the selfadjoint operator in a Hilbert space. Note that the classification of effectiveness and non-effectiveness is written in Reissig-Wirth [33] via the validity of Stricharz estimates which is different from our terminology. The critical exponent for the corresponding semilinear problem is also considered. The time-dependent version of energy methods with exponential weights can be found in Nishihara [28,29], Lin-Nishihara-Zhai [24] and D'Abbicco-Lucente-Reissig [5]. In D'Abbicco [3], the framework of Ikehata-Ohta [22] has been used. The method of scaling variables (originated by Galley-Raugel [10]) can be seen in Wakasugi [41,42]. For the study of blowup solutions with small initial data, it is discussed in D'Abbicco-Lucente [4] and in Ikeda-Sobajima-Wakasugi [17]. Detailed information for the case b(t) = 1 + t (that is, β = −1) is analysed in Ikeda-Inui [14]. If b(t) = μ(1 + t) −1 (the case β = 1), then the structures of the wave and heat equations appear. Therefore this is the threshold case for the In this case, it is expected that the critical exponent seems to depend on the constant μ, but it is still remained as open problems. We do not enter the detail of this case (for the detail, a recent research D'Abbicco [6] and the references therein). More general class of abstract evolution equations (in a Hilbert space) of the form is also treated in Yamazaki [46] under a weaker condition on b (and also c) compared with Wirth [43].
On the other hand, the problem of the semilinear damped wave equation in an exterior domain (with a smooth boundary ∂ ) is rather delicate because of the reflection at the boundary. Ono [31] discussed global existence for (1.8) with (1.2) via the result of Dan-Shibata [7]. In Ikehata [20,21], the global existence for the 2-dimensional case is proved via the weighted energy method when 2 < p < ∞. The blowup result for small solutions (when 1 < p ≤ 1 + 2 N ) can be found in Ogawa-Takeda [30], Fino-Ibrahim-Wehbe [8] and Ikeda-Sobajima [16]. Recently, in Sobajima-Wakasugi [36], a framework of weighted energy methods for (1.8) (with space-dependent damping) is proposed, which is applicable to solutions with the initial data satisfying , where we use the notation x = 1 + |x| 2 for x ∈ R N . As an application of the framework in [36], global existence for (1.8) ( f ≡ 0) is discussed in Sobajima [34]. More precisely, it is shown in [34] that if p satisfies then the corresponding solution of (1.8) exists uniquely when the initial data satisfy I λ (u 0 , u 1 ) 1 with λ ∈ [0, N 2 ) (the one-dimensional case is filled in Sobajima-Wakasugi [37]). In the recent paper Sobajima [35], an alternative approach of energy method with time-dependent dissipation is proposed, which enables us to find out the asymptotic expansion of linear solutions to the wave equation with time-dependent damping.
The goal of the present paper is to combine the framework of Sobajima-Wakasugi [36] and Sobajima [35]. More precisely, we aim to establish a framework of weighted energy methods for a (possibly wider) class of time-dependent damping, which is applicable to the initial data having a polynomial decay.
To state our result precisely, we give the definition of local-in-time and global-intime solutions to (1.1) which mainly focused in the present paper is the following.
. Then for every R > 0, there exists a positive constant T > 0 such that the following holds: if

then there exists a unique local-in-time weak solution u in (0, T ).
Here we introduce our definition of the class D diff of damping coefficients.
Define the sets D * , D over and D diff as follows; Moreover, one can see that the class D diff also contains the coefficient b satisfying Remark 1.2. The semilinear problem (1.7) has been mainly considered with b with the following assumption The above conditions are so-called effective essentially proposed in Wirth [43]. There are important to find a suitable estimate for linear solutions (like the Matsumura estimate) to apply the nonlinear term as an inhomogeneous term (see D'Abbicco-Lucente-Reissig [5] and the subsequent papers). The coefficient b in this class also belongs to D * via the computation with (ii) and (iv) with k = 1; one can choose The special class of the damping b ∈ D * with the choice ζ = b −1 has been discussed in Ikeda-Sobajima-Wakasugi [17]. In this case, one requires the differentiability for b which is essential to employ the technique of scaling variables. The definition of D * enables us to choose a certain non-differentiable damping.
The assumption for b in Yamazaki [46] is also covered. The damping of the form is applicable to the framework of Yamazaki [46] and also to our framework with ζ = b −1 . Our condition allows us to choose the (slightly generalized) damping The main novelty of this paper is the following. If the damping b belongs to the class D diff , then the following theorem asserts that the linear wave equation with the damping b has a kind of diffusive structure. The statement is formulated in the weighted energy inequality.
In particular, if b(t) = 1 + t (the threshold case for overdamping), then one has the logarithmic decay E (u; t) ≤ C(1 + log(1 + t)) −λ−1 . In the overdamping case b ∈ D * ∩ D over , Theorem 1.2 also provides an estimate for the weighed energy, but does not give new information.
We can also show global existence of weak solutions to (1.1) via the framework of the weighted energy method in Theorem 1.2.
. Then there exists a positive constant δ * > 0 such that if then there exists a unique global-in-time weak solution u of (1.1). Moreover, there exists a positive constant C > 0 such that for every t ≥ 0, From the viewpoint of global existence of (1.1) (with well-behaved damping b) for small initial data, it is proved in D'Abbicco-Lucente-Reissig [5] that if = R N and 1 + 2 N < p < p * (N ), then the problem (1.1) possesses a non-trivial global-in-time weak solution. In contrast, even if the damping is not so regular, Theorem 1.3 asserts that the existence of a suitable auxiliary function ζ immediately provides a global existence result with 1 + 2 N < p < p * (N ) for exterior domains. The statement is as follows.
then there exists a unique global-in-time weak solution of (1.1).
Remark 1.6. For the overdamping case b ∈ D * ∩ D over , global existence of small solutions to (1.1) with 1 < p < p * (N ) has been proved in Ikeda-Wakasugi [19]. Our proof can be modified for this cases. This is a consequence of the following fact (with λ = 0) Here we briefly show our idea of the treatment of damping terms. In the case of the usual damped wave equation ∂ 2 t u − u + ∂ t u = 0, the equation can be written by the alternative form ∂ t (∂ t u + u) = u. Although this modification is trivial, one can directly see important information of the asymptotic behavior in (1.5) in the above form. If we move to the problem with time dependent damping b(t), as an experience in Sobajima [35,Sect. 3] (written in an abstract formulation), we can find the alternative form with a multiplier m(t) which is required to be positive and satisfy the ordinary differential equation (1.10) note that the equation (1.10) already appears in Lin-Nishihara-Zhai [25] to study blowup phenomena. Note that existence of positive solutions to (1.10) is verified when b ∈ D * . Applying this consideration, we can see that the semilinear equation in (1.1) is rewritten as the alternative form This form plays a crucial role to apply an energy method, where the damping coefficient depends on t. In fact, the multiplier m(t) is effectively used in [35] to derive the energy estimates via the use of the following auxiliary functional We can see that all important quantities for the energy estimate come from the behavior of the multiplier m(t) (not directly from b(t)). Adopting the weighted energy method in Sobajima-Wakasugi [36,38], one can expect that functionals of the form seem to be reasonable to carry out the weighted energy methods with time-dependent damping b(t), where is a suitable function related to the parabolic equation Since the above structure is independent of the domain, we can also address the initial-boundary value problem (1.1). The present paper is originated as follows. In Sect. 2, we collect the fundamental facts (supersolutions of the linear heat equation and the Sobolev inequalities) to consider the problem (1.1). In Sect. 3, we explain how to derive the weighted energy estimates for the linear problem with inhomogeneous terms. In Sect. 4, the energy estimate (obtained in Sect. 3) is applied to discuss the semilinear problem (1.1).

Preliminaries
In this section, we collect some basic but important tools for the linear and nonlinear problems (1.1).

A supersolution of the heat equation
First we introduce a supersolution of the heat equation which has been used to construct a weight function in the energy method for the damped wave equation (see also Sobajima-Wakasugi [36] or Sobajima [34]). We use a refined version of weight functions in Sobajima-Wakasugi [39]. The detail of λ is slightly different from supersolutions in Sobajima-Wakasugi [39] but essentially equivalent to that. The important properties of λ is collected in the following lemma.
Then the following assertions hold: (i) There exists a positive constant η λ > 0 such that (ii) There exist two positive constants c λ > 0 and C λ > 0 such that (iii) There exists a positive constant C λ such that

Some functional inequalities
The following inequality is one of important tools to find out a "good term" in the calculation of energy methods via integration by parts.

Lemma 2.2. [34]
Let D be a bounded domain in R N with a smooth boundary. Assume that ∈ C 2 (D) is positive and δ ∈ (0, 1 2 ). Then for every v ∈ H 2 (D) ∩ H 1 0 (D), The next identity is nothing but the special case of the Gagliargo-Nirenberg inequality. The following modification is an adjusted version for weighted energy methods.

Lemma 2.3. Let D be a bounded domain in R N with a smooth boundary. Let
where (x, τ ) = τ + |x| 2 4 .

A weighted energy method for linear problem
In this section, we consider the inhomogeneous problem of the damped wave equation where D ⊂ R N is a bounded domain with a smooth boundary ∂ D. Here we only assume that Existence of the energy solution of (3.1) is also well-known (see e.g., Ikawa [13]).
Here we focus our attention to estimates for the functional where W is given in Theorem 1.2. Since several properties of solutions to (3.1) (in this section) will be applied to the solution of the semilinear problem (1.1), that is, w = u and g = f (u), it is necessary to discuss estimates for the weighted energy (3.3) with constants independent of D. Now we explain our strategy of weighted energy methods. Basically, we know that the usual energy equality holds: which is the fundamental property of solutions to the wave equation. Additionally, from the viewpoint (1.11), the equation in (3.1) can be reformulated as via the use of the multiplier m(t) satisfying existence of a suitable multiplier can be seen in Lemma 3.2 under the assumption b ∈ D * .

An auxiliary function describing time-scale
We first construct an auxiliary function m(t) which is reasonable to proceed the energy method for the damped wave equations with time-dependent damping. Now we consider the ordinary differential equation (3.5). The existence of positive solutions to (3.5) is characterized as follows.

(ii) there exists a positive solution m(t) of (3.5).
In this case, the minimal solution m(t) of (3.5) can be written by Proof. The solution of initial-value problem of (3.5) can be computed as This immediately yields the equivalence of (i) and (ii). In particular, there exist positive constants B 0 and B 1 such that for every t ≥ 0,   ([0, ∞)) admits a minimal positive solution m(t) of (3.5) satisfying (3.6) (we can take ζ = m). In this sense, we can assume the existence of m with (3.6) instead of that of ζ without loss of generality.
Then clearly, we have for every t ≥ t ε , On the one hand, integration by part implies This yields m * (t) ≤ ζ(t) 1−2ε . Similarly, we also have m * (t) ≥ ζ(t) 1+2ε . Consequently, we deduce and therefore we obtain lim t→∞ b(t)m(t) = 1. The other limit immediately verified via the equation m + 1 = bm. The proof is complete. when t ≥ t * . Then we have for every t ≥ t * , This implies that if a ≥ 2ε −1 m 2 L ∞ (0,t * ) , then we have the desired inequality.

A supersolution of ∂ t = m(t)
To derive weighted energy estimates for (3.1), we will employ the following idea of weighted estimates for the parabolic equation ∂ t v = m(t) v, which appears in [36] when b ≡ 1: By integration by parts, we formally see that for every positive-valued In view of the above equality, the construction of a positive supersolution to ∂ t = m(t) provides the uniform estimate for D v 2 −1 dx which can be understood as a weighted L 2 -estimate. To proceed the strategy explained above, we introduce a supersolution λ,a of ∂ t = m(t) by the trick in Sobajima-Wakasugi [38]. The following lemma immediately follows from Lemma 2.1.

Lemma 3.4. Let λ,a be as in Definition 3.2 and set
Then the following assertions hold: .
where the constants η λ , c λ , C λ and C λ are given in Lemma 2.1.

Remark 3.3.
The assumption b ∈ D diff = D * \D over provides that M(a; t) → ∞ as t → ∞ and therefore the solution of ∂ t = m(t) decays in some sense.

Weighted energy method for time-dependent damping
Now we derive the weight energy estimate for the solution w of (3.1). We will use the following weighted energy in the proof: where a is given in Lemma 3.4. To state the precise estimate for w, we put The following is the harvest of the above strategy.
Proposition 3.5. Assume (3.2) and b ∈ D * . Then there exist positive constants C 1 , C 2 , C 3 and a > 0 (independent of R and T ) such that for every t ∈ [0, T ], Proof. As explained in the beginning of Sect. 3, we introduce the auxiliary functional where We notice that since for every a ≥ a ε (given in Lemma 3.2), one has m(t)w∂ t w + w 2 the equivalence of the quantities is verified for every ν > 0 by taking a sufficiently large. In the final step, we will fix the parameter a. To shorten the notation, we use β = β λ , γ = γ λ and δ = δ λ (without subscripts). Let a ≥ 1 determined later. Using the equation in the alternative form (3.4), we have We see from Lemmas 2.2 and 3.4(i) that On the one hand, Lemma 3.4(iii) and the Young inequality yield Combining the above inequalities with Lemma 3.4(ii), we deduce (3.11) The estimate for E λ D (w; t) is derived as follows. Using integration by parts and the equation in (3.1), we see that Then Lemma 3.2 gives The relation M(a; t) + |∇ a | 2 = a with the Young inequality provides Summarizing the above estimates, we obtain by (3.11) and (3.12) we can deduce Here choosing a suitable parameter a ≥ 1 such that the two quantities X λ D,a (w; t) and ν M(a; t)E λ D,a (w; t) + E λ D,a * (w; t) are equivalent and (by virtue of Lemma 3.3), we obtain the desired inequality.

Proof of Theorem 1.2. Set
Let T > 0 be arbitrary. If supp u 0 ∪ supp u 1 is compact in , then we fix a constant R > R satisfying supp u 0 ∪ supp u 1 ⊂ B(0, R). By finite speed of propagation, we have supp u(·, t) ⊂ B(0, R + t) for every t ∈ [0, T ]. This means that u also satisfies (3.1) with g ≡ 0 and D = ∩ B(0, R + T ). Applying Proposition 3.5, we arrive the weighted energy estimate with the constant C which is independent of R and also T . Since T is arbitrary, we obtain the desired estimate. If supp u 0 ∪ supp u 1 is not compact, then the standard cut-off argument with the strong continuity of the C 0 -semigroup e tL in H 1 0 ( ) × L 2 ( ) provides the desired estimate.

Nonlinear estimates for global existence
Finally, we discuss the existence of global-in-time weak solutions to (1.1). As in the linear case, we additionally assume that supp u 0 ∪ supp u 1 ⊂ B(0, R) for some R > R . Let u be the solution of (1.1) in (0, T ) given in Proposition 1.1. Then by finite speed of propagation we also have supp u(·, t) ⊂ B(0, R + t) for every t ∈ [0, T ). As in the proof of Theorem 1.2, we consider the homogeneous Dirichlet boundary problem in bounded domain D = ∩ B(0, R + T ) and use the functionals (for the simplicity of the notations) introduced in (3.8) and (3.9), respectively. Then we define a non-decreasing continuous function note that in the linear case ( f ≡ 0), by Proposition 3.5, Z (t) (t > 0) is bounded.
The following proposition is crucial to obtain the uniform bound for Z in t.
. Assume further that supp u 0 ∪ supp u 1 is compact in . Let u be the localin-time solution of (1.1) in (0, T ). Then there exists a positive constant C (independent of R and T ) such that for every t ∈ (0, T ), The following lemma is the central part of the proof of Proposition 4.1 which treats the following quantity associated to the nonlinear term f (u); Proof. Here we put q = 1 + 4 N and divide the proof into two cases where p ≥ q and p(N , λ) ≤ p < q. (The case p ≥ q = 1+ 4 N ) Adopting the Gagliardo-Nirenberg inequality (Lemma 2.3 with τ = M(a; t), μ = λ), we deduce . Therefore the relation N 4 ( p − 1) ≥ 1 implies the following two kind of inequalities M(a; t) F(t) ≤ K λ, p M(a; t) The first inequality in the assertion can be verified the computation exactly the same as above. For the second inequality, the Hölder inequality and the Gagliardo-Nirenberg inequality yield These inequalities give the desired (second) inequality.
Proof of Proposition 4.1. Set F(ξ ) = ξ 0 f (θ ) dθ (ξ ∈ R) and the corresponding external functions in Proposition 3.5 as respectively. Then we see from (1.2) that Therefore combining Proposition 3.5 and the above inequalities together with Lemma 4.2, we obtain the desired inequality.
Proof of Theorem 1.3. For the case of initial data (u 0 , u 1 ) ∈ H having compact supports, then we see from Proposition 4.1 that where we have put X (0) = C p+1 p−1 (X (0) + X (0) p+1 2 ) and Z (t) = C 2 p−1 Z (t). By noticing the (non-)connectedness of {z ≥ 0; δ (z) = δ + z p+1 2 − z ≥ 0} with the convexity of δ , we can check that if which is nothing but the desired uniform estimate of the weighted energy By the argument with the blowup criteria, we can construct a global-in-time weak solution of (1.1) satisfying the above uniform estimate.
If we consider the case of initial data with non-compact supports, then the approximation procedure for the initial data via a family of cut-off functions with the previous step provides approximate (global-in-time weak) solutions u n with the uniform weighted energy estimate. Then we can show that the limit lim n→∞ u (taking a subsequence if necessary) is the global-in-time solution of (1.1) with the given initial data (u 0 , u 1 ). This means that the strategy of the proof of Theorem 1.3 also works when 1 < p < p * (N ) and b belongs to a class of overdamping coefficients.
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Conflict of interest
The author declares that he has no conflict of interest.
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