2. Preliminaries: Fourier analysis on compact Lie groups

In this section, we recall some basics of 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 [Reference Garetto and Ruzhansky7, Reference Ruzhansky and Turunen27, Reference Ruzhansky and Turunen28]. However, we mainly adopt the notation and terminology given in [Reference Ruzhansky and Turunen27].

Let us first recall the definition of a representation of a compact group $G.$ A unitary representation of $G$ is a pair $(\xi,\, \mathcal {H})$ such that the map $\xi :G \rightarrow U(\mathcal {H}),$ where $U(\mathcal {H})$ denotes the set of unitary operators on complex Hilbert space $\mathcal {H},$ such that it satisfies the following properties:

• • The map $\xi$ is a group homomorphism, that is, $\xi (x y)=\xi (x)\xi (y).$

• • The mapping $\xi :G \rightarrow U(\mathcal {H})$ is continuous with respect to strong operator topology (SOT) on $U(\mathcal {H}),$ that is, the map $g \mapsto \xi (g)v$ is continuous for every $v \in \mathcal {H}.$

The Hilbert space $\mathcal {H}$ is called the representation space. If there is no confusion, we just write $\xi$ for a representation $(\xi,\, \mathcal {H})$ of $G.$ Two unitary representations $\xi,\, \eta$ of ${G}$ are called equivalent if there exists a unitary operator, called intertwiner, $T$ such that $T \xi (x)=\eta (x) T$ for any $x \in {G}$. The intertwiner is an irreplaceable tool in the theory of representation of compact groups and helpful in the classification of representation. A (linear) subspace $V \subset \mathcal {H}$ is said to be invariant under the unitary representation $\xi$ of $G$ if $\xi (x) V \subset V$ for any $x \in {G}$. An irreducible unitary representation $\xi$ of $G$ is a representation such that the only closed and $\xi$-invariant subspaces of $\mathcal {H}$ are trivial once, that is, $\{0\}$ and the full space $\mathcal {H}$.

The set of all equivalence classes $[\xi ]$ of continuous irreducible unitary representations of $G$ is denoted by $\widehat {G}$ and called the unitary dual of $G.$ Since $G$ is compact, $\widehat {G}$ is a discrete set. It is known that an irreducible unitary representation $\xi$ of $G$ is finite dimensional, that is, the Hilbert space $\mathcal {H}$ is finite dimensional, say, $d_\xi$. Therefore, if we choose a basis $\mathfrak {B}:=\{e_1,\,e_2,\,\ldots,\, e_{d_\xi }\}$ for the representation space $\mathcal {H}$ of $\xi$, we can identify $\mathcal {H}$ as $\mathbb {C}^{d_\xi }$ and consequently, we can view $\xi$ as a matrix-valued function $\xi : G \rightarrow U(\mathbb {C}^{d_{\xi } \times d_{\xi }})$, where $U(\mathbb {C}^{d_{\xi } \times d_{\xi }})$ denotes the space of all unitary matrices. The matrix coefficients $\xi _{ij}$ of the representation $\xi$ with respect to $\mathfrak {B}$ are given by $\xi _{ij}(x):=\langle \xi (x) e_j,\, e_i \rangle$ for all $i,\, j \in \{1,\,2,\, \ldots,\, d_\xi \}.$ It follows from the Peter-Weyl theorem that the set

\[ \left\{\sqrt{d_{\xi}} \xi_{i j}: 1 \leq i, j \leq d_{\xi},[\xi] \in \widehat{G}\right\} \]

forms an orthonormal basis of $L^{2}(G)$.

The group Fourier transform of $f \in L^1(G)$ at $\xi \in \widehat {G},$ denoted by $\widehat {f}(\xi ),$ is defined by

\[ \widehat{f}(\xi):=\int_{G} f(x) \xi(x)^{*} \,{\rm d}x, \]

where ${\rm d}x$ is the normalised Haar measure on $G$. It is apparent from the definition that $\widehat {f}(\xi )$ is matrix valued and therefore, this definition can be interpreted as weak sense, that is, for $u,\,v \in \mathcal {H},$ we have

\[ \langle \widehat{f}(\xi)u, v \rangle:=\int_{G} f(x) \langle \xi(x)^{*}u, v \rangle \,{\rm d}x. \]

It follows from the Peter–Weyl theorem that, for every $f \in L^2(G),$ we have the following Fourier series representation:

\[ f(x)=\sum_{[\xi] \in \widehat{G}} d_{\xi} \operatorname{Tr}(\xi(x) \widehat{f}(\xi)). \]

The Plancherel identity for the group Fourier transform on $G$ takes the following form:

(2.1)\begin{equation} \|f\|_{L^{2}(G)}=\left(\sum_{[\xi] \in \widehat{G}} d_{\xi}\|\widehat{f}(\xi)\|_{\mathrm{HS}}^{2}\right)^{1 / 2}:=\|\widehat{f}\|_{\ell^2(\widehat{G})},\end{equation}

where $\|\cdot \|_{\mathrm {HS}}$ denotes the Hilbert–Schmidt norm of a matrix $A:=(a_{ij}) \in \mathbb {C}^{d_ xi \times d_\xi }$ defined as

\[ \|A\|_{\mathrm{HS}}^{2}=\operatorname{Tr}\left( A A^{*}\right)=\sum_{i, j=1}^{d_{\xi}}|a_{ij}|^2. \]

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 $\mathcal {L}$ be the Laplace–Beltrami operator on $G$. It is important to understand the action of the group Fourier transform on the Laplace–Beltrami operator $\mathcal {L}$ for developing the machinery for the proofs. For $[\xi ] \in \widehat {G}$, the matrix elements $\xi _{i j}$ are the eigenfunctions of $\mathcal {L}$ with the same eigenvalue $-\lambda _{\xi }^{2}$. In other words, we have, for any $x \in {G},$

\[ -\mathcal{L} \xi_{i j}(x)=\lambda_{\xi}^{2} \xi_{i j}(x), \qquad \text{for all } i, j \in\left\{1, \ldots, d_{\xi}\right\}. \]

The symbol $\sigma _{\mathcal {L}}$ of the Laplace–Beltrami operator $\mathcal {L}$ on $G$ is given by

(2.2)\begin{equation} \sigma_{\mathcal{L}}(\xi)={-}\lambda_{\xi}^{2} I_{d_{\xi}}, \end{equation}

for any $[\xi ] \in \widehat {G}$ and therefore, the following holds:

\[ \widehat{\mathcal{L} f}(\xi)=\sigma_{\mathcal{L}}(\xi) \widehat{f}(\xi)={-}\lambda_{\xi}^{2} \widehat{f}(\xi) \]

for any $[\xi ] \in \widehat { G}$.

For $s>0,$ the Sobolev space $H_{\mathcal {L}}^s(G)$ of order $s$ is defined as follows:

\[ H_{\mathcal{L}}^s(G):=\left\{u \in L^{2}(G):\|u\|_{H_{\mathcal{L}}^s(G)}<{+}\infty\right\}, \]

where $\|u\|_{H_{\mathcal {L}}^s(G)}=\|u\|_{L^{2}(G)}+\left \|(-\mathcal {L})^{s / 2} u\right \|_{L^{2}({G})}$ and $(-\mathcal {L})^{s / 2}$ is defined in terms of the group Fourier transform by the following formula:

\[ (-\mathcal{L})^{\alpha / 2} f :=\mathcal{F}^{{-}1}\left(\lambda_{\xi}^{2 \alpha }(\mathcal{F} u)\right), \quad \text{for all $[\xi] \in \widehat{G}$}. \]

Further, using Plancherel identity, for any $s>0$, we have that

\[ \left\|(-\mathcal{L})^{s / 2} f\right\|_{L^{2}({G})}^{2}=\sum_{[\xi] \in \widehat{G}} d_{\xi} \lambda_{\xi}^{2 s}\|\widehat{f}(\xi)\|_{\mathrm{HS}}^{2} . \]

We also recall the definition of the space $\ell ^\infty (\widehat G).$ We denote $\mathcal {S}^\prime (\widehat G)$ as the space of slowly increasing distributions on the unitary dual $\widehat G$ of $G$. Then the space $\ell ^\infty (\widehat G)$ is defined as

\[ \ell^\infty(\widehat G)=\{H=\{H([\xi])\}_{[\xi]\in\widehat G} : \|H\|_{\ell^\infty(\widehat G)} <\infty \}, \]

where $H([\xi ])\in \mathbb {C}^{d_\xi \times d_\xi } ~\text { for any } ~[\xi ]\in \widehat G$ and

(2.3)\begin{equation} \|H\|_{\ell^\infty(\widehat G)}:=\sup\limits_{[\xi]\in\widehat G}d_{\xi}^{-\frac{1}{2}}\|H(\xi)\|_{HS}<\infty. \end{equation}

Then $\ell ^\infty (\widehat G)$ is a subspace of $\mathcal {S}^\prime (\widehat G)$. Moreover, for any $f\in L^1(G)$, from the group Fourier transform it is true that

(2.4)\begin{equation} \|\widehat f\|_{\ell^\infty(\widehat G)}\leq\| f\|_{L^1(G)}. \end{equation}

We must mention that implementation of (2.4) is very important in order to use the $L^1(G)$-regularity for the Cauchy data. A detailed study on the construction of the space $\ell ^\infty (\widehat G)$ can be found in section 10.3.2 of [Reference Ruzhansky and Turunen27] (see also section 2.1.3 of [Reference Fischer and Ruzhansky6]).

3. $L^2$-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 $G:$

(3.1)\begin{equation} \begin{cases} \partial^2_tu-\mathcal{L}u+\partial_tu-\mathcal{L}\partial_tu=0, & x\in G, t>0,\\ u(0,x)=u_0(x), & x\in G,\\ \partial_tu(x,0)=u_1(x), & x\in G. \end{cases} \end{equation}

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 $u(t,\,),\, (-\mathcal {L})^{\frac {1}{2}}u(t,\, \cdot )$, $\partial _{t}u(t,\, )$ and $\partial _t (-\mathcal {L})^{1/2} u(t,\,\cdot )$.

Let $u$ be a solution to (3.1). Let $\widehat {u}(t,\, \xi )=(\widehat {u}(t,\, \xi )_{kl})_{1\leq k, l\leq d_\xi }\in \mathbb {C}^{d_\xi \times d_\xi },\, [\xi ]\in \widehat { G}$ 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 $\widehat {u}(t,\, \xi )$ is a solution to the following Cauchy problem for the system of ordinary differential equations (with size of the system that depends on the representation $\xi$)

(3.2)\begin{equation} \begin{cases} \partial^2_t\widehat{u}(t,\xi)- \sigma_{\mathcal{L}}(\xi)\widehat{u}(t,\xi)+\partial_t\widehat{u}(t,\xi)-\sigma_{\mathcal{L}}(\xi) \partial_t\widehat{u}=0, & [\xi]\in\widehat{ G},t>0,\\ \widehat{u}(0,\xi)=\widehat{u}_0(\xi), & [\xi]\in\widehat{ G},\\ \partial_t\widehat{u}(0,\xi)=\widehat{u}_1(\xi), & [\xi]\in\widehat{ G}, \end{cases} \end{equation}

where $\sigma _{\mathcal {L}}$ is the symbol of the Laplace–Beltrami operator $\mathcal {L}$ defined in (2.2). Using identity (2.2), system (3.2) is decoupled in $d_\xi ^2$ independent ODEs, namely,

(3.3)\begin{equation} \begin{cases} \partial^2_t\widehat{u}(t,\xi)_{kl}+(1+\lambda_\xi^2)\partial_t\widehat{u}(t,\xi)_{kl}+\lambda^2_\xi\widehat{u}(t,\xi)_{kl}=0, & [\xi]\in\widehat{ G},t>0,\\ \widehat{u}(0,\xi)_{kl}=\widehat{u}_0(\xi)_{kl}, & [\xi]\in\widehat{ G},\\ \partial_t\widehat{u}(0,\xi)_{kl}=\widehat{u}_1(\xi)_{kl}, & [\xi]\in\widehat{ G}, \end{cases} \end{equation}

for all $k,\,l\in \{1,\,2,\,\ldots,\,d_\xi \}.$

Then, the characteristic equation of (3.3) is given by

\[ \lambda^2+(1+\lambda_\xi^2)\lambda+\lambda_\xi^2=0, \]

and consequently, the characteristic roots of (3.3) are

\[ \lambda=\frac{-(1+\lambda_\xi^2)\pm| 1-\lambda_\xi^2| }{2}. \]

We note that if $\lambda _\xi ^2\neq 1,$ then there are two distinct roots, say, $\lambda ^+=-1$ and $\lambda ^-=-\lambda _\xi ^2,$ and if $\lambda _\xi ^2=1$ then both the roots are same and equal to $\lambda =-1.$ We analyse the following two cases for the solution to system (3.3).

Case I. Let $\lambda _\xi ^2\neq 1.$ The solution of (3.3) is given by

(3.4)\begin{equation} \widehat{u}(t,\xi)_{kl}=\mathcal{K}_0(t,\xi)\widehat{u}_0(\xi)_{kl}+\mathcal{K}_1(t,\xi)\widehat{u}_1(\xi)_{kl}, \end{equation}

where

(3.5)\begin{equation} \begin{cases} \mathcal{K}_0(t,\xi)=\frac{e^{-\lambda_\xi^2t}-\lambda_\xi^2e^{{-}t}}{1-\lambda_\xi^2},\\ \mathcal{K}_1(t,\xi)=\frac{e^{-\lambda_\xi^2t}-e^{{-}t}}{1-\lambda_\xi^2}. \end{cases} \end{equation}

Case II. Let $\lambda _\xi ^2=1.$ The solution of (3.3) is given by

(3.6)\begin{equation} \widehat{u}(t,\xi)_{kl}=\mathcal{K}_0(t,\xi)\widehat{u}_0(\xi)_{kl}+\mathcal{K}_1(t,\xi)\widehat{u}_1(\xi)_{kl}, \end{equation}

where

(3.7)\begin{equation} \begin{cases} \mathcal{K}_0(t,\xi)=(1+t)e^{{-}t},\\ \mathcal{K}_1(t,\xi)=te^{{-}t}. \end{cases} \end{equation}

Thus, we have

(3.8)\begin{equation} \widehat{u}(t,\xi)_{kl}= \begin{cases} \frac{e^{-\lambda_\xi^2t}-\lambda_\xi^2e^{{-}t}}{1-\lambda_\xi^2}\widehat{u}_0(\xi)_{kl}+\frac{e^{-\lambda_\xi^2t}-e^{{-}t}}{1-\lambda_\xi^2} \widehat{u}_1(\xi)_{kl}, & \lambda_\xi^2\neq1,\\ (1+t)e^{{-}t} \widehat{u}_0(\xi)_{kl}+te^{{-}t}\widehat{u}_1(\xi)_{kl}, & \lambda_\xi^2=1. \end{cases} \end{equation}

Also, we note that

\[ \partial_t^\ell\mathcal{K}_0(t,\xi)=\frac{(-\lambda_\xi^2)^\ell e^{-\lambda_\xi^2t}+({-}1)^{\ell+1}\lambda_\xi^2e^{{-}t}}{1-\lambda_\xi^2}, \]

and

\[ \partial_t^\ell\mathcal{K}_1(t,\xi)=\frac{(-\lambda_\xi^2)^\ell e^{-\lambda_\xi^2t}+({-}1)^{\ell+1}e^{{-}t}}{1-\lambda_\xi^2}. \]

First, we determine an explicit expression for the $L^2(G)$ norms of $u(t,\,\cdot ),\, (-\mathcal {L})^{1/2} u(t,\, \cdot ),\, \partial _t u(t,\,\cdot )$ and $\partial _t (-\mathcal {L})^{1/2} u(t,\,\cdot )$. 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 $\widehat { G}$ as:

\begin{align*} \mathcal{R}_1& =\{[\xi]\in\widehat{ G}:\lambda_\xi^2=0\},\\ \mathcal{R}_2& =\{[\xi]\in\widehat{ G}:0<\lambda_\xi^2<1\},\\ \mathcal{R}_3& =\{[\xi]\in\widehat{ G}:\lambda_\xi^2=1\}\text{ and }\\ \mathcal{R}_4& =\{[\xi]\in\widehat{ G}:\lambda_\xi^2>1\}. \end{align*}

Here, we note that some of the above sets may be empty.

3.1 Estimate for $\| u(t,\,\cdot )\|_{L^2( G)}$

By the Plancherel formula, we have

(3.9)\begin{equation} \| u(t,\cdot)\|_{L^2( G)}=\sum\limits_{[\xi]\in\widehat{ G}}d_\xi\sum\limits_{k,l=1}^{d_{\xi}}| \widehat{u}(t,\xi)_{kl}| ^2. \end{equation}

Estimate on $\mathcal {R}_1.$ Using $\lambda _\xi ^2=0$ in (3.5) we get

(3.10)\begin{equation} | \mathcal{K}_0(t,\xi)| ,~| \mathcal{K}_1(t,\xi)| \lesssim 1. \end{equation}

Hence, (3.4) implies that

(3.11)\begin{equation} | \widehat{u}(t,\xi)_{kl}| \lesssim | \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| . \end{equation}

Estimate on $\mathcal {R}_2.$ Since the set $\{\lambda _\xi ^2\}_{[\xi ]\in \widehat { G}}$ is a discrete set, there exist $\delta _1$ and $\delta _2$ such that

(3.12)\begin{equation} 0<\delta_1\leq\lambda_\xi^2\leq\delta_2<1,\quad[\xi]\in\mathcal{R}_2, \end{equation}

consequently $\frac {1}{1-\lambda _\xi ^2}$ is bounded on $\mathcal {R}_2$ and by (3.5) we have

(3.13)\begin{equation} | \mathcal{K}_0(t,\xi)| ,~| \mathcal{K}_1(t,\xi)| \lesssim e^{-\delta_1t}. \end{equation}

Hence, by (3.4) we get

(3.14)\begin{equation} | \widehat{u}(t,\xi)_{kl}| \lesssim e^{-\delta_1t}\left[| \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| \right]. \end{equation}

Estimate on $\mathcal {R}_3.$ By (3.7) we have

(3.15)\begin{equation} | \mathcal{K}_0(t,\xi)| ,~| \mathcal{K}_1(t,\xi)| \lesssim (1+t)e^{{-}t}. \end{equation}

Hence, by (3.6) we get

(3.16)\begin{equation} | \widehat{u}(t,\xi)_{kl}| \lesssim (1+t)e^{{-}t}\left[| \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| \right]. \end{equation}

Estimate on $\mathcal {R}_4.$ Again discreteness of the set $\{\lambda _\xi ^2\}_{[\xi ]\in \widehat { G}}$ implies that there exists $\delta _3$ such that

(3.17)\begin{equation} 1<\delta_3\leq\lambda_\xi^2, \qquad[\xi]\in\mathcal{R}_4. \end{equation}

Hence, $\frac {1}{\lambda _\xi ^2-1}$ and $\frac {\lambda _\xi ^2}{\lambda _\xi ^2-1}$ are bounded on $\mathcal {R}_4,$ consequently (3.5) yields

(3.18)\begin{equation} | \mathcal{K}_0(t,\xi)|,\quad | \mathcal{K}_1(t,\xi)| \lesssim e^{{-}t}.\end{equation}

Using (3.4) we obtain

(3.19)\begin{equation} | \widehat{u}(t,\xi)_{kl}| \lesssim e^{{-}t}\left[| \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| \right]. \end{equation}

Combining (3.11), (3.14), (3.16) and (3.19) we get

(3.20)\begin{equation} | \widehat{u}(t,\xi)_{kl}| \lesssim | \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| ,\quad[\xi]\in\widehat{ G}. \end{equation}

Substituting (3.20) into (3.9) we obtain

(3.21)\begin{equation} \| u(t,\cdot)\|_{L^2( G)} \leq C\left(\| u_0\|_{L^2( G)}+\| u_1\|_{L^2( G)}\right). \end{equation}

Remark Note that we do not get any decay on the right-hand side of (3.21) due to the fact that the set $\mathcal {R}_1$ is always nonempty (in fact singleton).

3.2 Estimate for $\|(-\mathcal {L})^{1/2} u(t,\,\cdot )\|_{L^2(G)}$

By the Plancherel formula, we get

\[ \|(-\mathcal{L})^{1/2} u(t,\cdot)||_{L^2(G)}=\sum\limits_{[\xi]\in\widehat{G}}d_\xi\|\sigma_{(-\mathcal{L})^{1/2}}(\xi)\widehat{u}(t,\xi)\|_{\operatorname{HS}}^2=\sum\limits_{[\xi]\in\widehat{G}}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2. \]

Estimate on $\mathcal {R}_1.$ We have

\[ \lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2=0. \]

Estimate on $\mathcal {R}_2.$ By (3.13) we obtain

(3.22)\begin{equation} \lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2\lesssim e^{{-}2\delta_1t}[|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2]. \end{equation}

Estimate on $\mathcal {R}_3.$ By (3.15) we have

(3.23)\begin{equation} \lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2\lesssim (1+t)^2e^{{-}2t}\left[|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right]. \end{equation}

Estimate on $\mathcal {R}_4.$ Again using the fact that $\frac {1}{\lambda _\xi ^2-1}$ and $\frac {\lambda _\xi ^2}{\lambda _\xi ^2-1}$ are bounded on $\mathcal {R}_4$ we obtain

(3.24)\begin{equation} \lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2\lesssim e^{{-}2t}\left[\lambda_\xi^2|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right]. \end{equation}

Therefore,

(3.25)\begin{align} \|(-\mathcal{L})^{1/2} u(t,\cdot)\|^2_{L^2(G)}& =\sum\limits_{[\xi]\in\mathcal{R}_1}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2+\sum\limits_{[\xi]\in\mathcal{R}_2}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2\nonumber\\ & \quad +\sum\limits_{[\xi]\in\mathcal{R}_3}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2+\sum\limits_{[\xi]\in\mathcal{R}_4}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2\nonumber\\ & \lesssim e^{{-}2\delta_1t}\sum\limits_{[\xi]\in\mathcal{R}_2}d_\xi\sum\limits_{k,l=1}^{d_\xi}\left(|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\nonumber\\ & \quad +(1+t)^2e^{{-}2t}\sum\limits_{[\xi]\in\mathcal{R}_3}d_\xi\sum\limits_{k,l=1}^{d_\xi}\left(|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\nonumber\\ & \quad+e^{{-}2t}\sum\limits_{[\xi]\in\mathcal{R}_4}d_\xi\sum\limits_{k,l=1}^{d_\xi}\left(\lambda_\xi^2|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\nonumber\\ & \lesssim (1+t)^2e^{{-}2\delta_1t}\left(\|u_0\|_{H^1_{\mathcal{L}}(G)}^2+\|u_1\|^2_{L^2(G)}\right)\nonumber\\ & \lesssim (1+t)^{{-}1}\left(\|u_0\|_{H^1_{\mathcal{L}}(G)}^2+\|u_1\|^2_{L^2(G)}\right). \end{align}

3.3 Estimate for $\|\partial _tu(t,\,\cdot )\|_{L^2(G)}$

By Plancherel theorem, we have

\[ \|\partial_tu(t,\cdot)\|^2_{L^2(G)}=\sum\limits_{[\xi]\in\widehat{G}}d_\xi\sum\limits_{k,l=1}^{d_\xi}|\partial_t\widehat{u}(t,\xi)_{kl}|^2. \]

We note that

(3.26)\begin{equation} \partial_t\widehat{u}(t,\xi)_{kl}= \begin{cases} \frac{\lambda_\xi^2}{1-\lambda_\xi^2}(e^{{-}t}-e^{-\lambda_\xi^2t})\widehat{u}_0(\xi)_{kl}+\frac{e^{{-}t}-\lambda_\xi^2e^{-\lambda_\xi^2t}}{1-\lambda_\xi^2}\widehat{u}_1(\xi)_{kl}, & \lambda_\xi^2\neq1,\\ - te^{{-}t}\widehat{u}_0(\xi)_{kl}+(1-t)e^{{-}t}\widehat{u}_1(\xi)_{kl}, & \lambda_\xi^2=1. \end{cases} \end{equation}

Estimate on $\mathcal {R}_1.$ From (3.26), we have

\[ |\partial_t\widehat{u}(t,\xi)_{kl}|=e^{{-}t}|\widehat{u}_1(\xi)_{kl}|. \]

Estimate on $\mathcal {R}_2.$ By (3.26) and (3.12) we obtain

\[ |\partial_t\widehat{u}(t,\xi)_{kl}|\lesssim e^{-\delta_1t}\left(|\widehat{u}_0(\xi)_{kl}|+|\widehat{u}_1(\xi)_{kl}|\right). \]

Estimate on $\mathcal {R}_3.$ By (3.26) we get

\[ |\partial_t\widehat{u}(t,\xi)_{kl}|\lesssim e^{{-}t}\left(t|\widehat{u}_0(\xi)_{kl}|+|1-t||\widehat{u}_1(\xi)_{kl}|\right)\lesssim (1+t)e^{{-}t}\left(|\widehat{u}_0(\xi)_{kl}|+|\widehat{u}_1(\xi)_{kl}|\right). \]

Estimate on $\mathcal {R}_4.$ Using (3.26) and (3.17) and the fact that $\frac {1}{\lambda _\xi ^2-1}$ and $\frac {\lambda _\xi ^2}{\lambda _\xi ^2-1}$ are bounded on $\mathcal {R}_4$, we obtain

\[ |\partial_t\widehat{u}(t,\xi)_{kl}|\lesssim e^{{-}t}\left(|\widehat{u}_0(\xi)_{kl}|+|\widehat{u}_1(\xi)_{kl}|\right). \]

Combining these, we get

\[ |\partial_t\widehat{u}(t,\xi)_{kl}|\lesssim (1+t)e^{-\delta_1t}\left(|\widehat{u}_0(\xi)_{kl}|+|\widehat{u}_1(\xi)_{kl}|\right),~\forall~[\xi]\in\widehat{G}. \]

Thus,

(3.27)\begin{align} \|\partial_tu(t,\cdot)\|^2_{L^2(G)}& \lesssim (1+t)^2e^{{-}2\delta_1t}\left(\|u_0\|^2_{L^2(G)}+\|u_1\|^2_{L^2(G)}\right)\nonumber\\ & \lesssim (1+t)^{{-}2}\left(\|u_0\|^2_{L^2(G)}+\|u_1\|^2_{L^2(G)}\right) . \end{align}

3.4 Estimate for $\|\partial _t(-\mathcal {L})^{1/2}u(t,\,\cdot )\|_{L^2(G)}$

By Plancherel theorem we have

\[ \|\partial_t(-\mathcal{L})^{1/2}u(t,\cdot)\|^2_{L^2(G)}=\sum\limits_{[\xi]\in\widehat{G}}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\partial_t\widehat{u}(t,\xi)_{kl}|^2. \]

Estimate on $\mathcal {R}_1.$ We have

\[ \lambda_\xi^2|\partial_t\widehat{u}(t,\xi)_{kl}|^2=0. \]

Estimate on $\mathcal {R}_2.$ By (3.13) and (3.26) we obtain

(3.28)\begin{equation} \lambda_\xi^2|\partial_t\widehat{u}(t,\xi)_{kl}|^2\lesssim e^{{-}2\delta_1t}[|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2]. \end{equation}

Estimate on $\mathcal {R}_3.$ By (3.26) we have

(3.29)\begin{equation} \lambda_\xi^2|\partial_t\widehat{u}(t,\xi)_{kl}|^2\lesssim (1+t)^2e^{{-}2t}\left[|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right]. \end{equation}

Estimate on $\mathcal {R}_4.$ Using (3.26), (3.17) and the fact that $\frac {1}{\lambda _\xi ^2-1}$ and $\frac {\lambda _\xi ^2}{\lambda _\xi ^2-1}$ are bounded on $\mathcal {R}_4$, we obtain

\[ \lambda_\xi^2|\partial_t\widehat{u}(t,\xi)_{kl}|\lesssim e^{{-}t}\lambda_\xi^2\left(|\widehat{u}_0(\xi)_{kl}|+|\widehat{u}_1(\xi)_{kl}|\right). \]

Therefore,

(3.30)\begin{align} \|\partial_t(-\mathcal{L})^{1/2} u(t,\cdot)\|^2_{L^2(G)}& =\sum\limits_{[\xi]\in\mathcal{R}_1}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2+\sum\limits_{[\xi]\in\mathcal{R}_2}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2\nonumber\\ & \quad+\sum\limits_{[\xi]\in\mathcal{R}_3}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2+\sum\limits_{[\xi]\in\mathcal{R}_4}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2|\widehat{u}(t,\xi)_{kl}|^2\nonumber\\ & \lesssim e^{{-}2\delta_1t}\sum\limits_{[\xi]\in\mathcal{R}_2}d_\xi\sum\limits_{k,l=1}^{d_\xi}\left(|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\nonumber\\ & \quad+(1+t)^2e^{{-}2t}\sum\limits_{[\xi]\in\mathcal{R}_3}d_\xi\sum\limits_{k,l=1}^{d_\xi}\left(|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\nonumber\\ & \quad+e^{{-}2t}\sum\limits_{[\xi]\in\mathcal{R}_4}d_\xi\sum\limits_{k,l=1}^{d_\xi}\lambda_\xi^2\left(|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\nonumber\\ & \lesssim (1+t)^2e^{{-}2\delta_1t}\left(\|u_0\|_{H^1_{\mathcal{L}}(G)}^2+\|u_1\|^2_{H^1_{\mathcal{L}}(G)}\right)\nonumber\\ & \lesssim (1+t)^{{-}3}\left(\|u_0\|_{H^1_{\mathcal{L}}(G)}^2+\|u_1\|^2_{H^1_{\mathcal{L}}(G)}\right). \end{align}

Now we are in a position to prove Theorem 1.1.

Proof of Theorem 1.1. Proof of theorem 1.1

The proof follows from the estimates (3.21), (3.25), (3.27) and (3.30) for $\|u(t,\, \cdot )\|_{L^{2}(G)}$, $\left \|(-\mathcal {L})^{1 / 2} u(t,\, \cdot )\right \|_{L^{2}(G)}$, $\left \|\partial _{t} u(t,\, \cdot )\right \|_{L^{2}(G)}$ and $\|\partial _t(-\mathcal {L})^{1/2} u(t,\,\cdot )\|_{L^2(G)}$, respectively.

4. $L^1( G)-L^2( G)$ estimates for the solution to the homogeneous problem

In this section, we show that there is no improvement of any decay rate for the norm $\|u(t,\,\cdot )\|_{L^2(G)}$ 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 a compact group, 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 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

\begin{align*} \sum\limits_{[\xi]\in\widehat G\backslash \mathcal{R}_1}d_\xi\sum\limits_{k,l=1}^{d_{\xi}}|\widehat{u}(t,\xi)_{kl}|^2& \lesssim (1+t)^2e^{{-}2\delta_1t}\sum\limits_{[\xi]\in\widehat G\backslash\mathcal{R}_1}d_\xi\sum\limits_{k,l=1}^{d_{\xi}}\left(|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\\ & \lesssim (1+t)^2e^{{-}2\delta_1t}\left(\|\widehat u_0(\xi)_{kl}\|_{L^2(G)}^2+\|\widehat u_1(\xi)_{kl}\|_{L^2(G)}^2\right) \end{align*}

for some suitable constant $\delta _1.$ Therefore, the contribution to the sum in (3.9) corresponding to $\mathcal {R}_1$ refrains us to get a decay rate for $\|u(t,\,\cdot )\|^2_{L^2(G)}.$ 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 $[\xi ]\in \mathcal {R}_1.$ Here, we must note that the set $\mathcal {R}_1$ is a singleton.

Note that for the multiplier in (3.5), the best estimate that one can obtain on the set $\mathcal {R}_1$ is

\[{\mid}\mathcal{K}_0(t,\xi)\mid,~\mid\mathcal{K}_1(t,\xi)\mid\lesssim 1. \]

Since the set $\mathcal {R}_1$ is singleton, using the definition defined in (2.3), we obtain

\begin{align*} \sum\limits_{[\xi]\in\mathcal{R}_1}d_\xi\sum\limits_{k,l=1}^{d_{\xi}}|\widehat{u}(t,\xi)_{kl}|^2& \lesssim d_\xi\sum\limits_{k,l=1}^{d_{\xi}}\left(|\widehat{u}_0(\xi)_{kl}|^2+|\widehat{u}_1(\xi)_{kl}|^2\right)\\ & \lesssim d_\xi\left(\|\widehat{u}_0(\xi)\|^2_{\operatorname{HS}}+\|\widehat{u}_1(\xi)\|^2_{\operatorname{HS}}\right)\\ & \lesssim \left(\sup\limits_{[\xi]\in\widehat G}d_\xi^{-\frac{1}{2}}\left(\|\widehat{u}_0(\xi)\|_{\operatorname{HS}}+\|\widehat{u}_1(\xi)\|_{\operatorname{HS}}\right)\right)^2\\ & \lesssim\left(\|\widehat{u}_0(\xi)_{kl}\|_{\ell^\infty(\widehat G)}+\|\widehat{u}_1(\xi)\|_{\ell^\infty(\widehat G)}\right)^2\\ & \lesssim \left(\|\widehat u_0(\xi)_{kl}\|_{L^1(G)}^2+\|\widehat u_1(\xi)_{kl}\|_{L^1(G)}^2\right). \end{align*}

This shows that even if we use $L^1(G)$-regularity, we are not able to get any decay rate for the norm $\|u(t,\,\cdot )\|_{L^2(G)}.$

The main reason behind this behaviour is that we cannot neglect the eigenvalue $0$ as the Plancherel measure on a compact Lie group turns out to be a weighted counting measure.

5. 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 $\mathcal {C}^1([0,\, T],\, H_{\mathcal {L}}^{1}(G))$. To present the proof of Theorem 1.4, first, we recall the notion of mild solutions in our setting.

Consider the space

\[ X(T):=\mathcal{C}^1 \left([0,T], H^1_{\mathcal{L}}(G)\right) , \]

equipped with the norm

(5.1)\begin{align} \|u\|_{X(T)}& :=\sup\limits_{t\in[0,T]}(\|u(t,\cdot)\|_{L^2(G)}+\|(-\mathcal{L})^{1/2}u(t,\cdot)\|_{L^2(G)}+\|\partial_tu(t,\cdot)\|_{L^2(G)}\nonumber\\ & \qquad +\|\partial_t(-\mathcal{L})^{1/2}u(t,\cdot)\|_{L^2(G)}). \end{align}

The solution to the nonlinear inhomogeneous problem

(5.2)\begin{equation} \begin{cases} \partial^2_tu-\mathcal{L}u+\partial_tu-\mathcal{L}\partial_tu=F(t, x), & x\in G,t>0,\\ u(0,x)= u_0(x), & x\in G,\\ \partial_tu(0, x)= u_1(x), & x\in G, \end{cases} \end{equation}

can be expressed, by using Duhamel's principle, as

\[ u(t, x):= u_{0}(x)*_{(x)}E_{0}(t, x)+u_{1}(x)*_{(x)}E_{1}(t, x) +\int_{0}^{t} F(s, x)*_{(x)} E_{1}(t-s, x) \,{\rm d}s, \]

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 $(u_{0},\, u_{1})=(\delta _{0},\, 0)$ and $(u_{0},\, u_{1})=$ $(0,\, \delta _{0})$, respectively. For any left-invariant differential operator $L$ on the compact Lie group ${G}$, we applied the property that $L(v*_{(x)} E_{1}(t,\, \cdot ))=v *_{(x)} L(E_{1}(t,\, \cdot ))$ and the invariance by time translations for the viscoelastic wave operator $\partial ^2_t-\mathcal {L}+\partial _t-\mathcal {L}\partial _t$ 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 $N: u \in X(T) \rightarrow N u(t,\, x)$ defined as

(5.3)\begin{align} N u(t, x)& = \varepsilon u_{0}(x) *_{(x)} E_{0}(t, x)+\varepsilon u_{1}(x) *_{(x)} E_{1}(t, x)\nonumber\\ & \quad +\int_{0}^{t}|u(s, x)|^{p} *_{(x)} E_{1}(t-s, x) \,{\rm d}s\end{align}

in the evolution space $\mathcal {C}^1 ([0,\, T],\, H_{\mathcal {L}}^{1}(G))$, 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 $T=T(\varepsilon )$ is based on Banach's fixed point theorem with respect to the norm on $X(T)$ as defined above. More importantly, for $\left \|(u_{0},\, u_{1})\right \|_{H_{\mathcal {L}}^{1}(G) \times H_{\mathcal {L}}^{1}(G) }$ small enough, if we can show the validity of the following two inequalities:

\begin{align*} \|N u\|_{X(T)} & \leq C\left\|\left(u_{0}, u_{1}\right)\right\|_{H_{\mathcal{L}}^{1}(G) \times H_{\mathcal{L}}^{1}(G) }+C\|u\|_{X(T)}^{p},\\ \|N u-N v\|_{X(T)} & \leq C\|u-v\|_{X(T)}\left(\|u\|_{X(T)}^{p-1}+\|v\|_{X(T)}^{p-1}\right), \end{align*}

for any $u,\, v \in X(T)$ and for some suitable constant $C>0$ independent of $T$. Then by Banach's fixed point theorem, we can ensure 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 [Reference Ruzhansky and Yessirkegenov29]. We also refer [Reference Ruzhansky and Yessirkegenov29] for the detailed proof of this inequality for more general connected unimodular Lie groups.

Lemma 5.2 Let $G$ be a (connected) compact Lie group with topological dimension $n\geq 3.$ Assume that $q\ge 2$ such that $q\leq \frac {2n}{n-2}$. Then the following Gagliardo–Nirenberg type inequality holds

(5.4)\begin{equation} \|f\|_{L^q(G)}\lesssim \|f\|^{\theta(n, q)}_{H^{1}_\mathcal{L}(G)}\|f\|^{1-\theta(n, q)}_{L^{2}(G)} \end{equation}

for all $f\in H^{1}_\mathcal {L}(G)$, where $\theta (n,\, q)=n(\frac {1}{2}-\frac {1}{q} )$.

One can also consult [Reference Palmieri21, Reference Ruzhansky and Yessirkegenov29] for several immediate important remarks.

Proof of Theorem 1.4. Proof of theorem 1.1

Expression (5.3) can be written as $N u=u^\sharp +I[u]$, where

\[ u^\sharp(t,x)=\varepsilon u_{0}(x) *_{(x)} E_{0}(t, x)+\varepsilon u_{1}(x) *_{(x)} E_{1}(t, x) \]

and

\[ I[u](t,x):=\int\limits_0^t |u(s,x)|^p*_x E_1(t-s, x)\,{\rm d}s. \]

Now for the part $u^\sharp$, from Theorem 1.1, immediately it follows that

(5.5)\begin{equation} \|u^\sharp\|_{X(T)}\lesssim\varepsilon\|(u_0,u_1)\|_{{H}_{\mathcal{L}}^1 (G)\times{H}_{\mathcal{L}}^1 (G)}. \end{equation}

On the other hand, for the part $I[u]$, using Minkowski's integral inequality, Young's convolution inequality, Gagliardo–Nirenberg type inequality (5.4), Theorem 1.1 and by time translation invariance property of the Cauchy problem (1.1), we get

(5.6)\begin{align} \|\partial_t^j(-\mathcal{L})^{i/2}I[u]\|_{L^2(G)}& =\left(\int_{G} \big |\partial_t^j(-\mathcal{L})^{i /2} \int\limits_0^t |u(s,x)|^p*_x E_1(t-s, x)\,{\rm d}s\big |^2 dg\right)^{\frac{1}{2}}\nonumber\\ & =\left(\int_{G}\big | \int\limits_0^t |u(s,x)|^p*_x \partial_t^j(-\mathcal{L})^{i /2}E_1(t-s, x)\,{\rm d}s\big|^2 dg\right)^{\frac{1}{2}} \nonumber\\ & \lesssim \int\limits_0^t \| |u(s,\cdot )|^p*_x \partial_t^j(-\mathcal{L})^{i /2}E_1(t-s, \cdot)\|_{L^2(G)}\,{\rm d}s\nonumber\\ & \lesssim \int\limits_0^t \| u(s,\cdot)^p\|_{L^2(G)} \|\partial_t^j(-\mathcal{L})^{i /2}E_1(t-s, \cdot)\|_{L^2(G)}\,{\rm d}s\nonumber\\ & \lesssim \int\limits_0^t (1+t-s)^{{-}j-\frac{i}{2}} \|u(s,\cdot)\|^p_{L^{2p}(G)}\,{\rm d}s\nonumber\\ & \lesssim\int\limits_0^t \|u(s,\cdot)\|^{p\theta(n,2p, )}_{H^1_{\mathcal{L}}(G)}\|u(s,\cdot)\|^{p(1-\theta(n,2p ))}_{L^2(G)}\,{\rm d}s \nonumber\\ & \lesssim t \|u\|^p_{X(t)}, \end{align}

for all $(i,\,j) \in \{(0,\, 0),\, (1,\, 0),\, (0,\,1),\, (1,\, 1)\}$. Again for $(i,\,j) \in \{(0,\, 0),\, (1,\, 0),\, (0,\,1), (1,\, 1)\}$, similar calculations as in (5.6) together with Holder's inequality, we get

(5.7)\begin{align} & \|\partial_t^j(-\mathcal{L})^{i /2}\left(I[u]-I[v]\right)\|_{L^2(G)}\nonumber\\ & \quad\lesssim \int\limits_0^t (1+t-s)^{{-}j-\frac{i}{2}} \|u(s,\cdot)|^p-|v(s,\cdot)|^p\|_{L^{2}(G)}\,{\rm d}s\nonumber\\ & \quad\lesssim\int\limits_0^t \|u(s,\cdot)-v(s,\cdot)\|_{L^{2p}(G)}\left(\|u(s,\cdot)\|^{p-1}_{L^{2p}(G)}+\|v(s,\cdot)\|^{p-1}_{L^{2p}(G)}\right)\,{\rm d}s\nonumber\\ & \quad\lesssim t \|u-v\|_{X(t)}\left(\|u\|^{p-1}_{X(t)}-\|v\|^{p-1}_{X(t)}\right). \end{align}

Thus, combining (5.5), (5.6) and (5.7), we have

(5.8)\begin{equation} \|N u\|_{X(t)} \leq D \varepsilon\left\|\left(u_{0}, u_{1}\right)\right\|_{H_{\mathcal{L}}^{1 }(G) \times H_{\mathcal{L}}^{1 }(G)}+DT\|u\|_{X(t)}^{p} \end{equation}

and

(5.9)\begin{equation} \|Nu-Nv\|_{X(T)}\leq DT \|u-v\|_{X(t)}\left(\|u\|^{p-1}_{X(T)}-\|v\|^{p-1}_{X(T)}\right). \end{equation}

Thus, for sufficiently small $T$, the map $N$ turns out to be a contraction in some neighbourhood of $0$ in the Banach space $X(T).$ Therefore, it follows from Banach's fixed point theorem that there exists a uniquely determined fixed point $u$ of the map $N.$ This fixed point $u$ is the mild solution to system (1.1) on $[0,\, t] \subset [0,\, T]$.