Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf A:=(A^1,\dots,A^d)\in (\mathbb R^{n\times n})^d$ be a matrix-vector, where $A^j\in\mathbb R^{n\times n}$, and let $B\in \mathbb R^{n\times n}$ be not required to be symmetric but have one single eigenvalue zero, we consider the Cauchy problem for linear $n\times n$ systems having the form \begin{equation*} \partial_{t}u+\mathbf A\cdot \nabla_{\mathbf x} u+Bu=0,\qquad (\mathbf x,t)\in \mathbb R^d\times \mathbb R_+. \end{equation*} Under appropriate assumptions, we show that the solution $u$ is decomposed into $u=u^{(1)}+u^{(2)}$, where $u^{(1)}$ has the asymptotic profile which is the solution, denoted by $U$, of a parabolic equation and $u^{(1)}-U$ decays at the rate $t^{-\frac d2(\frac 1q-\frac 1p)-\frac 12}$ as $t\to +\infty$ in any $L^p$-norm, and $u^{(2)}$ decays exponentially in $L^2$-norm, provided $u(\cdot,0)\in L^q(\mathbb R^d)\cap L^2(\mathbb R^d)$ for $1\le q\le p\le \infty$. Moreover, $u^{(1)}-U$ decays at the optimal rate $t^{-\frac d2(\frac 1q-\frac 1p)-1}$ as $t\to +\infty$ if the system satisfies a symmetry property. The main proofs are based on asymptotic expansions of the solution $u$ in the frequency space and the Fourier analysis.


Introduction
Consider the Cauchy problem for partially dissipative linear hyperbolic systems where A = (A 1 , . . . , A d ) ∈ (R n×n ) d and B ∈ R n×n , not required to be symmetric. The system (1.1) can be regarded as discrete-velocity models where A determines the velocities of moving particles and B gives the transition rates of the velocities after collisions among the particles in the system. For instance, this type of dissipative linear systems arises in the Goldstein-Kac model [6,8] and the model of neurofilament transport in axons [5]. The large-time behavior of the solution u to (1.1) in terms of decay estimates has been established for years. It follows from [18] that under appropriate assumptions on A and B, if u is the solution to (1.1) with the initial data u 0 ∈ L 1 (R d ) ∩ L 2 (R d ), then one has for some positive constants c and C. Moreover, the estimate (1.2) was generalized in [3], where B can be written in the conservative-dissipative form B = diag (O, D) with D, a positive definite matrix not required to be symmetric. The authors in [3] also showed that where δ ∈ {1/2, 1}, U solving a parabolic system arising in the low-frequency analysis decays diffusively, and V solving a hyperbolic system arising in the high-frequency analysis decays exponentially. The decay estimate (1.5) is remarkable since it holds for general p and q ranging over [1, ∞]. Such kind of decay estimates is very well-known e.g. the L p -L q decay estimate for the linear damped wave equation as in [7,11,14,15].
To obtain (1.5) in one dimension d = 1, one primarily considers the asymptotic expansions of the fundamental solution to the system (1.1) in the Fourier space, divided into the low frequency, the intermediate frequency and the high frequency which naturally produce the time-asymptotic profile. Then, by an interpolation argument once the L ∞ -L 1 estimate and the L p -L p estimate for 1 ≤ p ≤ ∞ are accomplished, one obtains the desired L p -L q estimate for any 1 ≤ q ≤ p ≤ ∞. The same strategy will be applied to the system (1.1) in several dimensions d ≥ 2 in this paper. Nevertheless, difficulties occur as the dimension d increases. For instance, as mentioned in [3], one cannot expect the estimate (1.6) u L 1 ≤ C u 0 L 1 hold in general since for large time, L 0 u, where L 0 is the left eigenvector associated with the eigenvalue 0 of B, behaves as the solution ω to the reduced system where R 0 is the right eigenvector associated with the eigenvalue 0 of B, and thus, it is known in [4] that (1.6) is not true in general. The estimate (1.6) in fact depends strongly on a uniform parabolic operator. Nonetheless, this obstacle can be defeated if d = 1 as in [13] or if 0 is a simple eigenvalue of B since the system (1.7) then becomes scalar and it allows us to obtain (1.6) as we will see in this paper. Another difficulty arises in the highfrequency analysis due to the loss of integrability and the fact that one cannot perform a uniform expansion of the fundamental solution as the dimension d increases. Hence, the corrector V as in (1.5) cannot be obtained trivially. The aim of this paper is to study the L p -L q decay estimate for the conservative part u (1) of the solution u to the system (1.1) in several dimensions d ≥ 2 for general p and q in [1, ∞] in order to generalize (1.2), (1.3) and (1.4), where B is not required to be symmetric but has one single eigenvalue zero. The L p -L q estimate as in (1.5) for the multi-dimensional case d ≥ 2 is still a challenge for the author. A j x j , where A = (A 1 , . . . , A d ) ∈ (R n×n ) d and B ∈ R n×n . We start with the following reasonable assumptions.
Condition A.
[Hyperbolicity] A = A(w) for w ∈ S d−1 is uniformly diagonalizable with real linear eigenvalues i.e. there is an invertible matrix R = R(w) for w ∈ S d−1 satisfying for any matrix norm, such that R −1 AR is a diagonal matrix whose nonzero entries are real linear in w ∈ S d−1 .

Condition R. [Diagonalizing matrix]
There is a matrix R uniformly diagonalizing A such that R −1 BR is a constant matrix. Moreover, the requisite condition for the decay of the solution u to (1.1), strictly related to the Shizuta-Kawashima condition: the eigenvectors of A(x) do not belong to the kernel of B for any x = 0 (see [10,17,19] and therein), is given by Condition D. [Uniform dissipation] There is a constant θ > 0 such that for any eigenvalue λ = λ(ik) of E = E(ik) in (1.8) for k ∈ R d , one has Re λ(ik) ≥ θ|k| 2 1 + |k| 2 , ∀k = 0 ∈ R d .
Remark 1.1 (Relaxing the conditions A and R). The requirement of the linearity of the eigenvalues of the matrix A satisfying the condition A and the existence of the matrix R satisfying the condition R can be omitted by considering the dissipative structures proposed in [3,18]. Nonetheless, the structures in [3,18] require that the system (1.1) is Friedrich symmetrizable while in our case, the matrix A is only uniformly diagonalizable. The advantage of the linearity of the eigenvalues of the matrix A and the existence of the matrix R is that one can construct the high-frequency asymptotic expansion of E in (1.8) after subtracting a suitable Lebesgue measure zero set.
We now construct the asymptotic parabolic-limit U of the solution u to (1.1). Let Γ be an oriented closed curve in the resolvent set of B such that it encloses zero except for the other eigenvalues of B. One sets (1.9) P where c = (c h ) ∈ R d and D = (D hℓ ) ∈ R d×d is positive definite with scalar entries Under the assumptions A, R, B and D, the solution u is decomposed into where and u (2) is the remainder, where P 0 is the eigenprojection associated with the eigenvalue of E in (1.8) converging to 0 as |k| → 0 and χ is a cut-off function with support contained in the ball B(0, ε) ⊂ R d , valued in [0, 1], for small ε > 0. Moreover, for any 1 ≤ q ≤ p ≤ ∞ and t ≥ 1, one has where U is the solution to (1.10) with the initial data U 0 ∈ L q (R d ), and one has (1.14) u (2) L 2 ≤ Ce −ct u 0 L 2 for some constant c > 0 and for all t ≥ 1.

Remark 1.3 (Finite speed of propagation).
In the case where the solution u to the system (1.1) has finite speed of propagation, since the fundamental solution associated with u has compact support contained in the wave cone {(x, t) ∈ R d × R : |x/t| ≤ C} for some constant C > 0, one can decompose u into u = u (1) + u (2) , where and u (2) is the remainder, where χ is a cut-off function with support contained in the ball B(0, ρ) ⊂ R d , valued in [0, 1], for any ρ > 0, and the estimates (1.13) and (1.14) still hold for t ≥ 1. This fact will be proved in the subsequent sections. For instance, it is the case where the system (1.1) is Friedrich symmetrizable. Nonetheless, in one dimension d = 1, the case |x/t| > C can be treated since the Cauchy integral theorem holds for the whole complex plane, and thus, one can use the estimates for the asymptotic expansion of the fundamental solution in the high frequency after changing paths of integrals of holomorphic functions (see [13]).
Moreover, consider the one-dimensional 2 × 2 linear Goldstein-Kac system It can be checked easily that w := u 1 + u 2 satisfies the linear damped wave equation where w 0 and w 1 are appropriate initial data. It then follows from [11] that for any 1 ≤ q ≤ p ≤ ∞ and t ≥ 1, where φ is the solution to the heat equation Without regarding the exponentially decaying term in (1.15), there is a difference of a quantity of 1/2 between the decay rates (1.15) and (1.13). The difference can be explained by a symmetry property that the one-dimensional 2 × 2 linear Goldstein-Kac system possesses. Such kind of symmetry properties is already studied in [13] based on the existence of an invertible matrix S commuting with B and anti-commuting with the matrix A = A of one-dimensional dissipative linear hyperbolic systems. More general, in several dimensions d ≥ 2, the symmetry property is given by We will show that under the conditions B, D and S, the decay rate in the estimate (1.13) increases. We primarily refine the asymptotic profile U .
With the coefficients P Theorem 1.5 (Optimal decay rate). Under the same hypotheses of Theorem 1.2, if the condition S holds in addition, the solution u is also decomposed into u = u (1) + u (2) as in (1.12) such that for any 1 ≤ q ≤ p ≤ ∞ and t ≥ 1, one has where U is the solution to (1.16) with the initial data U 0 .
The paper is organized as follows. Section 2 is devoted to proofs and examples of Theorem 1.2 and Theorem 1.5, where the proofs are based on the estimates obtained in Section 5. In order to prove these estimates in Section 5, we primarily invoke some useful tools of the Fourier analysis and the perturbation analysis in Section 3. With these tools, we construct the asymptotic expansions of the operator E in (1.8) in Section 4 in order to obtain the asymptotic expansions of the fundamental solution to the system (1.1) to be able to prove the estimates in Section 5.
Notations and Definitions. We introduce here the notations and definitions which will be used frequently through out this paper. See [1,2] for more details. Definition 1.6. Let u be a function from R d to a Banach space equipped with norm | · |, we define the Lebesgue spaces L p (R d ) for 1 ≤ p ≤ ∞ consisting of functions u satisfying and satisfying u L ∞ := ess sup Let α ∈ N d be the multi-index α := (α 1 , . . . , α d ) with α j ∈ N. One denotes by where |α| := α 1 + · · · + α d , the partial derivatives of a smooth function f on R d . Then, for smooths functions f and g on R d , we have the Leibniz rule (1 + |x|) k |∂ α u(x)| < +∞.
One denotes by S ′ (R d ) the dual space of S(R d ) and u ∈ S ′ (R d ) is called a tempered distribution. For u ∈ S, the Fourier transformû(k) = F(u(x)) is defined bŷ where x · k is the usual scalar product on R d , and the inverse Fourier transform ofû also denoted by u(x) = F −1 (û(k)) is given by On the other hand, we can define the Fourier transform of tempered distributions u ∈ S ′ (R d ) by the inner product ·, · L 2 on L 2 (R d ), namely Definition 1.8. Let s ∈ R, the Sobolev space H s (R d ) consists of tempered distributions u such thatû ∈ L 2 loc (R d ) and The linear space of all such ρ is denoted by M p (R d ) equipped with norm · Mp .
2. Proofs and Examples of Theorem 1.2 and Theorem 1.5 For k ∈ R d , let E = E(ik) ∈ R n×n be in (1.8). Let c ∈ R d and D ∈ R d×d be in (1.11). Let P (0) 0 ∈ R n×n be in (1.9) and P (1) the kernel associated with the system (1.1), andΦ t (x) :=Φ(x, t) = F −1 (e −c·ikt−k·Dkt ) ∈ R, the kernel associated with the system (1.10). Note that Consider also the kernelΨ t (x) :=Ψ(x, t) = F −1 (e −k·Dkt ) ∈ R associated with the system (1.16). One has 0 · ik)) ∈ R n×n . We are now able to give the proofs of Theorem 1.2 and Theorem 1.5 by using the estimates which will be proved later in Section 5.
Proof of Theorem 1.2. Let u ∈ R n be the solution to (1.1) with the initial data u 0 and U ∈ R n be the solution to (1.10) with the initial data U 0 . One has Moreover, by the relation (2.1), one has where Φ t is given by (2.2). On the other hand, we decompose (2) is the remainder, where P 0 is the eigenprojection associated with the eigenvalue of E in (1.8) converging to 0 as |k| → 0 and χ 1 is a cut-off function with support contained in the ball B(0, ε) ⊂ R d , valued in [0, 1], for small ε > 0. Therefore, by Proposition 5.6, Proposition 5.8 and Proposition 5.10, for 1 ≤ q ≤ p ≤ ∞, there is a constant C > 0 such that we have where χ 2 := 1 − χ 1 − χ 3 and χ 3 is a cut-off function with support contained in {k ∈ R d : |k| > ρ}, valued in [0, 1], for large ρ > 0. Finally, by Proposition 5.6, Proposition 5.7 and Proposition 5.9, one also has for some constants c > 0 and C > 0. The proof is done.
where v j i ∈ R for i, j ∈ {1, 2, 3} and a, b, c > 0, and the initial data is Moreover, the initial data is chosen as Theorem 1.2 then implies that the solution u to the three-dimensional 3 × 3 Goldstein-Kac system can be decomposed into u = u (1) + u (2) such that the difference u (1) − U decays in L p (R d ) at the rate t − 3 with respect to u 0 in L q (R d ) as t → +∞ for any 1 ≤ q ≤ p ≤ ∞, where U is the solution to the above system (1.10). The formulas of c and D in fact coincide the formulas obtained by using the graph theory as in Example 3.3 p. 412 in [12].
We give the proof of Theorem 1.5.
Proof of Theorem 1.5. The proof is similar to the proof of Theorem 1.2 whereΦ t and Φ t are substituted byΨ t and Ψ t respectively once considering U to be the solution to (1.16). We finish the proof.
Example 2.2. Consider the two-dimensional linearized isentropic Euler equations with damping which can be written in the vectorial form Moreover, the matrix R satisfying the condition R and the matrix S satisfying the condition S are given by Then, Theorem 1.5 implies that u = u (1) + u (2) , where u (1) has the asymptotic profile, which is the solution U ∈ R 3 to the Cauchy problem This result is comparable with [7] since ρ ∈ R satisfying (2.5) also satisfies the linear damped wave equation Remark 2.3 (Proof of the case of finite speed of propagation). In the case where Γ t has compact support contained in the wave cone {(x, t) ∈ R d × R : |x/t| ≤ C} for some constant C > 0, also by Proposition 5.6 -Proposition 5.10, u (1) can be refined by where χ 1 is a cut-off function with support contained in the ball B(0, ρ) ⊂ R d , valued in [0, 1], for any ρ > 0. The proof is then similar to the above proofs. Moreover, this property holds for the above two examples since they are in fact symmetric hyperbolic systems.

Useful lemmas
This section is devoted to some useful facts of the Fourier analysis in [1,2] and the perturbation analysis in [9]. They will be used in Section 4 and Section 5.
3.1. Fourier analysis. We introduce here the two well-known inequalities which are the Young inequality and the complex interpolation inequality. On the other hand, we also introduce a powerful Fourier multiplier estimate which is the estimate (3.1) given by Lemma 3.3. The multiplier estimates are very helpful to study the L p -L p estimate for 1 ≤ p ≤ ∞.

Lemma 3.2 (Complex interpolation inequality). Consider a linear operator T which continuously maps
Proof. See the proof of Corollary 1.12 p. 12 in [1].
for some constant C > 0, one has the estimate Proof. See the proof of Lemma 6.1.5 p.135 in [2].

Perturbation analysis.
We consider the perturbation theory for linear operators in [9] that will be used for studying the asymptotic expansions of the fundamental solution to the system (1.1).
Consider the operator T (z) for z ∈ C having the form Exceptional points of the analytic operator T (z) in (3.2) for z ∈ C are defined to be points in where the the eigenvalues of T (z) intersect. Nonetheless, they are of finite number in the plane. In the domain excluding these points, the operator T (z) has p holomorphic distinct eigenvalues with constant algebraic multiplicities. Moreover, the p eigenprojections and the p eigennilpotents associated with them are also holomorphic. In fact, the eigenvalues of T (z) are solutions to the dispersion polynomial det(T (z) − µI) = 0 with holomorphic coefficients. The eigenvalues of T (z) are then branches of one or more than one analytic functions with algebraic singularities of at most order n. As a consequence, the number of eigenvalues of T (z) is a constant except for a number of points which is finite in each compact set of the plane. The exceptional points can be either regular points of the analytic functions or branch-points of some eigenvalues of T (z). In the former case, the eigenprojections and the eigennilpotents associated with the eigenvalues are bounded while in the latter case, they have poles at the exceptional points even if the eigenvalues are continuous there (see [9]). We study the behavior of the eigenvalues of T (z) and the associated eigenprojections and eigennilpotents near an exceptional point. Without loss of generality, we assume that the exceptional point is the point 0 ∈ C. Let λ (0) be an eigenvalue of T (0) with algebraic multiplicity m ≥ 1 and let P (0) and N (0) be the associated eigenprojection and eigennilpotent. One has The eigenvalue λ (0) is in general split into several eigenvalues of T (z) for small z = 0. The set of these eigenvalues is called the λ (0) -group. The total projection of this group, denoted by P (z), is holomorphic at z = 0 and is approximated by where P (j) can be computed in terms of the coefficients T (j) in (3.2) and the coefficients N (0) , P (0) and Q (0) given respectively by where Γ, in the resolvent set of T (0) , is an oriented closed curve enclosing λ (0) except for the other eigenvalues of T (0) . In fact, from [9] (eq. (2.13) p. 76), one has (3.5) Moreover, the subspace ran P (z) := P (z)C n is m-dimensional and invariant under T (z).
The λ (0) -group eigenvalues of T (z) are identical with all the eigenvalues of T (z) in ran P (z). In order to determine the λ (0) -group eigenvalues, therefore, we have only sole an eigenvalue problem in the subspace ran P (z), which is in general smaller than the whole space C n . The eigenvalue problem for T (z) in ran P (z) is equivalent to the eigenvalue problem for Thus, the λ (0) -group eigenvalues of T (z) are exactly those eigenvalues of T r (z) which are different from 0, provided |λ (0) | is large enough to ensure that these eigenvalues do not vanish for the small z under consideration. The last condition does not restrict the generality, for T (0) could be replaced by T (0) + α with a suitable scalar α without changing the nature of the problem (see [9]). We also have the following result in [9]. (1) and λ (0) is a simple eigenvalue of T (0) , the eigenvalue λ(z) of T (z) converging to λ (0) as |z| → 0 and its associated eigenprojection P (z) are holomorphic at z = 0. Moreover, for small z = 0, P (z) is approximated by (3.3) with the coefficients P (j) for j = 0, 1, 2, . . . and λ(z) is approximated by On the other hand, the eigennilpotent associated with λ(z) which is N (z) = T (z) − λ(z)I P (z) vanishes identically.
Proof. For any eigenvalue λ (0) of T (0) with algebraic multiplicity m ≥ 1, one considers the weighted mean of the λ (0) -group defined bŷ where P (z) is the total projection associated with the λ (0) -group.
We study the asymptotic expansions ofλ(z) and P (z) for small z = 0. The expansion of P (z) is given by (3.3) and following [9] (eq. (2.8) p. 76), the coefficient and Γ is a small positively-oriented circle around λ (0) . On the other hand, following [9] (eq. (2.21) p.78 and eq. (2.30) p.79), the weighted meanλ(z) of the λ (0) -group is approximated by where the coefficientλ (j) is given by where the relative coefficients are introduced before.
Moreover, one obtains the following result from Lemma 3.4. Proof. Recall T (z) = T (0) + zT (1) , one can study the eigenvalue problem for T (z) by considering the operator S . It implies that the eigenvalue λ S (z) of T S (z) converging to λ (0) as |z| → 0 and the associated eigenprojection P S (z) are holomorphic at z = 0. Moreover, for small z = 0, the expansion of P S (z) is given by (3.3) with coefficients denoted by P (j) S for j = 0, 1, 2, . . . and λ S (z) is approximated by On the other hand, the eigennilpotent N S (z) associated with λ S (z) vanishes identically. Consider the total projection P S (z) associated with the λ (0) -group of T S (z) in (3.3) with the coefficients P (j) S . We also consider the formula (3.10) of P (j) S and Γ is a small positively-oriented circle around λ (0) .
Finally, since λ S (z) ≡ λ(z) due to (3.14) and the fact that they are single eigenvalues, we deduce from (3.17) that λ (j) = −λ (j) = 0 for all j odd. We finish the proof.
Let σ(T, D) be the spectrum of T considered in the domain D, we finish this section by introducing the reduction method in [9] which can be applied for the semi-simpleeigenvalue case. Lemma 3.6 (Reduction process). Let T (z) be in (3.2) with the coefficients T (i) for i = 0, 1, 2, . . . and let λ (0) be a semi-simple eigenvalue of T (0) . Let P (z) in (3.3) with the coefficients P (i) for i = 0, 1, 2, . . . be the total projection of the λ (0) -group. The following holds for small z = 0 where T j (z) commutes with P j (z) and P j (z) satisfies The expansions of T j (z) and P j (z) are with the associated eigenprojection P j . Let T j (z) :=T (z)P j (z) and using (3.19), (3.23) and the fact that T (z)P (z) = zT (z), one obtains (3.18) and (3.20). We finish the proof.

Preliminaries to Section 5
In this section, we study the asymptotic expansions of E(ik) = B + A(ik) in (1.8) for k ∈ R d , which will be used in Section 5. One has where ζ := |k| ∈ [0, +∞) and w := k/|k| ∈ S d−1 . Moreover, since S d−1 is compact, ζ = 0 is an isolated exceptional point of E(ζ, w) uniformly for w ∈ S d−1 while there is a finite number of exceptional curves of E(ζ, w) for 0 < ζ < +∞. The exceptional point ζ = +∞ is not a uniform exceptional point for w ∈ S d−1 in general (see [3,9]). Nonetheless, we can approximate E(ζ, w) near ζ = +∞ by subtracting a suitable Lebesgue measure zero set taken advantage of the conditions A and R. In this paper, we are only interested in the asymptotic expansions of E(ζ, w) near ζ = 0 and ζ = +∞. As a consequence of Lemma 3.4 and Lemma 3.6, we obtain the followings.
Proposition 4.1 (Low-frequency approximation). If the assumptions B and D hold, then for small k ∈ R d , E(ik) is approximated by where c = (c h ) ∈ R d and D = (D hℓ ) ∈ R d×d is positive definite with scalar entries 0 , and E j (ik) commutes with P j (ik) and one has j > 0 is the j-th nonzero eigenvalue of B with the associated eigenprojection P  Proof. We primarily consider the 0-group of E(ζ, w) in (4.1) for small ζ > 0 and w ∈ S d−1 .
Recall the spectrum σ(B) of B. Since 0 ∈ σ(B) is simple if the assumption B holds, the eigennilpotent N 0 associated with 0 ∈ σ(B) is a null matrix and one obtains from (3.3), (3.5) and (3.6) that the total projection P 0 (ζ, w) of the 0-group is approximated by is the eigenprojection associated with 0 ∈ σ(B) and On the other hand, by (3.8) and (3.9) in Lemma 3.4, the 0-group of E(ζ, w) consists of one single eigenvalue λ 0 (ζ, w) approximated by (4.12) λ 0 (ζ, w) = iζλ We consider the other groups of E(ζ, w) for small ζ > 0. Let λ (0) j ∈ σ(B)\{0} be the j-th nonzero eigenvalue of B for j ∈ {1, . . . , s}, one deduces directly from (3.3) that the approximation of the total projection P j (ζ, w) of the λ (0) j -group is given by where P (0) j is the eigenprojection associated with λ (0) j ∈ σ(B)\{0}. Moreover, due to the discussion above (3.7), the study of the λ (0) j -group of E(ζ, w) is equivalent to the study of the eigenvalues of E j (ζ, w) = E(ζ, w)P j (ζ, w) in ran P j (ζ, w). Furthermore, one has On the other hand, by definition, one also has E j (ζ, w) commutes with P j (ζ, w).
Finally, since s j=0 P j (ζ, w) = I, the identity matrix, one has We thus obtain (4.2) -(4.8) once considering (4.10) -(4.17) in the coordinates k ∈ R d except for the fact that the matrix D in (4.4) is positive definite. We now prove that D is positive definite. Consider the eigenvalue λ 0 (ik) in (4.3) of E(ik) for k ∈ R d with the coefficients c ∈ R d and D ∈ R d×d given by (4.4). If the assumption D holds, then since c · k ∈ R, there is a constant θ > 0 such that for small k = 0 ∈ R d , one has As |k| → 0, one has Re (w · Dw) ≥ θ > 0 for all w ∈ S d−1 . Therefore, for any x = 0 ∈ R d , one has Re (x T Dx) = |x| 2 Re (w · Dw) > 0, where x T is the transpose of the vector x.
Finally, since the condition S implies that for w ∈ R d , there is an invertible matrix S = S(w) satisfying S(w)A(w) = −A(w)S(w) and S(w)B = BS(w), we obtain (4.9) directly from Corollary 3.5. The proof is done.
Note that under the assumption A, there is an invertible matrix R = R(w) for w ∈ S d−1 such that R −1 AR is a diagonal matrix with nonzero entries are real linear eigenvalues of A = A(w) for w ∈ S d−1 . Hence, one can consider the ℓ-th diagonal element of R −1 AR as the linear function where the coefficients ν  There is a Lebesgue measure zero set contained in S d−1 such that except for this set, the number of distinct eigenvalues of A(w) for w ∈ S d−1 is r and the algebraic multiplicities associated with them are r j for j ∈ {1, . . . , r}.
Proof. Recall the partition S = {S 1 , . . . , S r } with cardinality r. Assume that there are i, j ∈ {1, . . . , r} such that i = j and ν [i] (w 0 ) = ν [j] (w 0 ) for some w 0 ∈ S d−1 . We prove that w 0 belongs to a Lebesgue measure zero set in R d−1 . In fact, w 0 belongs to the intersection of the affine hyperplane for any i = j by definiton, and the unit sphere S d−1 . Moreover, the dimension of the intersection is at most d − 2 and it is therefore a Lebesgue measure zero set in R d−1 . Thus, (w) for any i = j and for w ∈ S d−1 subtracted a Lebesgue measure zero set. Finally, since the repeated eigenvalues of A(w) are ν ℓ (w) determined by the coefficient vectors ν ℓ for ℓ ∈ {1, . . . , n}, it follows immediately that the number of distinct eigenvalues of A(w) for w ∈ S d−1 is r and the algebraic multiplicities associated with them are r j , the cardinality of S j , for j ∈ {1, . . . , r} excluding a Lebesgue measure zero set. We finish the proof.
One sets, for j ∈ {1, . . . , r}, the projection Let R = R(w) for w ∈ S d−1 be the matrix satisfying the conditions A and R. One has Proposition 4.3 (High-frequency approximation). If the assumptions A, R and D hold, then for large k ∈ R d , E(ik) is almost everywhere approximated by where the constant s j ≤ r j which is also constant as well as r, Υ jm (ik) commutes with Π jm (ik) and one has Proof. Based on Lemma 4.2, if the condition A holds, the spectrum of R −1 AR(w) for w ∈ S d−1 is the set {α 1 (w), . . . , α r (w)} where α j (w) = ν [j] (w) given by (4.18) for j ∈ {1, . . . , r} with finite constant r, the cardinality of S, and [j] is the representation of the elements of S j , for almost everywhere. Thus, from here in this proof, we consider always for almost everywhere and we drop w in the coefficients written in below if they are in fact constant for almost everywhere.

Decay estimates (Core of the paper)
In this section, we prove the estimates used in the proofs of Theorem 1.2 and Theorem 1.5. We primarily give a priori estimates for the principal parabolic part of the fundamental solution Γ t to the system (1.1). Then, we estimate Γ t by diving the frequency space into: the low frequency, the intermediate frequency and the high frequency. The main proofs are related to the interpolation between the L ∞ -L 1 estimate and the L p -L p estimate for 1 ≤ p ≤ ∞. Moreover, the L ∞ -L 1 estimate is obtained directly while the L p -L p estimate is obtained based on the Carlson-Beurling inequality (3.1) in Lemma 3.3. Moreover, since the Carlson-Beurling inequality (3.1) depends on the analysis of partial derivatives, one considers the followings.
Lemma 5.1 (Partial derivative). Let α ∈ N d with |α| ≥ 0, for any scalar smooth functions q = q(x, t) on R d × R + , we have where {I j : j = 1, . . . , r} is any possible partition of the index-set I α determined by α.
Proof. We prove by induction. Let α ∈ N d , if |α| = 0, then since I α = ∅, there is no partition of I α to be considered, and thus, ∂ 0 e q(x,t) = e q(x,t) . If |α| = 1, by the definition of ∂ α , we have where {I j : j = 1, . . . , r} is any possible partition of the index-set I α determined by α. We then consider all of possible partitions of I β . The first possibilities are the partitions {{I j : j = 1, . . . , r}, {i}} since I β has α i + 1 indices i. The last choices are that for each partition {I j : j = 1, . . . , r} of I α , we generate the partition {I ′ j : j = 1, . . . , r} of I β by putting i into I ℓ and let I ′ j = I j for all j = ℓ for ℓ ∈ {1, . . . , r}. Thus, since r varies, there is no other possible partition of I β to take part in. Therefore, we obtain from (5.3) that where the sum is made on all possible partitions {I ′ j : j = 1, . . . , r ′ } of I β determined by β. We thus proved (5.1).
Remark 5.2. Lemma 5.1 is applied only to the case where q = q(x, t) is scalar for (x, t) ∈ R d × R + , the matrix case is a challenge as the loss of commutativity of q and its partial derivatives.

Proposition 5.3 (Parabolic estimate).
If D ∈ R d×d is positive definite, for 1 ≤ q ≤ p ≤ ∞, there is a constant C > 0 such that for any U 0 ∈ L q (R d ), one has Proof. We primarily study the L ∞ -L 1 estimate. By the Young inequality and since D is positive definite, there are constants c > 0 and C > 0 such that for t > 0, we have We study the L p -L p estimate for 1 ≤ p ≤ ∞. Let α ∈ N d with |α| ≥ 0, by the formula (5.1) in Lemma 5.1, we have where {I j : j = 1, . . . , r} is any possible partition of the index-set I α determined by α.
On the other hand, by the definition of ∂ I j , there is a constant C > 0 such that where |I j | is the number of elements of I j with possible repeated indices for j ∈ {1, . . . , r}. We are then not interested in the cases where |I j | > 2 for some j ∈ {1, . . . , r}. Thus, we can consider only the partitions {I j : j = 1, . . . , r} of I α where 1 ≤ |I j | ≤ 2. Hence, we have where m ≥ 0 is the cardinality of the set {j ∈ {1, . . . , r} : |I j | = 1} and ℓ ≥ 0 is the cardinality of the set {j ∈ {1, . . . , r} : |I j | = 2}. Moreover, by definition, one has m + 2ℓ = |I α | = |α|, where |I α | = r j=1 |I j |, the number of elements of the index-set I α determined by α with possible repeated indices.
Thus, since D is positive definite, there are constants c > 0 and C > 0 such that Hence, since m + 2ℓ = |α|, we have By the Carlson-Beurling inequality (3.1) in Lemma 3.3, one has for any integer s > d/2, 1 ≤ p ≤ ∞ and t > 0. Therefore, by the definition of the M p -norm, we have the L p -L p estimate Finally, by applying the interpolation inequality and the estimates (5.5) and (5.8), we obtain (5.4). The proof is done.
Remark 5.4. Note that the derivative estimate (5.6) is true for all k ∈ R d .
Let χ j for j = 1, 2, 3 be cut-off functions on R d , valued in [0, 1], such that supp χ 1 ⊂ {k ∈ R d : |k| ≤ ε} and supp χ 3 ⊂ {k ∈ R d : |k| ≥ ρ} for small ε > 0 and large ρ > 0, and We are now going to study the large-time behavior of the fundamental solution Γ t to the system (1.1) in each partition of the frequency space.
For k ∈ R d , we recall the Fourier transform of the fundamental solution Γ t to the system (1.1), namely where E is given in (1.8). We also recall where c, D are given by (1.11), P (0) 0 is given by (1.9) and P (1) 0 is given by (1.17). 5.1. Low-frequency analysis. The aim of this subsection is to study the L p -L q estimate for the low-frequency part of Γ t for any 1 ≤ q ≤ p ≤ ∞. One thus considersΓ t χ 1 .
Lemma 5.5 (Derivative estimate). Let p(x) be a scalar polynomial on R d such that the lowest order of p(x) is h ≥ 1 and let α ∈ N d with |α| ≥ 0. There is a constant C > 0 such that for small x ∈ R d and t > 0, we have Proof. Let α ∈ N d with |α| ≥ 0 and p(x) be a polynomial on R d such that the lowest order of p(x) is h ≥ 1. For any partition {I j : j = 1, . . . , r} of I α determined by α, by the definition of ∂ I j , there is a constant C(j) > 0 such that for any k ∈ {0, . . . , h − 1} and small x ∈ R d , where |I j | is the number of elements of the index-set I j with possible repeated indices. Note that r j=1 |I j | = |I α | = |α| by definition. It implies that there is a constant C(r) = max j C(j) > 0 such that for small x ∈ R d and t > 0, we have where m k ≥ 0 is the cardinality of {j ∈ {1, . . . , r} : |I j | = h − k} for k ∈ {0, . . . , h − 1} and ℓ ≥ 0 is the cardinality of J := {j ∈ {1, . . . , r} : |I j | > h}. Moreover, we have We thus obtain (5.11) and (5.12) with C = max r C(r) > 0 from (5.1), (5.13) and (5.14). The proof is done.
Let P 0 be given by (4.5), we have the following.
Proposition 5.6 (Low-frequency estimate). If the assumptions B and D hold, then for 1 ≤ q ≤ p ≤ ∞, there is a constant C > 0 such that for t > 0, we have If the condition S holds in addition, then we have On the other hand, for 1 ≤ q ≤ 2 ≤ p ≤ ∞, there are constants c > 0 and C > 0 such that for t > 0, we have Proof. Under assumptions B and D, from (4.2) -(4.8) in Proposition 4.1, for small k ∈ R d , one has where c ∈ R d and D ∈ R d×d is positive definite given by (4.4), P (0) 0 is the eigenprojection associated with 0 ∈ σ(B), and λ We now prove Proposition 5.6 by primarily establishing the L ∞ -L 1 estimate. Then, by constructing the L p -L p estimate for 1 ≤ p ≤ ∞, we apply the interpolation inequality.
By changing the coordinates (x, t) → (x − ct, t), one can always assume that c = 0 without loss of generality. We study the L ∞ -L 1 estimate. Consider Then, there are constants c > 0 and C > 0 such that By the Young inequality, we have j > 0, with the associated eigenprojection P  [16], for any ε > 0, there is an induced norm such that |N (0) j | ≤ ε and due to the fact that every norms in finite-dimensional space are equivalent, one deduces that since |k| small and Re λ Hence, we obtain Step 2. L p -L p estimates.
Under the symmetry property S.
Moreover, if in addition the condition S holds, then for small k, from (4.2) -(4.9) in Proposition 4.1, one hasΓ t χ 1 =Γ where c ∈ R d and D ∈ R d×d is positive definite given by (4.4), P (0) 0 is the eigenprojection associated with 0 ∈ σ(B), P 0 ∈ (R n×n ) d is in (4.6), and λ 0 · ik)χ 1 (k), J := e −k·Dkt+O(|k| 4 )t O(|k| 2 )χ 1 (k). The estimates are then similar to the previous case. We omit the details. We thus obtain for 1 ≤ q ≤ p ≤ ∞ and t > 0 that The proof is done since the others are also similar to before.
Proof. Recall E(ik) = B + A(ik) in (1.8) for k ∈ R d . We considerΓ t whereΓ t (k) = e −E(ik)t . Since the condition D holds, Re λ(ik) > 0 for any eigenvalue λ(ik) of E(ik) and k = 0 ∈ R d . Thus, the operator e −E(ik) has the spectral radius rad(e −E(ik) ) < 1 for almost everywhere. It follows from the Householder theorem in [16] that there is an induced norm such that 0 < ϕ := ess sup R d |e −E(ik) | < 1.
Moreover, if Γ t has compact support contained in {(x, t) ∈ R d × R : |x/t| ≤ C} for some constant C > 0. From (5.48) and the Young inequality, there are c ′ , c > 0 and C > 0 such that for 1 ≤ p ≤ ∞, one has We finish the proof of (5.47) by applying the interpolation inequality and by using the L ∞ -L 1 estimate (5.49), the L 2 -L 2 estimate (5.50) and the L p -L p estimates (5.51).
Proposition 5.8. If the conditions B and D hold, then for 1 ≤ q ≤ p ≤ ∞, there are constants c > 0 and C > 0 such that for t ≥ 1, one has Similarly, we have Proof. We estimate F −1 (Φ t (k)χ 2 (k)) * u 0 and the other is similar. Recall that Φ t (k) = e −c·ikt−k·Dkt P (0) 0 , where c ∈ R d and D ∈ R d×d is positive definite given by (4.4) under the assumptions B and D.

5.3.
High-frequency analysis. The aim of this part is to give an L 2 -L 2 estimate of the high-oscillation part of Γ t , which isΓ t χ 3 in the Fourier space, whereΓ t is given by (5.9). Proposition 5.9 (High-frequency estimate). If the conditions A, R and D hold, then there are constants c > 0 and C > 0 such that one has the estimate F −1 (Γ t (k)χ 3 (k)) * u 0 L 2 ≤ Ce −ct u 0 L 2 , ∀t > 0.
Proposition 5.10. If the conditions B and D hold, then for 1 ≤ q ≤ p ≤ ∞, there are constants c > 0 and C > 0 such that for t ≥ 1, one has Similarly, we have Proof. Similarly to the proof of Proposition 5.8 where χ 2 is substituted by χ 3 .