Blow-up solutions of damped Klein–Gordon equation on the Heisenberg group

We prove the blow-up of solutions of the semilinear damped Klein–Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper Ruzhansky and Tokmagambetov (J Differ Equ 265(10):5212–5236, 2018), where the global in time well-posedness was proved for the small energy solutions.

1. Introduction 1.1.Setting of the problem.This note is devoted to study the blow up of solutions of the Cauchy problem for the semilinear damped Klein-Gordon equation for the sub-Laplacian L on the Heisenberg group with the damping term determined by b > 0 and the mass m > 0. A total energy of problem (1.1) is defined as where we assume the following 2020 Mathematics Subject Classification.35L71; 35R03, 35B44.Key words and phrases.Blow-up, sub-Laplacian, Heisenberg group, damped Klein-Grodon equation.
The first and second authors were supported by EPSRC grant EP/R003025/2.The first author was also supported by FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021).
Then we have that The conservation of energy law follows from this gives where Also, let us define the Nehari functional We assume that the nonlinear term f (u) satisfies the condition where α > 2. In particular, this includes the case 1.2.Literature overview.The study of the damped wave equation on the Heisenberg group started in Bahouri-Gerrard-Xu [1] to prove the dispersive and Strichartz inequalities based on the analysis in Besov-type spaces.Later, Greiner-Holcman-Kannai [4] explicitly computed the wave kernel for the class of second-order subelliptic operators, where their class contains degenerate elliptic and hypoelliptic operators such as the sub-Laplacian and the Grushin operator.Also, Müller-Stein [15] established L p -estimates for the wave equation on the Heisenberg group.Recently, Müller-Seeger [16] obtained the sharp version of L p estimates on the H-type groups.
The blow-up solutions of evolution equations on the Heisenberg group were considered by Georgiev-Palmieri [5] where they proved the global existence and nonexistence results of the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with the power nonlinear term.The proof of blow-up solutions is based on the test function method.The first author and Yessirkegenov [22] established the existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds.In [23], by using the comparison principle they obtain blow-up type results and global in t-boundedness of solutions of nonlinear equations for the heat p-sub-Laplacian on the stratified Lie groups.The global existence and nonexistence for the nonlinear porous medium equation were studied by the authors in [19] on the stratified Lie groups.
This work is motivated by the paper [21] of the first author and Tokmagambetov where the global existence of solutions for small data of problem (1.1) was shown on the Heisenberg group and on general graded Lie groups.In the sense of the potential wells theory, we can understand this result in the sense that when the initial energy is less than the mountain pass level E(0) < d and the Nehari functional is positive I(u 0 ) > 0, there exists a global solution of the problem (1.1).A natural question arises when the solution of problem (1.1) blows up in a finite time or E(0) > 0 and I(u 0 ) < 0.
The main aim of this paper is to obtain the blow-up solutions of problem (1.1) in a finite time for arbitrary positive initial energy.Our proof is based on an adopted concavity method, which was introduced by Levine [9] to establish the blow-up solutions of the abstract wave equation of the form P u tt = −Au + F (u) (including the Klein-Gordon equation) for the negative initial energy.It was also used for parabolic type equations (see [10,12,11,13,14]).Modifying the concavity method, Wang [29] proved the nonexistence of global solutions to nonlinear damped Klein-Gordon equation for arbitrary positive initial energy under sufficient conditions.Later, Yang-Xu [27] extended this result by introducing a new auxiliary function and the adopted concavity method.
1.3.Preliminaries on the Heisenberg group.Let us give a brief introduction of the Heisenberg group.Let H n be the Heisenberg group, that is, the set R 2n+1 equipped with the group law where ξ := (x, y, t) ∈ H n , x := (x 1 , . . ., x n ), y := (y 1 , . . ., y n ), and ξ −1 = −ξ is the inverse element of ξ with respect to the group law.The dilation operation of the Heisenberg group with respect to the group law has the following form (see e.g.[7], [20]) δ λ (ξ) := (λx, λy, λ 2 t) for λ > 0. The Lie algebra h of the left-invariant vector fields on the Heisenberg group H n is spanned by and with their (non-zero) commutator The horizontal gradient of H n is given by so the sub-Laplacian on H n is given by L (H n ) and a.e.t ∈ [0, T 1 ) with u(0) = u 0 (x) and u t (0) = u 1 (x) represents a weak solution of problem (1.1).
Note that T 1 denotes the lifespan of the solution u(x, t) and •, • is the duality between H −1 L (H n ) and H 1 L (H n ).Here H 1 L (H n ) denotes the sub-Laplacian Sobolev space, analysed by Folland [6], see also [7].

Main Result
We now present the main result of this paper.

2). Assume that the Cauchy data u
and Then the solution of equation (1.1) blows up in finite time T * such that Re(u 0 , u 1 ) , where the blow-up time T * ∈ (0, T 0 ) with T 0 < +∞.
The global in time well-posedness of problem (1.1) was proved by the first author and Tokmagambetov [21] for the small energy solutions and the nonlinearity f (u) satisfying Proof of Theorem 2.1 .First, recall the Nehari functional Then the proof includes two steps.
Step I.In this step, we claim that I(u(t)) < 0, and for 0 ≤ t < T 1 where µ = max{b, m, α} and In the last line we have used that Re u tt , u = Re Now let us suppose by contradiction that I(u(t)) < 0 for all 0 ≤ t < t 0 , and I(u(t 0 )) = 0.
Step II.Define the functional We observe the following estimates Using the above inequalities, we calculate for all 0 ≤ t ≤ T 0 .The second derivate with respect to time of M(t) is for all 0 ≤ t ≤ T 0 , where we used the equality from (2.4).Then we construct the differential inequality as follows In the second line that we have used (2.7).By selecting ω = α − 1 − m(α−2) µ+1 which satisfies ω > 1 since µ + 1 > m and using the argument from Step I, we obtain µ + 1 2Re(u 0 , u 1 ) − 2αE(0) ≥ 0, Note that we have used the fact A ′ (t) > 0 and the expression (2.5) with A(t) = 2Re(u, u t ) + b||u|| 2 L 2 (H n ) , and the condition (2.3) in the last line, respectively.So we obtain the inequality M ′′ (t)M(t) − ω + 3 4 (M ′ (t)) 2 > 0. Then