Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More preciously, we investigate some $L^2$-estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$ utilizing the group Fourier transform on $G$. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\cdot)\|_{L^2(G)}$ by further assuming the $L^1(G)$-regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal{C}^1([0,T],H^1_{\mathcal L}(G)).$


Introduction
Let G be a compact Lie group and let L be the Laplace-Beltrami operator on G (which also coincides with the Casimir element of the enveloping algebra of the Lie algebra of G).In this paper we derive decay estimates for the solution to the Cauchy problem for a nonlinear wave equation with two types of damping terms, namely, x ∈ G, ∂ t u(x, 0) = εu 1 (x), x ∈ G, where ε is a positive constant describing the smallness of Cauchy data.Here, for the moment, we assume that u 0 and u 1 are taken from the energy space H 1 L (G) and concerning the nonlinearity of f (u), we shall deal only with the typical case such as f (u) := |u| p , p > 1 without loosing the essence of the problem.The equation (1.1) is known as the viscoelastic dumped wave equation associted with the Laplace-Beltrami operators on compact Lie groups.
The linear viscoelastic damped wave equation in the setting of the Euclidean space has been well studied in the literature.Several prominent researchers have devoted considerable attention to the following Cauchy problem for linear damped wave equation due to its application of this model in the theory of viscoelasticity and some fluid dynamic.
In his seminal work, Matsumura [18] first established basic decay estimates for the solution to the linear equation (1.2) and afterthat, many researchers have concentrated on investigated a typical important nonlinear problem, namely, the following semilinear damped wave equation x ∈ R n , t > 0, u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x), x ∈ R n . (1.3) In this case, there exists a real number p F ∈ (1, ∞) such that if p > p F , then for some range of p the corresponding Cauchy problem (1.3) has a small global in time solution u(t, x) for the small initial data u 0 and u 1 .On the other hand, when p ∈ (1, p F ], under some condition on the initial data ( R n u i (x)dx > 0, i = 0, 1), the corresponding problem (1.3) does not have any nontrivial global solutions.In general, such a number p F is called as the Fujita critical exponent.For a detailed study related to Fujita exponent, we refer to [1,13,9,20,33] and references therein.
Further, the study of the semilinear damped wave equation (1.3) is further generalized by the following strongly damped wave equation by several researchers recently.When µ = 0, in the case, for the dissipative structures of the Cauchy problem (1.4), Ponce [24] and Shibata [31] derived some L p (R n ) − L q (R n ) decay estimates for the solution to (1.4) with µ = 0.In the last decade, some )-regularity were also derived by several authors in [8,11,4,2].In the same period, the authors of [8] proved global (in time) existence of small data solution to the corresponding semilinear Cauchy problem to (1.4) with power nonlinearity on the right-hand side.Recently, Ikehata-Todorova-Yordanov [15] and Ikehata [10] have caught an asymptotic profile of solutions to problem (1.4) which is well-studied in the field of the Navier-Stokes equation case.
The study of the semilinear wave equation has also been extended in the non-Euclidean framework.Several papers are devoted for studying linear PDE in non-Euclidean structures in the last decades.For example, the semilinear wave equation with or without damping has been investigated for the Heisenberg group [19,26].In the case of graded groups, we refer to the recent works [25,30,32].Concerning the damped wave equation on compact Lie groups, we refer to [22,21,23,7] (see also [5] for the fractional wave equation).Here, we would also like to highlight that estimates for the linear viscoelastic damped wave equation on the Heisenberg groupin was studied in [16].
Recently, Ikehata-Sawada [14] and Ikehata-Takeda [12] considered and studied the following Cauchy problem which has two types of damping terms Such type of related problem with slight variants extensively investigated in by authors [2,17,3].An interesting and viable problem is to consider such types of (i.e., Cauchy problem 1.7) viscoelastic damped wave equations in the setting of non-Euclidean spaces, in particular, compact Lie groups.So far to the best of our knowledge, in the framework of compact Lie group, the viscoelastic damped wave equation have not been studied yet.Our main aim of this this article is to study the Cauchy problem for nonlinear wave equation with two types of damping terms on the compact Lie group G, namely, x ∈ G.
1.1.Main results.Throughout the paper we denote L q (G), the space of q-integrable functions on G with respect to the normalized Haar measure for 1 ≤ q < ∞ (respectively, essentially bounded for q = ∞) and for s > 0 and q ∈ (1, ∞) the Sobolev space H s,q L (G) is defined as the space . We simply denote H s L (G) as the Hilbert space H s,2 L (G).By employing the tools from the Fourier analysis for compact Lie groups, our first result below is concerned with the existence of the global solution to the homogeneous Cauchy problem (1.1) (i.e., when f = 0) satisfying the suitable decay properties.More precisely, our goal is to derive L 2 (G)-decay estimates for the Cauchy data as it is stated in the following theorem.
x ∈ G. (1.7) Then, u satisfies the following L 2 -estimates for any t ≥ 0, where C is a positive multiplicative constant.
Remark 1.2.From the statement of Theorem 1.1 one can find that the regularity u 1 ∈ H 1 L (G) is necessary to remove the singularity of ∂ t (−L) 1/2 u(t, •) L 2 (G) near t = 0. Remark 1.3.We also show that there is no improvement of any decay rate for the norm u(t, •) L 2 (G) in Theorem 1.1 even if we assume L 1 (G)-regularity for u 0 and u 1 .
Next we prove the local well-posedness of the Cauchy problem (1.1) in the energy evolution space C 1 [0, T ], H 1 L (G) .In particular, a Gagliardo-Nirenberg type inequality (proved in [29]) will be used in order to estimate the power nonlinearity in L 2 (G).The following result is about the local existence for the solution of the Cauchy problem (1.1).
Theorem 1.4.Let G be a compact, connected Lie group and let n be the topological dimension of G. Assume that n ≥ 3. Suppose that u 0 , u 1 ∈ H 1 L (G) and p > 1 such that p ≤ n n−2 .Then, there exists T = T (ε) > 0 such that the Cauchy problem (1.1) admits a uniquely determined mild solution u in the space C 1 ([0, T ], H 1 L (G)).As in [22], we note that, in the statement of Theorem 1.4, the restriction on the upper bound for the exponent p which is p ≤ n n−2 is necessary in order to apply Gagliardo-Nirenberg type inequality (5.4) in (5.6) the the proof of Theorem 1.4.The other restriction n ≥ 3 is also technical and is made to fulfill the assumptions for the employment of such inequality.This could be avoided if one look for solution in a different space such as It is customary to study the corresponding nonlinear homogeneous problem, that is, when f = 0 prior to investigate the nonhomogneous problem (1.1).In this process, we first establish a L 2 -energy estimates for the solution to the homogeneous viscoelastic damped wave equation on the compact Lie group G. Having these estimates on our hand, we implement a Gagliardo-Nirenberg type inequality on compact Lie group ( [29,22,21,23]) to prove the local well-posedness result for the solution to (1.1).We also show that, even if we assume L 1 (G)-regularity for u 0 and u 1 , there is no additional decay rate can be gained for the L 2 norm of the solution of the corresponding homogeneous Cauchy problem.
Apart from introduction the paper is organized as follows.In Section 2, we recall some essentials from the Fourier analysis on compact Lie groups which will be frequently used throughout the paper.In Section 3, we prove Theorem 1.1 by deriving some L 2 decay estimates for the solution of the homogeneous nonlinear viscoelastic damped wave equation on the compact Lie group G.We also show that, there is no additional gain in the decay rate of the L 2 norm of the solution to the corresponding homogeneous Cauchy problem even if we assume L 1 (G)-regularity for u 0 and u 1 in Section 4. Finally, in Section 5, we briefly recall the notion of mild solutions in our framework and prove the local well-posedness of the Cauchy problem (1.1) in the energy evolution space C 1 ([0, T ], H α L (G)).1.2.Notations.Throughout the article, we use the following notations: • f g : There exists a positive constant C (whose value may change from line to line in this manuscript) such that f ≤ Cg.
• dx : The normalized Haar measure on the compact group G.
• L : The Laplace-Beltrami operator on G.
• C d×d : The set of matrices with complex entries of order d.
The identity matrix of order d.

Preliminaries: Fourier analysis on compact Lie groups
In this section, we recall some basics of the Fourier analysis on compact (Lie) groups to make the manuscript self-contained.A complete account of representation theory of the compact Lie groups can be found in [7,28,27].However, we mainly adopt the notation and terminology given in [27].
Let us first recall the definition of a representation of a compact group G.A unitary representation of G is a pair (ξ, H) such that the map ξ : G → U (H), where U (H) denotes the set of unitary operators on complex Hilbert space H, such that it satisfies following properties: • The map ξ is a group homomorphism, that is, ξ(xy) = ξ(x)ξ(y).
• The mapping ξ : G → U (H) is continuous with repsect to strong operator topology (SOT) on U (H), that is, the map g → ξ(g)v is continuous for every v ∈ H.The Hilbert space H is called the representation space.If is there is no confusion, we just write ξ for a representation (ξ, H) of G. Two unitary representations ξ, η of G are called equivalent if there exists an unitary operator, called intertwiner, T such that T ξ(x) = η(x)T for any x ∈ G.The intertwiner is a irreplaceable tool in the theory of representation of compact groups and helpful in the classification of representation.A (linear) subspace V ⊂ H is said to be invariant under the unitary representation ξ of G if ξ(x)V ⊂ V for any x ∈ G.An irreducible unitary representation ξ of G is a representation such that the only closed and ξ-invariant subspaces of H are trivial once, that is, {0} and the full space H.
The set of all equivalence classes [ξ] of continuous irreducible unitary representations of G is denoted by G and called the unitary dual of G. Since G is compact, G is a discrete set.It is known that an irreducible unitary representation ξ of G is finite dimensional, that is, the Hilbert space H is finite dimensional, say, d ξ .Therefore, if we choose a basis B := {e 1 , e 2 , . . ., e d ξ } for the representation space H of ξ, we can identify H as C d ξ and consequently, we can view ξ as a matrix-valued function ξ : G → U (C d ξ ×d ξ ), where U (C d ξ ×d ξ ) denotes the space of all unitary matrices.The matrix coefficients ξ ij of the representation ξ with respect to B are given by ξ ij (x) := ξ(x)e j , e i for all i, j ∈ {1, 2, . . ., d ξ }.It follows from the Peter-Weyl theorem that the set The group Fourier transform of f ∈ L 1 (G) at ξ ∈ G, denoted by f (ξ), is defined by where dx is the normalised Haar measure on G.It is apparent from the definition that f (ξ) is matrix valued and therefore, this definition can be interpreted as weak sense, that is, for u, v ∈ H, we have It follows from the Peter-Weyl theorem that, for every f ∈ L 2 (G), we have the following the Fourier series representation: The Plancherel identity for the group Fourier transform on G takes the following form where • HS denotes the Hilbert-Schmidt norm of a matrix A := (a ij ) ∈ C dxi×d ξ defined as We would like to emphasize here that the Plancherel identity is one of the crucial tools to establish L 2 -estimates of the solution to PDEs.
Let L be the Laplace-Beltrami operator on G.It is important to understand the action of the group Fourier transform on the Laplace-Beltrami operator L for developing the machinery of the proofs.For [ξ] ∈ G, the matrix elements ξ ij , are the eigenfunctions of L with the same eigenvalue −λ 2 ξ .In other words, we have, for any The symbol σ L of the Laplace-Beltrami operator L on G is given by for any [ξ] ∈ G and therefore, the following holds: For s > 0, the Sobolev space H s L (G) of order s is defined as follows: and (−L) s/2 is defined in terms of the group Fourier transform by the follwoing formula Further, using Plancherel identity, for any s > 0, we have that We also recall the definition of the space ℓ ∞ ( G).We denote S ′ ( G) as the space of slowly increasing distributions on the unitary dual G of G. Then the space ℓ ∞ ( G) is defined as Moreover, for any f ∈ L 1 (G), from the group Fourier transform it is true that We must mention that implementation of (2.4) very important in order to use the L 1 (G)regularity for the Cauchy data.A detailed study on the construction of the space ℓ ∞ ( G) can be found in Section 10.3.2 of [27] (see also Section 2.1.3 of [6]).

L 2 -estimates for the solution to the homogeneous problem
In this section, we derive L 2 (G)-L 2 (G) estimates for the solutions to (1.7) when f = 0, namely, the homogeneous problem on G : x ∈ G. (3.1) We employ the group Fourier transform on the compact Lie group G with respect to the space variable x together with the Plancherel identity in order to estimate L 2 -norms of u(t, ), (−L) ), and ∂ t (−L) 1/2 u(t, •).Let u be a solution to (3.1).Let u(t, ξ) = ( u(t, ξ) kl ) 1≤k,l≤d ξ ∈ C d ξ ×d ξ , [ξ] ∈ G denote the Fourier transform of u with respect to the x variable.Invoking the group Fourier transform with respect to x on (3.1), we deduce that u(t, ξ) is a solution to the following Cauchy problem for the system of ODE's (with size of the system that depends on the representation ξ) where σ L is the symbol of the of the Laplace-Beltrami operator operator L defined in (2.2).Using the identity (2.2), the system (3.2) is decoupled in for all k, l ∈ {1, 2, . . ., d ξ }.
Then, the characteristic equation of (3.3) is given by and consequently the characteristic roots of (3.3) are We note that if λ 2 ξ = 1, then there are two distinct roots, say, λ + = −1 and λ − = −λ 2 ξ , and if λ 2 ξ = 1 then both the roots are same and equal to λ = −1.We analyze the following two cases for the solution to the system (3.3).
Case I. Let λ 2 ξ = 1.The solution of (3.3) is given by where where Thus we have (3.8) Also we note that First we determine an explicit expression for the L 2 (G) norms of u(t, •), (−L) 1/2 u(t, •), ∂ t u(t, •), and ∂ t (−L) 1/2 u(t, •).We apply the group Fourier transform with respect to the spatial variable x together with the Plancherel identity in order to determine the L 2 (G) norms.
To simplify the presentation we introduce the following partition of the unitary dual G as: Here we note that some of the above sets may be empty.

Estimate for u(t, •) L 2 (G)
. By Plancherel formula we have (3.9) Estimate on R 1 .Using λ 2 ξ = 0 in (3.5) we get is bounded on R 2 and by (3.5) we have Hence by (3.4) we get Substituting (3.20) in (3.9) we obtain Remark.Note that we do not get any decay on the R.H.S of (3.21) due to the fact that the set R 1 is always non empty (in fact singleton) .

Estimate for
. By Plancherel formula we get Estimate on R 1 .We have Estimate on R 2 .By (3.13) we obtain Estimate on R 3 .by (3.15) we have Estimate on R 4 .Again using the fact that 1 Therefore, . By Plancherel theorem we have We note that Estimate on R 1 .From (3.26), we have Estimate on R 2 .By (3.26) and (3.12) we obtain Estimate on R 3 .By (3.26) we get Estimate on R 4 .Using (3.26), (3.17) and the fact that 1 . By Plancherel theorem we have Estimate on R 1 .We have Estimate on R 2 .By (3.13) and (3.26) we obtain Estimate on R 3 .By (3.26) we have Estimate on R 4 .Using (3.26), (3.17) and the fact that 1  In this section we show that there is no improvement of any decay rate for the norm u(t, •) L 2 (G) when further we assume L 1 (G)-regularity for u 0 and u 1 .Note that in Theorem 1.1 we employed data on L 2 (G) basis.Since G is compact group it follows that the Haar measure of G is finite.This implies that L 2 (G) is continuously embedded in L 1 (G) and therefore, one might be curious to know that which changes will occur if we further implement L 1 (G)-regularity for u 0 and u 1 .
From (3.14), (3.16), and (3.19) it immediately follows that for some suitable constant δ 1 .Therefore, the contribution to the sum in (3.9) corresponding to R 1 refrain us to get a decay rate for u(t, •) 2 L 2 (G) .Thus, if we want to employ L 1 (G)regularity rather than L 2 (G)-regularity, then we must apply it to obtain the estimation of the terms with [ξ] ∈ R 1 .Here, we must note that the set R 1 is a singleton.
Note that for the multiplier in (3.5), the best estimate that one can obtain on the set R 1 are Since the set R 1 is singleton, using the definition defined in (2.3), we obtain This shows that even if we use L 1 (G)-regularity we are not able to get any decay rate for the norm u(t, •) L 2 (G) .
One can easily observe that the main reason behind this behaviour is that we can not neglect the eigenvalue 0 as the Plancherel measure on a compact Lie group turns out to be a weighted counting measure.Remark 4.1.In the noncompact setting such as the Euclidean space and the Heisenberg group, one can get a global existence result for a non empty range for p by asking an additional L 1 -regularity for the initial data.Consequently, we get an improved decay rates for the estimates of the L 2 -norm of the solution to the corresponding linear homogeneous problem.One can see [22,16] for the illustration and discussion on this matter.

Local existence
This section is devoted to prove Theorem 1.4, i.e., the local well-posedness of the Cauchy problem (1.1) in the energy evolution space C 1 [0, T ], H 1 L (G) .To present the proof of Theorem 1.4, first we recall the notion of mild solutions in our setting.
Consider the space X(T ( The solution to the nonlinear inhomogeneous problem can be expressed, by using Duhamel's principle, as where * (x) denotes the convolution with respect to the x variable, E 0 (t, x) and E 1 (t, x) are the fundamental solutions to the homogeneous problem (5.2), i.e., when F = 0 with initial data (u 0 , u 1 ) = (δ 0 , 0) and (u 0 , u 1 ) = (0, δ 0 ), respectively.For any left-invariant differential operator L on the compact Lie group G, we applied the property that L v * and the invariance by time translations for the viscoelastic wave operator ∂ 2 t − L + ∂ t − L∂ t in order to get the previous representation formula.Definition 5.1.The function u is said to be a mild solution to (5.2) on [0, T ] if u is a fixed point for the integral operator N : u ∈ X(T ) → N u(t, x) defined as N u(t, x) = εu 0 (x) * (x) E 0 (t, x) + εu 1 (x) * (x) E 1 (t, x) + t 0 |u(s, x)| p * (x) E 1 (t − s, x) ds (5.3) in the evolution space C 1 [0, T ], H 1 L (G) , equipped with the norm defined in (5.1).As usual, the proof of the fact that, the map N admits a uniquely determined fixed point for sufficiently small T = T (ε), is based on Banach's fixed point theorem with respect to the norm on X(T ) as defined above.More preciously, for (u 0 , u 1 ) H 1 L (G)×H 1 L (G) small enough, if we can show the validity of the following two inequalities N u X(T ) ≤ C (u 0 , u 1 ) H 1 L (G)×H 1 L (G) + C u p X(T ) , N u − N v X(T ) ≤ C u − v X(T ) u p−1 X(T ) + v p−1 X(T ) , for any u, v ∈ X(T ) and for some suitable constant C > 0 independent of T .Then by Banach's fixed point theorem we can assure that the operator N admits a unique fixed point u.This function u will be the mild solution to (5.2) on [0, T ].
In order to prove the local existence result, an important tool is the following Gagliardo-Nirenberg type inequality which can be derived from the general version of this inequality given in [29].We also refer [29] for the detailed proof of this inequality for more general connected unimodular Lie groups.
One can also consult [29,22] for several immediate important remarks.

4 .
L 1 (G) − L 2 (G) estimates for the solution to the homogeneous problem