Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^n$ with power type nonlinearity $|u|^p$ and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space $\dot H^{-\gamma}_{\mathcal{L}}(\mathbb{H}^n), \gamma>0$, on $\mathbb{H}^n$. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent $p_{\text{crit}}(Q, \gamma)=1+\frac{4}{Q+2\gamma},$ for some $\gamma\in (0, \frac{Q}{2})$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. More precisely, we establish a global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{crit}}(Q, \gamma)$ in the energy evolution space; a finite time blow-up of weak solutions for $1<p<p_{\text{crit}}(Q, \gamma)$ under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.


Introduction and discussion on main results
1.1.Description of problem and background.In this study, our main aim is to determine a new critical exponent for the Cauchy problem for a semilinear damped wave equation with the power type nonlinearities as follows: $ ' & ' % u tt ´Lu `ut " |u| p , g P H n , t ą 0, up0, gq " εu 0 pgq, g P H n , u t p0, gq " εu 1 pgq, where L is the sub-Laplacian on the Heisenberg group H n , 1 ă p ă 8, and the initial data pu 0 , u 1 q with its size parameter ε ą 0 belongs to subelliptic (or Folland-Stein) homogeneous Sobolev spaces of negative order pu 0 , u 1 q P 9 H ´γ L pH n q ˆ9 H ´γ L pH n q with γ ą 0. In other words, we study the global-in-time existence of small data solutions and the blow-up in finite time of solutions to the Cauchy problem (1.1).
To discuss the classical Euclidean scenario, let us consider the semilinear damped wave equation on R n with the power type nonlinearities as follows: $ ' & ' % u tt ´∆u `ut " |u| p , x P R n , t ą 0, up0, xq " u 0 pxq, x P R n , u t p0, xq " u 1 pxq, where 1 ă p ă 8 and ∆ is the Laplacian on R n .When the initial data belongs additionally to L 1 -space, the global existence or a blow-up result to (1.2), depending on the critical exponent has been studied in [21,25,38,40] and references therein.The critical exponent signifies the threshold condition on the exponent p for the global-in-time Sobolev solutions and the blow-up of local-in-time weak solutions with small data.The critical exponent for solutions to (1.2) is the so-called Fujita exponent given by p Fuji pnq :" 1 `2 n (see [21,25,38,40]).More precisely, ‚ When n " 1, 2, Matsumura in his seminal paper [25] proved the global-in-time existence of small data solutions for p ą p Fuji pnq.‚ For any n ě 1, a global existence for p ą p Fuji pnq (by assuming compactly supported initial data) and blow-up of the local-in-time solutions in the subcritical case 1 ă p ă p Fuji pnq was explored by Todorova and Yordanov [38].‚ For p " p Fuji pnq, the blow-up result was obtained by Zhang [40].Later, Ikehata and Tanizawa [21] removed the restriction of compactly supported data for the supercritical case p ą p Fuji pnq.Moreover, the sharp lifespan estimates for the Cauchy problem (1.2) with additional L 1 -data assumption are given by where C is a positive constant independent of ε.We cite [17,18,19,23,24] for a detailed study on the sharp lifespan estimates for the Cauchy problem (1.2).We also refer to the excellent book [7] for the global-in-time small data solutions for the semilinear damped wave equations in the Euclidean framework.
In recent years, considerable attention has been devoted by numerous researchers to finding new critical exponents for the classical semilinear damped wave equations on R n in different contexts.For instance, considering the Cauchy problem (1.2) with initial data additionally belonging to L m -spaces with m P p1, 2q, the critical exponent is changed and the new modified Fujita exponent becomes p Fuji pnq :" p Fuji ´n m ¯" 1 `2m n .
Unlike the L 1 -data case, here with additional L m -regular data, the global-in-time solution exists uniquely at the critical point p Fuji `n m ˘" 1 `2m n .This is the main difference between L 1 and L m regular data.We refer to [16,20,27] and references therein for a detailed study related to the critical exponent p Fuji `n m ˘for the solutions to the semilinear wave equations with the L m -regular data.
The study of the semilinear damped wave equation has also been extended in the non-Euclidean framework.Several papers have studied nonlinear PDEs in non-Euclidean settings in the last decades.For example, the semilinear wave equation with or without damping has been investigated for the Heisenberg group in [12,26,34].In the case of graded groups, we refer to the recent works [13,33,34,36].We refer to [10,28,30,31,3,6] concerning the wave equation on compact Lie groups and [2,39,22,1] on a Riemannian symmetric space of non-compact type.In particular, for the semilinear damped wave equation on the Heisenberg group H n , namely $ ' & ' % u tt ´Lu `ut " |u| p , g P H n , t ą 0, up0, gq " u 0 pgq, g P H n , u t p0, gq " u 1 pgq, where p ą 1, L is the sub-Laplacian on the Heisenberg group H n , it was shown in [12] that the critical exponent is given by where Q :" 2n `2 represents the homogeneous dimension of H n .It is interesting to note that (1.4) is also the Fujita exponent for the semilinear heat equation # u t ´Lu " |u| p , g P H n , t ą 0, up0, gq " u 0 pgq, g P H n , on H n , where p ą 1.This topic has been discussed in [12,13,35].However, in the case of a compact Lie group G, it has been shown in [28] that p Fuji p0q :" 8 is the critical exponent for the solution to the semilinear damped wave equation on G.We cite [34] for a global existence result with small data in the more general setting of graded Lie groups for the semilinear damped wave equation involving a Rockland operator with an additional mass term.
Recently, Chen and Reissig [5] considered the following semilinear damped wave equation on R n , where p ą 1 and the initial data additionally belonging to homogeneous Sobolev spaces of negative order 9 H ´γ pR n q with γ ą 0. They obtained a new critical exponent p " p crit pn, γq :" 1 `4 n`2γ for some γ P p0, n 2 q in this framework.More specifically, the authors proved that: ‚ For p ą p crit pn, γq, the problem (1.5) admits a global-in-time Sobolev solution for sufficiently small data of lower regularity.‚ For 1 ă p ă p crit pn, γq, the solutions to (1.5) blow-up in a finite time.In other words, there exists T ą 0 such that the solution to (1.5) satisfies }u p¨, t m q} 8 Ñ 8 as t m Ñ T .Further, the authors also investigated sharp lifespan estimates for weak solutions to (1.5), in which the sharpness of lifespan estimates is given by where C is a positive constant independent of ε and p 1 is the Lebesgue exponent conjugate of p such that 1 p `1 p 1 " 1.We remark that some other technical assumptions on p and γ are also required to find the sharp lifespan estimate in the sub-critical case.
To the best of our knowledge, in the non-Euclidean framework, the subelliptic damped wave equation on the Heisenberg group with initial data localized in Sobolev spaces of negative order has not been considered in the literature so far, even for the linear Cauchy problem.Therefore, an interesting and viable problem is to study several qualitative properties such as global-in-time well-posedness, blow-up criterion, decay rate, asymptotic profiles to solutions for the subelliptic damped wave equations on the Heisenberg group H n with initial data additionally belonging to subelliptic Sobolev space 9 H ´γ L pH n q with γ ą 0. The main aim of this paper is to investigate and determine a critical exponent for the Cauchy problem for semilinear damped wave equation (1.1) with the initial data additionally belonging to subelliptic homogeneous Sobolev spaces 9 H ´γ L pH n q of negative order ´γ.More specifically, ‚ Under additional assumptions for the initial data in 9 H ´γ L pH n q, we obtain a new critical exponent to (1.1) given by with γ P ˆ0, ? ˙.
‚ We derive sharp lifespan estimates for weak solutions to the semilinear Cauchy problem (1.1).Define the lifespan T ε as the maximal existence time for solution of (1.1), i.e., T ε :" !T ą 0 : there exists a unique local-in-time solution to the Cauchy problem (1.1) on r0, T q with a fixed parameter ε ą 0 If the initial data is from 9 H ´γ L pH n q with γ P p0, γq and the exponent p satisfies 1 `2γ Q ď p ď Q Q´2 , then the new sharp lifespan estimates are given by where the positive constant C is independent of ε.
1.2.Main results: Comprehensive review.Throughout the paper, we denote L q pH n q, 1 ď q ă 8, the space of q-integrable functions on H n with respect to the Haar measure dg on H n , which is nothing but the Lebesgue measure of R 2n`1 , and the space of all essentially bounded functions on H n for q " 8.The fractional subelliptic (Folland-Stein) Sobolev space H s L pH n q, s P R, associated to the sub-Laplacian L on H n , is defined as Similarly, we denote by 9 H s L pH n q, the homogeneous Sobolev defined as the space of all f P D 1 pH n q such that p´Lq s{2 f P L 2 pH n q.At times, we also write H s L and 9 H s L for H s L pH n q and 9 H s L pH n q , respectively from here on.For pu 0 , u 1 q P A s,´γ Utilizing techniques derived from the noncommutative Fourier analysis on the Heisenberg group H n , we present our initial finding regarding time decay estimates in the 9 H s L -norm of solutions to the homogeneous version of the Cauchy problem (1.1).This result is detailed below.
Theorem 1.1.Let H n be the Heisenberg group with the homogeneous dimension Q.Let pu 0 , u 1 q P ´Hs L X 9 H ´γ L ¯such that s ě 0 and γ P R such that s `γ ě 0. Consider the following linear Cauchy problem $ ' & ' % u tt ´Lu `ut " 0, g P H n , t ą 0, up0, gq " u 0 pgq, g P H n , u t p0, gq " u 1 pgq, g P H n . (1.8) Then, the solution u satisfies the following 9 H s L -decay estimate }upt, ¨q} 9 for any t ě 0.
The next result is about the global-in-time well-posedness of the Cauchy problem (1.1) in the energy evolution space C pr0, T s, H s L q , s P p0, 1s.In this case, a version of a Gagliardo-Nirenberg type inequality on H n (see Section 2) will play a crucial role in estimating the power type nonlinearity in L 2 pH n q.Let us first make clear what do we mean by a solution of (1.1).For the global existence result we will work with the mild solutions of (1.1).
We say that a function u is a mild solution to (1.1) on r0, T s if u is a fixed point for the integral operator N : u P X s pT q Þ Ñ Nupt, gq, given by Nupt, gq :" u lin pt, gq `unon pt, gq, (1.10) in the energy evolution space X s pT q ." C pr0, T s, H s L pH n qq , s P p0, 1s, equipped with the norm }u} XspT q :" sup Here ˚pgq is the group convolution product on H n with respect to the g variable and E 0 pt, gq and E 1 pt, gq represent the fundamental solutions to the homogeneous problem (1.8) with initial data pu 0 , u 1 q " pδ 0 , 0q and pu 0 , u 1 q " p0, δ 0 q, respectively.
We prove the global-in-time existence and uniqueness of small data Sobolev solutions to the semilinear damped wave equation (1.1) of low regularity by finding a unique fixed point to the operator N, i.e., Nu P X s pT q for all T ą 0. It means that there exists a unique global solution u to the equation Nu " u P X s pT q which also gives the solution to (1.1).In order to prove that N has a uniquely determined fixed point, we use Banach's fixed point argument on the space X s pT q defined above.
Keeping p crit pQ, γq " 1 `4 Q`2γ in mind, we state the following global-in-time existence result.
Theorem 1.2.Let s P p0, 1s and γ P `0, Q 2 ˘.Assume that the exponent p satisfies where γ denotes the positive root of the quadratic equation 2γ 2 `Qγ´2Q " 0.Then, there exists a small positive constant ε 0 such that for any pu 0 , u 1 q P A s,´γ L satisfying }pu 0 , u 1 q} A s,´γ L " ε P p0, ε 0 s, the Cauchy problem for the semilinear damped wave equation (1.1) has a uniquely determined Sobolev solution u P C pr0, 8q, H s L q .Therefore, the lifespan of the solution is given by T ε " 8.Moreover, the solution satisfies the following two estimates listed below: }upt, ¨q} L 2 À p1 `tq ´γ 2 }pu 0 , u 1 q} A s,´γ L , and }upt, ¨q} 9 It is crucial to emphasize that there is no loss in the decay of the solution when transitioning from the linear to the nonlinear problem.In other words, the decay rates outlined in the preceding theorem align precisely with the decay rates established for the corresponding linearized damped wave equation, as presented in Theorem 1.1.However, the restriction 1 ă p ď Q Q´2s in the above theorem is necessary in order to apply a Gagliardo-Nirenberg type inequality on H n .Remark 1.3.Some examples for the admissible range of exponents p for the global-in-time existence result in the low dimensions Heisenberg group H n with n " 1 and 2, that is, Q " 4 and 6, respectively, are as follows: ‚ When Q " 4, we take s P p0, 1s and γ P p0, 2q and the exponent satisfies ‚ When Q " 6, we take s P p0, 1s and γ P p0, 3q and the exponent satisfies Note that the positive root γ of 2γ 2 `Qγ ´2Q " 0 is always strictly less than 2 for any homogeneous dimension Q.
Based on the aforementioned illustrations, it is clear that the global existence result, as stated in Theorem 1.2, is only pertinent to specific lower homogeneous dimensions.This constraint is due to the technical stipulation that 1 ă p ď Q Q´2s .For global existence results in higher homogeneous dimensions, one can study Sobolev solutions by considering initial data from subelliptic Sobolev spaces with an appropriate degree of higher/large regularity.
Our next result is about a blow-up (in-time) result in the subcritical case to the Cauchy problem (1.1) for certain values of p regardless of the size of the initial data.Before the blowup result, we first introduce a suitable notion of a weak solution to the Cauchy problem (1.1).
Definitions 1.4.For any T ą 0, a weak solution of the Cauchy problem (1.1) in r0, T q ˆHn is a function u P L p loc pr0, T q ˆHn q that satisfies the following integral relation: t φpt, gq ´Lφpt, gq ´Bt φpt, gq ˘dg dt ´ε ż |upt, gq| p φpt, gq dg dt, (1.12) for any φ P C 8 0 pr0, T q ˆHn q.If T " 8, we call u to be a global-in-time weak solution to (1.1), otherwise u is said to be a local in time weak solution to (1.1).
Let |¨| be any homogeneous norm on the Heisenberg group H n , while we denote p1 `|g| 2 q 1 2 by the Japanese bracket xgy for g P H n .Then, under some additional assumptions on the initial data, we have the following blow-up result.
Theorem 1.5.Let γ P `0, Q 2 ˘and let the exponent p satisfy 1 ă p ă p crit pQ, γq.We also assume that the non-negative initial data pu 0 , u where C 0 ą 0 is a fixed constant.Then, there is no global (in time) weak solution to the Cauchy problem (1.1).Moreover, the lifespan T w,ε of local (in time) weak solutions to the Cauchy problem (1.1) satisfies the following upper bound estimate where C is a positive constant independent of ε.
From Theorems 1.2 and Theorem 1.5, we can conclude that the critical exponent for the semilinear damped wave equation (1.1) is p crit pQ, γq :" 1`4 Q`2γ (see (1.6)) when initial data are additionally taken from the negative order Sobolev space 9 H ´γ L , γ ą 0. It gives us a new way to look at the critical exponent for the semilinear damped wave equation in Sobolev space with a negative order.Indeed, we can interpret the exponent p crit pQ, γq as a modification of the exponent p Fujita `Q 2 ˘" 1 `4 Q (which would correspond to the critical exponent with only L 2 regularity for the Cauchy data) by working with even weaker regularity (H ´γ L regularity), the critical exponent becomes even smaller p crit pQ, γq " 1 `4 Q`2γ .Since, we are working with less regularity for the data, the decay rates in the estimates for the solutions of the homogeneous problem improve and the Banach fixed point theorem holds for a larger range of p.
The behavior of the Cauchy problem at the critical case p " p crit pQ, γq remains uncertain, as it is unclear whether a global (in time) small data Sobolev solutions exist or the weak solutions will blow-up.
In the next remark, particularly, we consider s " 1 for Theorem 1.2 and 1.5 to give a rough idea about the critical exponent.
Remark 1.6.Due to the applications of Gagliardo-Nirenberg inequality, we have the technical restriction on the exponent 1 ă p ď Q Q´2 .Consequently, the ranges of exponent p for globalin-time existence and blow-up are as follows.
‚ When Q " 4, 6 -blow-up of weak solutions if 1 ă p ă p crit pQ, γq, -and global-in-time existence of Sobolev solutions if ‚ When Q ě 8, from Theorem 1.2, the set of admissible exponents p is completely empty.
For a detailed analysis of the critical exponent which depends on the parameter γ and Q, we can describe it by the pγ, pq plane in Figure 1.In Figure 1, with an increase of the homogeneous dimension Q of H n , the curve p " p crit pQ, γq and the segment p " 1 `2γ Q will move following the direction of Õ and Ô lines arrows, respectively.
The formulations of Theorem 1.2 and Theorem 1.5 for s " 1 also give the critical index for regularity of initial data that belongs additionally to 9 Regarding the initial data that also additionally belongs to 9 H ´γ L with γ ą 0, ‚ then local (in-time) weak solutions in general blow up in finite time for 0 ă γ ă γ crit pp, Qq; ‚ then the global (in-time) small data Sobolev solutions exists uniquely for γ crit pp, Qq ă γ ă mintγ, Q 2 u.From Theorem 1.5, for 1 ă p ă p crit pQ, γq, we know that the non-trivial local (in time) weak solution blow up in finite time and we have the following upper bound estimates for the lifespan Now, it is important to investigate the lower bound estimates for the lifespan.
To thoroughly examine the lifespan, it is essential to consider the precise definition of mild solutions to the Cauchy problem (1.1) on r0, T q with T ą 0 for u P Cpr0, T q, H 1 L q.Let T m,ε be the lifespan of a mild solution u.Then, we have the following result regarding the lower bound for the lifespan.
Theorem 1.7.Let γ P p0, γq and let the exponent p satisfy 1 ă p ă p crit pQ, γq such that We also assume that pu 0 , u 1 q P A L 1 .Then, there exists a constant ε 0 such that for every ε P p0, ε 0 s, the lifespan T m,ε of mild solutions u to the Cauchy problem (1.1) satisfies the following lower bound condition: where D is a positive constant independent of ε, but may depends on p, Q, γ as well as }pu 0 , u Since u in Theorem 1.2 is a mild solution to (1.1), this mild solution is also a weak solution to (1.1) (according to a density argument) with timespan T m,ε ď T ε .Once more drawing on Theorems 1.5 and 1.7, for 1 ă p ă p crit pQ, γq, we claim a sharp estimate for the lifespan T ε as stated below Moreover, the lifespan estimate mentioned above agrees with the one for additional L 1 -regular data when particularly we consider γ " Q 2 .
˝Range for global existence of solution ˝Range for blow-up of solution Figure 1.Description of the critical exponent in the pγ, pq plane Remark 1.8.One interesting observation is to discuss the acceptable ranges for p so that we can get sharp lifespan estimates.This can be summed up as follows: , we can achieve sharp lifespan estimates; , sharp lifespan estimates can be achieved; ‚ when Q ě 8, for 1 `2γ Q ď p ď Q Q´2 and 0 ă γ ď Q Q´2 , we can achieve sharp lifespan estimates.
We conclude the introduction with a brief outline of the organization of the paper.In Section 1, we discuss the main results and their explanations related to the critical exponent p crit pQ, γq and sharp lifespan estimates for weak solutions to Cauchy problem (1.1) when initial data taken additionally from 9 H ´γ L .Section 2 is devoted to recalling some basics of the Fourier analysis on the Heisenberg group H n to make the paper self-contained.Using the Fourier analysis on the Heisenberg group H n , we derive 9 H s L pH n q-decay estimates for the solution to the linear damped wave equation (1.1) with vanishing right-hand side in Section 3. Using the 9 H s L pH n q-decay estimates for the solution to a linear damped wave equation, we demonstrate global-in-time well-posedness for the semilinear Cauchy problem (1.1) if p ą p crit pQ, γq with the help of Banach's fixed point argument in Section 4. In order to estimate the nonlinear term in 9 H ´γ L pH n q, we use Sobolev inequality and the Gagliardo-Nirenberg inequality on the Heisenberg group H n .In Section 5, by employing the test function method, we conclude the optimality by showing the blow-up of weak solutions even for small data for the range 1 ă p ă p crit pQ, γq.As a byproduct, we also obtain upper bound estimates for the lifespan.We conclude our paper by deriving the sharp lower bound estimates for the lifespan of mild solutions in Section 6 by using the method of contradiction.

Preliminaries: Analysis on the Heisenberg group H n
In this section, we recall some basics of the Fourier analysis on the Heisenberg groups H n to make the manuscript self-contained.A complete account of the representation theory on H n can be found in [12,37,34,8,13,32].However, we mainly adopt the notation and terminology given in [8] for convenience.We commence this section by establishing the notations that will be employed consistently throughout the paper.
2.1.Notations.Throughout the article, we use the following notations: ‚ f À g : There exists a positive constant C (whose value may change from line to line in this manuscript) such that f ď Cg. ‚ f » g: Means that f À g and g À f .‚ H n : The Heisenberg group.‚ Q: The homogeneous dimension of H n .‚ dg : The Haar measure on the Heisenberg group H n .‚ L : The sub-Laplacian on H n .‚ H ´γ L : The subelliptic Sobolev spaces of negative order with γ ą 0 on H n .
2.2.The Hermite operator.In this subsection, we recall some definitions and properties of Hermite functions which we will use frequently in order to study Schrödinger representations and sub-Laplacian on the Heisenberg group H n .We start with the definition of Hermite polynomials on R.
Let H k denote the Hermite polynomial on R, defined by and h k denote the normalized Hermite functions on R defined by The Hermite functions th k u are the eigenfunctions of the Hermite operator (or the onedimensional harmonic oscillator) H " ´d2 `x2 with eigenvalues 2k`1, k " 0, 1, 2, ¨¨¨.These functions form an orthonormal basis for L 2 pRq.The higher dimensional Hermite functions denoted by e k are then obtained by taking tensor products of one dimensional Hermite functions.Thus for any multi-index k " pk 1 , ¨¨¨, k n q P N n 0 and x " px 1 , ¨¨¨, x n q P R n , we define e k pxq " ś n j"1 h k j px j q.The family te k u kPN n 0 is then an orthonormal basis for L 2 pR n q.They are eigenfunctions of the Hermite operator H " ´∆ `|x| 2 , namely, we have an ordered set of natural numbers tµ k u kPN n 0 such that He k pxq " µ k e k pxq, x P R n , for all k P N n 0 and x P R n .More precisely, H has eigenvalues corresponding to the eigenfunction e k for k P N n 0 .Given f P L 2 pR n q, we have the Hermite expansion where P m denotes the orthogonal projection of L 2 pR n q onto the eigenspace spanned by te k : |k| " mu.Then the spectral decomposition of H on R n is given by Since 0 is not in the spectrum of H, for any s P R, we can define the fractional powers H s by means of the spectral theorem, namely The canonical basis for the Lie algebra h n of H n is given by the left-invariant vector fields: which satisfy the commutator relations rX i , Y j s " δ ij T, for i, j " 1, 2, . . .n.Moreover, the canonical basis for h n admits the decomposition h n " V ' W, where V " spantX j , Y j u n j"1 and W " spantT u.Thus, the Heisenberg group H n is a step 2 stratified Lie group and its homogeneous dimension is Q :" 2n `2.The sublaplacian L on H n is defined as where ∆ R 2n is the standard Laplacian on R 2n .

2.4.
Fourier analysis on the Heisenberg group H n .We start this subsection by recalling the definition of the operator-valued group Fourier transform on H n .By Stone-von Neumann theorem, the only infinite-dimensional unitary irreducible representations (up to unitary equivalence) are given by π λ , λ in R ˚, where the mapping π λ is a strongly continuous unitary representation defined by π λ pgqf puq " e iλpt`1 2 xyq e i ?λyu f pu `a|λ|xq, g " px, y, tq P H n , for all f P L 2 pR n q.We use the convention For each λ P R ˚, the group Fourier transform of f P L 1 pH n q is a bounded linear operator on L 2 pR n q defined by Let BpL 2 pR n qq be the set of all bounded operators on L 2 pR n q.As the Schrödinger representations are unitary, for any λ P R ˚, we have If f P L 2 pH n q, then p f pλq is a Hilbert-Schmidt operator on L 2 pR n q and satisfies the following Plancherel formula where }.} S 2 stands for the norm in the Hilbert space S 2 , the set of all Hilbert-Schmidt operators on L 2 pR n q and dµpλq " c n |λ| n dλ with c n being a positive constant.Therefore, with the help of the orthonormal basis te k u kPN n 0 for L 2 pR n q and the definition of the Hilbert-Schmidt norm, we have The above expression allows us to write the Plancherel formula in the following way: For f P SpH n q, the space of all Schwartz class functions on H n , the Fourier inversion formula takes the form where TrpAq denotes the trace of the operator A. Furthermore, the action of the infinitesimal representation dπ λ of π λ on the generators of the first layer of the Lie algebra h n is given by dπ λ pX j q " a |λ|B x j for j " 1, ¨¨¨, n, dπ λ pY j q " i signpλq a |λ|x j for j " 1, ¨¨¨, n.
Since the action of dπ λ can be extended to the universal enveloping algebra of h n , combining the above two expressions, we obtain where H " ´∆ `|x| 2 is the Hermite operator on R n .Thus the operator valued symbol σ L pλq of L acting on L 2 pR n q takes the form σ L pλq " ´|λ|H.Furthermore, for s P R, using the functional calculus, the symbol of p´Lq s is |λ| s H s , where the notion of H s is defined in (2.1).
The Sobolev spaces H s L , s P R, associated to the sublaplacian L, are defined as Similarly, we denote by 9 H s L pH n q, the homogeneous Sobolev defined as the space of all f P D 1 pH n q such that p´Lq s{2 f P L 2 pH n q.More generally, we define 9 H p,s L pH n q as the homogeneous Sobolev defined as the space of all f P D 1 pH n q such that p´Lq s{2 f P L p pH n q for s ă Q p .Then we recall the following important inequalities, see e.g.[34,8,4,14,15] for a more general graded Lie group framework.However, we will state those in the Heisenberg group setting.Theorem 2.1 (Hardy-Littlewood-Sobolev inequality).Let H n be the Heisenberg group with the homogeneous dimension Q :" 2n `2.Let a ě 0 and 1 ă p ď q ă 8 be such that a Q " 1 p ´1 q .
Then we have the following inequality }f } 9 H q,´a L pH n q À }f } L p pH n q .We have the Gagliardo-Nirenberg inequality on H n as follows: Theorem 2.2 (Gagliardo-Nirenberg inequality).Let Q be the homogeneous dimension on the Heisenberg group H n .Assume that s P p0, 1s, 1 ă r ă Q s , and 2 ď q ď rQ Q ´sr .
Then we have the following Gagliardo-Nirenberg type inequality, 3. Linear damped wave equations: 9 H s L pH n q-norm estimates In this section, as a preliminary step in preparing to investigate the local and global wellposedness of the nonlinear Cauchy problem (1.1), we examine its associated linear counterpart, which involves a vanishing right-hand side.Specifically, our attention is directed towards the decay properties of solutions, in which our proofs are slightly different from the known results.Let us consider the Cauchy problem $ ' & ' % B 2 t u ´Lu `Bt u " 0, g P H n , t ą 0, up0, gq " u 0 pgq, g P H n , B t up0, gq " u 1 pgq, where u 0 pgq and u 1 pgq are the initial data additionally belonging to subelliptic Sobolev space 9 H ´γ L pH n q of negative order.We will derive 9 H s L pH n q-norm estimates for the solution upt, ¨q to the homogeneous problem (3.1).We make use of the group Fourier transform on the Heisenberg group H n , specifically with respect to the spatial variable g, and combine it with the Plancherel identity to estimate the 9 H s L pH n q-norm.We refer to [29], where a similar approach has been carried out for the L 2 -estimates of the solution to the damped wave equation (3.1) on the Heisenberg group H n .
Given that the group Fourier transform of a function f P L 2 pH n q is no longer a function but a family of bounded linear operators t p f pλqu λPR ˚on L 2 pR n q, the proofs are more involved and do not follow as in the Euclidean setup.We overcome this obstacle by using a trick to project these operators by using the orthonormal basis te k u kPN n of L 2 pR n q, namely, by working with the Fourier coefficients , respectively.Observe that, for |β k,λ | ! 1, we get Again, for |β k,λ | " 1, we obtain We collect the above easy observations as the following relations: Therefore, the solution to the homogeneous problem (3.3) is given by p upt, λq kl " A 0 pt, λq kℓ p u 0 pλq kl `A1 pt, λq kℓ p u 1 pλq kℓ , where and Thus from the above collected observations and asymptotic expression, we deduce the following pointwise estimates and for some suitable positive constant c.Before finding the Sobolev norm of upt, ¨q, first we notice that, for β 2 k,λ " |λ|µ k ă ε 2 , we have ˇˇp|λ|µ k q s`γ e ´ct|λ|µ k ˇˇÀ p1 `tq ´ps`γq , for s `γ ě 0. (3.11) Now by using the Plancherel formula and the fact that te k u kPN n is an orthonormal basis of L 2 pR n q, we have where pIq :" |λ|µ k and using (3.9) and (3.11), we get Similarly, using (3.10) and (3.11) we can find that Case 2: When |β k,λ | ą N " 1: Again using (3.9) we get Also, with the help of (3.10) we find that Case 3: When ε ď |β k,λ | ď N: From (3.9) and (3.10) , we get and for any t ě 0.

Global-in-time well-posedness
In this section, we will prove Theorem 1.2, that is, the global-in-time well-posedness of the Cauchy problem (1.1) in the energy evolution space C pr0, T s, H s L pH n qq.
Proof of Theorem 1.2.Recall that for s ě 0 and γ P R such that s`γ ě 0, from the estimate (1.9) of Theorem 1.1, we have }upt, ¨q} 9 In particular, for γ " 0 and s ě 0, we get }upt, ¨q} 9 From (4.1), in particular for s P r0, 1s and γ ą 0 we have the following estimate for the Sobolev solutions of the linear Cauchy problem }upt, ¨q} 9 where we have used the Sobolev embedding L 2 Ă H s´1 L for s ď 1. Particularly, for s " 0 in (4.3), using the Sobolev embedding H s L Ă L 2 for s ě 0, we obtain }upt, ¨q} Now from (4.3) and (4.4), for s P p0, 1s, we can write Thus from above, we can claim that u P X s pT q and Our next aim is to prove }u non } XspT q À }u} p XspT q , under some conditions for p.First we will estimate L 2 and 9 H ´γ L norm of |upt, ¨q| p .Applying Gagliardo-Nirenberg inequality (Theorem 2.2), we have for σ P r0, T s with θ 1 " Q 2s p1 ´1 p q P r0, 1s, provided that On the other hand, using the Hardy-Littlewood-Sobolev inequality (Theorem 2.1), we have }|upσ, ¨q| p } 9 Since m P p1, 2q, we have to restrict 0 ă γ ă Q 2 .Again, the Gagliardo-Nirenberg inequality (Theorem 2.2) implies that }|upσ, ¨q| p } 9 }upσ, ¨q} 9 " p1 `σq ´p 2 rγ`Qp }upσ, ¨q} 9 for σ P r0, T s with θ 2 " Q s p 1 2 ´1 mp q P r0, 1s.Using the fact that θ 2 P r0, 1s, we get Since m " 2Q Q`2γ P p1, 2q, from (4.9), we obtain Therefore, from (4.7) and (4.10), if we consider then from (4.6) and (4.8), for σ P r0, T s, we assert that }|upσ, ¨q| p } L 2 X 9 Using the pL 2 X 9 H ´γ L q ´L2 estimate given in (4.3) (with u 0 " 0, u 1 " |upσ, ¨q| p ) as well as from the statement (4.11), we have the following L 2 -estimate of the solution XspT q (4.12)  Therefore from (4.17) and (4.15), we obtain }u non pt, ¨q} XspT q À }u} p XspT q , under the conditions on p as follows: , from the integrability and decay estimates of solution.‚ p ě 1 `2γ Q , from the application of the Gagliardo-Nirenberg inequality.First, we consider the case γ ą 2. Then we observe that max Now for γ ď 2, we have to compare between 1 `4 Q`2γ and 1 `2γ Q .Notice that 1 `4 Q`2γ and 1 `2γ Q intersects at a point γ, which is the positive root of the quadratic equation 2γ2 `Qγ ´2Q " 0.Moreover, it is easy to check that the positive root γ ă 2 for all Q ě 4.This shows that Finally, for any γ P p0, Q 2 q, the condition for the exponent p is reduced to Furthermore, we have Now we will calculate }Nu ´N ū} XspT q .In order to do that, we first notice that L pH n q }upσ, ¨q ´ūpσ, ¨q} }upσ, ¨q ´ūpσ, ¨q} 9 for σ P r0, T s with θ 3 " Q s p 1 2 ´1 mp q P r0, 1s. 2 ´1 mp qq }upσ, ¨q ´ūpσ, ¨q} XspT q ´}u} p´1 XspT q `}ū} p´1 XspT q À p1 `σq ´p 2 p1´1 p qpγ`Q 2 q }upσ, ¨q ´ūpσ, ¨q} XspT q ´}u} p´1 XspT q `}ū} p´1 XspT q À p1 `σq Now using the given range of p, we get }Nu ´N ū} XspT q ď C}u ´ū} XspT q ´}u} p´1 XspT q `}ū} p´1 XspT q ¯, ( for any u, ū P X s pT q and for some C ą 0. Also from (4.18), we obtain for some D ą 0 with initial data space A s,´γ L :" pH s L X 9 H ´γ L q ˆpL 2 X 9 H ´γ L q. Therefore, by Banach's fixed point theorem, there exists a uniquely determined fixed point u ˚of the operator N, which means u ˚" Nu ˚P X s pT q for all positive T .This fixed point u ẘill be our mild solution to (1.1) on r0, T s.This implies that there exists a global (in-time) small data Sobolev solution u ˚of the equation u ˚" Nu ˚in X s pT q, which also gives the solution to the semilinear damped wave equation (1.1) and this completes the proof of the theorem.

Blow-up analysis of solutions: a test function method
Proof of Theorem 1.5.In order to prove this result, we apply the so-called test function method.By contradiction, we assume that there exists a global in time weak solution u to (1.1).Let us consider two bump functions α P C 8 0 pR n q and β P C 8 0 pRq such that  with supp β Ă p´1, 1q.Now from (1.12) with keeping in mind the fact B t ϕ R ď 0 (by choosing the factor β appropriately, for example, let β be even and non-increasing in r0, 8q), an application of Hölder's and Young's inequalities yields where in (5.1), we used the fact that measpD R q « R Q .Therefore Now using our assumption (1.13), it remains to find estimate for ż H n pu 0 pgq `u1 pgqq ϕ R p0, gqdg.
Before that, we have to justify that the set of all initial data pu 0 , u 1 q in 9 H ´γ L ˆ9 H ´γ L with the assumptions (1.13), is non-empty.For that, we denote the set D Q,γ as D Q,γ :" tpu 0 , u 1 q P L 1 loc pH n q ˆL1 loc pH n q : u 0 pgq `u1 pgq ě C 0 xgy ´Qp 1 2 `γ Q q plogpe `|g|qq ´1u, In particular, consider the functions Then using the polar decomposition (see Proposition (1.15) in [9]) for H n , we have ż (5.4) By the assumption p P ´1, 1 `4 Q`2γ ¯, we have Q `2 ´2p 1 ´Q 2 `γ ă 0 and therefore, it is easy to see that for R " 1.Hence, from (5.4) and (5.5), we obtain which is a contradiction.This completes the proof of the blow-up result.Since the scaling factor R 2 , appeared in the bump function β with respect to the time varibale has to be dominated by the lifespan T w,ε of the weak solution in order to guarantee ϕ R P C 8 0 pr0, T qˆH n q, to generate upper bound estimate for the lifespan, we consider R Ò T w,ε in (5.4).As a result, a contradiction (similar to (5.4)) exists if w,ε plog T w,ε q ´1.
In other words T w,ε ď Cε ´p Q 4 `γ 2 qq ´1 , which is the desired lifespan for the local in time weak solutions to the Cauchy problem (1.1), where the constant C is positive and independent of ε and p.

Sharp lifespan estimates of solutions
The aim of this section is to find the lower bound estimates of lifespan.We will make use of certain notations, specifically, the evolution space X 1 pT q and the data space A L 1 , introduced in Section 1.As we are now considering the case where 1 ă p ă p crit pQ, γq, in order to obtain lower limit estimates for the lifespan, we will use a different nonlinear inequality instead of (4.5).
Its important to note that the function G " GpT q is continuous for T P p0, T m,ε q.Nonetheless, (6.5) demonstrates that there exists a time T 0 P pT ˚, T m,ε q such that G pT 0 q ď Mε, which contradicts to the assumption that T ˚is the supremum.To put it another way, we must enforce the following condition: D p1 `T ˚qαpp,Q,γq M p´1 ε p´1 ě 1.
This implies that 1 `T ˚ě D ´1 αpp,Q,γq M ´p´1 αpp,Q,γq ε ´p´1 αpp,Q,γq .This implies that we can deduce the blow-up time estimate as This completes the proof of the theorem regarding lower bound estimates of lifespan.Remark 6.1.We believe that the results obtained in this paper on the Heisenberg group can be generalized on a general stratified Lie group G, where the formula for p crit pQ, γq will be the same as in the case of Heisenberg group, with Q now the homogeneous dimension of G.The global existence result can be proved by following the approach of [34, Section 4], while the blow-up result can be proved using again the test function method considered in [11].
lin pt, gq " u 0 pgq ˚pgq E 0 pt, gq `u1 pgq ˚pgq E 1 pt, gq is the solution to the corresponding linear Cauchy problem (1.8) and .1) 2.3.The Heisenberg group.One of the simplest examples of a non-commutative and non-compact group is the famous Heisenberg group H n .The theory of the Heisenberg group plays a crucial role in several branches of mathematics and physics.The Heisenberg group H n is a nilpotent Lie group whose underlying manifold is R 2n`1 and the group operation is defined by px, y, tq ˝px 1 , y 1 , t 1 q " px `x1 , y `y1 , t `t1 `1 2 pxy 1 ´x1 yqq, where px, y, tq, px 1 , y 1 , t 1 q are in R n ˆRn ˆR and xy 1 denotes the standard scalar product in R n .Moreover, H n is a unimodular Lie group on which the left-invariant Haar measure dg is the usual Lebesgue measure dx dy dt.
Invoking the group Fourier transform with respect to g on (3.1), we get a Cauchy problem related to a parameter-dependent functional differential equation for p upt, λq, namely, L pλq is the symbol of the sub-Laplacian L on H n .In fact, we know that σ L pλq " ´|λ|H.For any k, ℓ P N n , let us introduce the notation p upt, λq k,ℓ." pp upt, λqe ℓ , e k q L 2 pR n q ,where te k u kPN n is the system of Hermite functions forming an orthonormal basis of L 2 pR n q.Since He k " µ k e k , p upt, λq k,ℓ solves an ordinary differential equation with respect to the variable t depending on parameters λ P R ˚and k, ℓ P N n , ℓPN n for the Fourier transform p f pλq for each λ P R ˚. k .Then, the characteristic equation of (3.3) is given by m 2 `m `β2 k,λ " 0, and so the characteristic roots m 1 and m 2 are .18) ´σq ´γ 2 dσ}u} p p1 `t ´σq ´γ 2 dσ À p1 `tq ´γ 2 .