Exponential decay of solutions of damped wave equations in one dimensional space in the $L^p$ framework for various boundary conditions

We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants. The key ingredient in the study of these systems is the use of the internal dissipation energy to estimate the difference of solutions with their mean values in an average sense.

The decay of the solutions of (1.1) equipped with either (1.2), or (1.3), or (1.4) has been extensively investigated in the case where a is independent of t, i.e., a(t, x) = a(x) and mainly in the L 2 -framework, i.e. within an Hilbertian setting.In this case, concerning the Dirichlet boundary condition, under the additional geometric multiplier condition on a, by the multiplier method, see e.g., [20,24], one can prove that the solution decays exponentially, i.e., there exist positive constants C and γ independent of u such that The assumption that a satisfies the geometric multiplier condition is equivalent to the requirement that a(x) ≥ λ > 0 on some neighbourhood of 0 or 1.Based on more sophisticate arguments in the seminal work of Bardos, Lebeau, and Rauch on the controllability of the wave equation [3], Lebeau [23] showed that (1.6) also holds without the geometric multiplier condition on a, see also the work of Rauch and Taylor [31].When the damping coefficient a is also time-dependent, similar results have been obtained recently by Le Rousseau et al. in [22].It is worth noticing that strong stabilization, i.e., the energy decay to zero for each trajectory, has been established previously using LaSalle's invariance argument [14,15].The analysis of the nonlinear setting associated with (1.1) can be found in [6,17,26,27,34] and the references therein.Similar results holds for the Neumann boundary condition [3,22,26,34].Concerning the dynamic boundary condition without interior damping effect, i.e., a ≡ 0, the analysis for L 2 -framework was previously initiated by Quinn and Russell [30].They proved that the energy exponentially decays in L 2 -framework in one dimensional space.The exponential decay for higher dimensional space was proved by Lagnese [21] using the multiplier technique (see also [30]).The decay hence was established for the geometric multiplier condition and this technique was later extended in [25], see also [1] for a nice account on these issues.
Much less is known about the asymptotic stability of (1.1) equipped with either (1.2), or (1.3), or (1.4) in L p -framework.This is probably due to the fact that for linear wave equations considered in domains of R d with d ≥ 2 is not a well defined bounded operator in general in L p framework with p = 2, a result due to Peral [29].As far as we know, the only work concerning exponential decay in the L p -framework is due to Kafnemer et al. [19], where the Dirichlet boundary condition was considered.For the damping coefficient a being time-independent, they showed that the decay holds under the additional geometric multiplier condition on a for 1 < p < +∞.Their analysis is via the multiplier technique involving various non-linear test functions.In the case of zero damping and with a dynamic boundary condition, previous results have been obtained in [7] where the problem has been reduced to the study of a discrete time dynamical system over appropriate functional spaces.
The goal of this paper is to give a unified approach to deal with all the boundary considered in (1.2), (1.3), and (1.4) in the L p -framework for 1 < p < +∞ under the condition (1.5).Our results thus hold even in the case where a is a function of time and space.The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants for which new insights are required.
Concerning the Dirichlet boundary condition, we obtain the following result.
As in [16,19], we set One can check that for a smooth solution u of (1.1) and (1.2), the pair of functions (ρ, ξ) defined in (1.8) satisfies the system (1.9) One cannot hope the decay of a general solutions of (1.9) since any pair (c, c) where c ∈ R is a constant is a solution of (1.9).Nevertheless, for (ρ, ξ) being defined by (1.9) for a solution u of (1.1), one also has the following additional information Concerning System (1.9) itself (i.e., without necessarily assuming (1.10)), we prove the following result, which takes into account (1.10).
In the L p -framework, the Neumann boundary condition and its corresponding hyperbolic systems are discussed in Section 5 and the dynamic boundary condition and its corresponding hyperbolic systems are discussed in Section 6. Concerning the dynamic boundary condition, the decay holds even under the assumption a ≥ 0. The analysis for the Neumann case shares a large part in common with the one of the Dirichlet boundary condition.The difference in their analysis comes from taking into account differently the boundary condition.The analysis of the dynamic condition is similar but much simpler.
The study of the wave equation in one dimensional space via the corresponding hyperbolic system is known.The controllability and stability of hyperbolic systems has been also investigated extensively.This goes back to the work of Russel [32,33] and Rauch and Taylor [31].Many important progress has been obtained recently, see, e.g., [4] and the references therein.It is worth noting that many works have been devoted to the L 2 -framework.Less is studied in the L p -scale.In this direction, we want to mention [9] where the exponential stability is studied for dissipative boundary condition.
Concerning the wave equation in one dimensional space, the exponential decay in L 2 -setting for the dynamic boundary condition is also established via its corresponding hyperbolic systems [30].However, to our knowledge, the exponential decay for the Dirichlet and Neumann boundary conditions has not been established even in L 2 -framework via this approach.Our work is new and quite distinct from the one in [30] and has its own interest.First, the analysis in [30] uses essentially the fact that the boundary is strictly dissipative, i.e., κ > 0 in (1.4).Thus the analysis cannot be used for the Dirichlet and Neumann boundary conditions.Moreover, it is not clear how to extend it to the L p -framework.Concerning our analysis, the key observation is that the information of the internal energy allows one to control the difference of the solutions and its mean value in the interval of time (0, T ) in an average sense.This observation is implemented in two lemmas (Lemma 3.2 and Lemma 3.3) after using a standard result (Lemma 3.1) presented in Section 3.These two lemmas are the main ingredients of our analysis for the Dirichlet and Neumann boundary conditions.The proof of the first lemma is mainly based on the characteristic method while as the proof of the second lemma is inspired from the theory of functions with bounded mean oscillations due to John and Nirenberg [18].As seen later that, the analysis for the dynamic boundary condition is much simpler for which the use of Lemma 3.1 is sufficient.
An interesting point of our analysis is the fact that these lemmas do not depend on the boundary conditions used.In fact, one can apply it in a setting where a bound of the internal energy is accessible.This allows us to deal with all the boundary conditions considered in this paper by the same way.Another point of our analysis which is helpful to be mentioned is that the asymptotic stability for hyperbolic systems in one dimensional space has been mainly studied for general solutions.This is not the case in the setting of Theorem 1.2 where the asymptotic stability holds under condition (1.10).It is also worth noting that the time-dependent coefficients generally make the phenomena more complex, see [13] for a discussion on the optimal null-controllable time.
The analysis in this paper cannot handle the cases p = 1 and p = +∞.Partial results in this direction for the Dirichlet boundary condition can be found in [19] where a is constant and in some range.These cases will be considered elsewhere by different approaches.
The paper is organized as follows.The well-posedness of (1.1) equipped with one of the boundary conditions (1.2) and (1.3) is discussed in Section 2, where a slightly more general context is considered (the boundary condition (1.4) is considered directly in Section 6; comments on this point is given in Remark 6.3).Section 4 is devoted to the proof of Theorem 1.1 and Theorem 1.2.We also relaxed slightly the non-negative assumption on a in Theorem 1.1 and Theorem 1.2 there (see Theorem 4.1 and Theorem 4.2) using a standard perturbative argument.The Neumann boundary condition is studied in Section 5 and the Dynamic boundary condition is considered in Section 6.

The well-posedness in L p -setting
In this section, we give the meaning of the solutions of the damped wave equation (1.1) equipped with either the Dirichlet boundary condition (1.2) or the Neumann boundary condition (1.3) and establish their well-posedness in the L p -framework with 1 ≤ p ≤ +∞.We will consider a slightly more general context.More precisely, we consider the system (2.1) Here a, b, c ∈ L ∞ ((0, T ) × (0, 1)) and f ∈ L p ((0, T ) × (0, 1)).
We begin with the Dirichlet boundary condition.
in the distributional sense in (0, T ) for all v ∈ C 1 c (0, 1).Definition 2.1 can be modified to deal with the case p = +∞ as follows.
Concerning the Neumann boundary condition, we have the following definition.
Definition 2.1 can be modified to deal with the case p = +∞ as follows.
Concerning the well-posedness of the Dirichlet system (2.1) and (2.2), we establish the following result.
1 By interpolation, one can use )) for any 1 ≤ q < +∞.This condition is used to give the meaning of the initial conditions.
2 By interpolation, one can use 0, 1)) for any 1 ≤ q < +∞.This condition is used to give the meaning of the initial conditions.
Concerning the well-posedness of the Neumann system (2.1) and (2.3), we prove the following result.
Remark 2.1.The definition of weak solutions and the well-posedness are stated for p = 1 and p = +∞ as well.The existence and the well-posedness is well-known in the case p = 2.The standard analysis in the case p = 2 is via the Galerkin method.
The rest of this section is devoted to the proof of Proposition 2.1 and Proposition 2.2 in Section 2.1 and Section 2.2, respectively.
2.1.Proof of Proposition 2.1.The proof is divided into two steps in which we prove the uniqueness and the existence.
and, if p = +∞ then the following two facts hold u 0,n ⇀ u 0 weakly star in W 1,∞ 0 (0, 1) and u 1,n ⇀ u 1 weakly star in L ∞ (0, 1), and, for 1 ≤ q < +∞, The existence of (a n , b n , c n ) and the existence of u 0,n and u 1,n follows from the standard theory of Sobolev spaces, see, e.g., [5].Let u n be the weak solution corresponding to (a Extend u n , g n , and f in (0, T ) × R by first odd refection in (−1, 0) and so on, and still denote the extension by u n and g n , and f .We then have (2.17) The d'Alembert formula gives As in the proof of the uniqueness, we then have, for 1 ≤ p < +∞ and 0 < t < T , (2.18) and, for p = +∞, .
The proof is complete.
Remark 2.2.Our proof on the well-posedness is quite standard and is based on the d'Alembert formula.This formula was also used previously in [19].
Remark 2.3.There are several ways to give the notion of weak solution even in the case p = 2, see, e.g., [2,8].The definitions given here is a nature modification of the case p = 2 given in [2].
2.2.Proof of Proposition 2.2.The proof of Proposition 2.2 is similar to the one of Proposition 2.1.To apply the d'Alembert formula, one just needs to extend various function appropriately and differently.For example, in the proof of the uniqueness, one extend u and g in (0, T ) × R by appropriate reflection in x first by even extension in (−1, 0), i.e., u(t, x) = u(t, −x) and g(t, x) = g(t, −x) in (0, T ) × (−1, 0) and so on.The details are left to the reader.

Some useful lemmas
In this section, we prove three lemmas which will be used through out the rest of the paper.The first one is quite standard and the last two ones are the main ingredients of our analysis for the Dirichlet and Neumann boundary condition.We begin with the following lemma.Lemma 3.1.Let 1 < p < +∞, 0 < T < T0 , and a ∈ L ∞ ((0, T ) × (0, 1)) be such that a ≥ 0 in (0, T ) × (0, 1).There exists a positive constant C depending only on p, T0 , and a L ∞ such that, for (ρ, ξ) ∈ L p (0, T ) × (0, 1) 2 , Proof.The proof of Lemma 3.1 is quite standard.For the convenience of the reader, we present its proof.There exists a positive constant C p depending only on p such that • for 2 ≤ p < +∞, it holds, for α, β ∈ R, Using this, we derive that This yields and, using Hölder's inequality, one gets The conclusion follows from (3.3) and (3.4).
Then there exists z ∈ (x 0 − ε 0 /2, x 0 + ε 0 /2) such that for some positive constant C depending only on ε 0 , λ, p, T 0 , T0 , and a L ∞ . Proof.Set By choosing y = z in (3.11) and (3.12), then by using (3.13) and the fact that (itself consequence of (1.5)) for some positive constant C(a, p) only depending on a, p, one gets the conclusion.
The next lemma is also a main ingredient of our analysis for the Dirichlet and Neumann boundary conditions.Lemma 3.3.Let 1 ≤ p < +∞ and L > l > 0, and let u ∈ L p (0, L + l).Then there exists a positive constant C depending only on p, L, and l such that Here and in what follows, ffl b a means 1 b−a ´b a for b > a. Proof.By scaling, one can assume that L = 1.Fix n ≥ 2 such that 2/n ≤ l ≤ 2/(n − 1).
Remark 3.1.Related ideas used in the proof of Lemma 3.3 was implemented in the proof of Caffarelli-Kohn-Nirenberg inequality for fractional Sobolev spaces [28].

Exponential decay in L p -framework for the Dirichlet boundary condition
In this section, we prove Theorem 1.1 and Theorem 1.2.We begin with the proof Theorem 1.2 in the first section, and then use it to prove Theorem 1.1 in the second section.We finally extend these results for a which might be negative in some regions using a standard perturbation argument in the third section.4.1.Proof of Theorem 1.2.We will only consider smooth solutions (ρ, ξ) 4 .The general case will follow by regularizing arguments.Moreover, replacing (ρ, ξ) by (ρ − c 0 , ξ − c 0 ), where the constant c 0 is defined in (1.12), we can assume that Multiplying the equation of ρ with ρ|ρ| p−2 , the equation of ξ with ξ|ξ| p−2 , and integrating the expressions with respect to x, after using the boundary conditions, we obtain, for t > 0, This implies Integrating the equations of ρ and ξ, summing them up and using the boundary conditions, we obtain d dt ˆ1 0 ρ(t, x) + ξ(t, x) dx = 0 for t > 0.
It follows that By (4.2) and (4.3), to derive (1.11), it suffices to prove that there exists a constant c > 0 depending only on a L ∞ (R + ×(0,1)) , ε 0 , γ, and p such that for any T > 2, there exists c T > 0 only depending on p, T, a so that (4.4) .
By scaling, without loss of generality, one might assume that We next prove the following estimates (4.12) The arguments being similar, we only provide that of (4.12).For x ∈ (0, 1), one has, by using the boundary condition at x = 0, i.e., ρ(•, 0) = ξ(•, 0), which yields, after substracting A ξ to both sides of the above equality, We use the characteristics method and (3.9),(3.10) to upper bound the first two integrals in the right-hand side of (4.14) by C(m p + m 2/p p ).As for the third integral in the right-hand side of (4.14), we perform the change of variables t = x + z to obtain which is upper bounded by C(m p + m 2/p p ) according to (4.11).The proof of (4.12) is complete.We now resume the argument for (4.4).We start by noticing that, for every t ∈ (0, T ) Taking the p-th power, integrating over t ∈ (0, T ) and using (4.11), one gets that Similarly, for every x ∈ (0, 1), Integrating over x ∈ (0, 1) and using (4.3), then taking the p-th power and using (4.12) and (4.13) yield (4. 16) Integrating over x ∈ (0, 1) and using (4.5), one gets (4.17 p ) and hence m p ≥ c 3 for some positive constant depending only on a L ∞ (R + ×(0,1)) , ε 0 , γ, and p (after fixing for instance T = 3).The proof of the theorem is complete.4.2.Proof of Theorem 1.1.Using Theorem 1.2, we obtain the conclusion of Theorem 1.1 for smooth solutions.The proof in the general case follows from the smooth case by density arguments.4.3.On the case a not being non-negative.In this section, we first consider the following perturbed system of (1.9): We establish the following result.
It follows that By (4.21) and (4.22), to derive (4.20), it suffices to prove that there exists a constant c > 0 depending only on a L ∞ (R + ×(0,1)) , ε 0 , γ, and p such that for T = 35 , it holds Using the facts that a ≥ 0 and b is bounded, a simple application of Gronwall's lemma to (4.21) yields the existence of α > 0 depending only on b L ∞ (R + ×(0,1)) so that Let (ρ 1 , ξ 1 ) be the unique solution of the system (4.25) Thus −b(ρ − ξ) and b(ρ − ξ) can be considered as source terms for the system of (ρ 1 , ξ 1 ).We then derive from (4.24) that Then (4.27) Applying Theorem 1.2, we have for some positive constant c depending only on a L ∞ , ε 0 , and λ.The conclusion now follows from (4.26) and (4.27).

Exponential decay in L p -framework for the Neuman boundary condition
In this section, we study the decay of the solutions of the damped wave equation equipped the Neumann boundary condition and the solutions of the corresponding hyperbolic systems.Here is the first main result of this section concerning the wave equation.
One can check that for smooth solutions u of (1.1), the pair of functions (ρ, ξ) defined in (1.8) satisfies the system (5.3) Concerning (5.3), we prove the following result.
The rest of this section is organized as follows.The first subsection is devoted to the proof of Theorem 5.2 and the second subsection is devoted to the proof of Theorem 5.1.5.1.Proof of Theorem 5.2.The argument is in the spirit of that of Theorem 1.2.As in there, we will only consider smooth solutions (ρ, ξ).Multiplying the equation of ρ with ρ|ρ| p−2 , the equation of ξ with ξ|ξ| p−2 , and integrating the expressions with respect to x, after using the boundary conditions, we obtain, for t > 0, (5.5) This implies .
5.2.Proof of Theorem 5.1.The proof of Theorem 5.1 is in the same spirit of Theorem 1.1.However, instead of using Theorem 1.2, we apply Theorem 5.2.In fact, as in the proof of Theorem 1.1, we have Assertion (5.1) follows with two different appropriate positive constants C and γ.
Remark 5.1.We can also consider the setting similar to the one in Section 4.3 and establish similar results.This allows one to deal with a class of a for which a is not necessary to be non-negative.The analysis for this is almost the same lines as in Section 4.3 and is not pursued here.
6. Exponential decay in L p -framework for the dynamic boundary condition In this section, we study the decay of the solution of the damped wave equation equipped the dynamic boundary condition and of the solutions of the corresponding hyperbolic systems.Here is the first main result of this section concerning the wave equation.Theorem 6.1.Let 1 < p < +∞, κ > 0, and a ∈ L ∞ R + × (0, 1) non negative.Then there exist positive constants C, γ depending only on p, κ, and a L ∞ R + ×(0, 1) such that for all u 0 ∈ W 1,p (0, 1) and u 1 ∈ L p (0, 1), there exists a unique weak solution u ∈ C([0, +∞); W 1,p (0, 1)) ∩ C 1 ([0, +∞); L p (0, 1)) such that ∂ t u, ∂ x u ∈ C([0, 1]; L p (0, T )) for all T > 0 of (6.1) Remark 6.1.In Theorem 6.1, a weak considered solution of (6.1) means that ∂ tt u(t, x) − ∂ xx u(t, x) + a(t, x)∂ t u = 0 holds in the distributional sense, and the boundary and the initial conditions are understood as usual thanks to the regularity imposing condition on the solutions.
Regarding System (6.4) with c 0 , c 1 not necessarily equal, we prove the following result.
The rest of this section is organized as follows.The proof of Theorem 6.2 is given in the first section and the proof of Theorem 6.1 is given in the second section.
The conclusion then follows using the characteristics method.
6.2.Proof of Theorem 6.1.We first deal with the well-posedness of the system.The uniqueness follows as in the proof of Proposition 2.1 via the d'Alembert formula.The existence can be proved by approximation arguments.First deal with smooth solutions (with smooth a) using Theorem 6.2 and then pass to the limit.The details are omitted.The proof of (6.5) is in the same spirit of (1.7).However, instead of using Theorem 1.2, we apply Theorem 6.2.The details are left to the reader.Remark 6.3.One can prove the well-posedness of (1.1) and (1.4) directly in L p -framework.Nevertheless, to make the sense for the boundary condition, one needs to consider regular solutions and then a is required to be more regular than just L ∞ .We here take advantage of the fact that such a system has a hyperbolic structure as given in (6.4).This give us the way to give sense for the solution by imposing the fact ∂ t u, ∂ x u ∈ C([0, 1]; L p (0, T )) for all T > 0.