In the Euclidean case, pseudo-hyperbolic equations served as models for the unidirectional propagation of nonlinear dispersive long waves [2], creep buck- ling [5] for example

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

NONEXISTENCE RESULTS FOR A PSEUDO-HYPERBOLIC EQUATION IN THE HEISENBERG GROUP

MOKHTAR KIRANE, LAKHDAR RAGOUB

Abstract. Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear pseudo-hyperbolic equation

utt−∆_Hutt−∆_Hu=|u|^p, (η, t)∈H×(0,∞), p >1,

where ∆_His the Kohn-Laplace operator on the (2N+ 1)-dimensional Heisen- berg groupH. Then, this result is extended to the case of a 2×2-system of the same type. Our technique of proof is based on judicious choices of the test functions in the weak formulation of the sought solutions.

1. Introduction

In this article, we are concerned with the nonexistence of weak solutions to the nonlinear pseudo-hyperbolic equation

u_tt−∆_Hu_tt−∆_Hu=|u|^p, (η, t)∈H×(0,∞), p >1, (1.1) under the initial conditions

u(η,0) =u0(η), ut(η,0) =u1(η), η∈H, (1.2) where ∆_H is the Kohn-Laplace operator on the (2N+ 1)-dimensional Heisenberg group H. In the Euclidean case, pseudo-hyperbolic equations served as models for the unidirectional propagation of nonlinear dispersive long waves [2], creep buck- ling [5] for example. For further applications, one is referred to the valuable book [1]

where a sizeable number of pseudo-hyperbolic equations are studied. Our proofs rely on the test function method [8, 12]. For the reader convenience, some back- ground facts used in the sequel are recalled.

The (2N+ 1)-dimensional Heisenberg groupHis the spaceR^2N+1equipped with the group operation

η◦η⁰= (x+x⁰, y+y⁰, τ+τ⁰+ 2(x·y⁰−x⁰·y)),

for allη = (x, y, τ), η⁰ = (x⁰, y⁰, τ⁰)∈R^N×R^N ×R, where ·denotes the standard scalar product in R^N. This group operation endowsHwith the structure of a Lie group.

2010Mathematics Subject Classification. 47J35, 34A34, 35R03.

Key words and phrases. Nonexistence; nonlinear pseudo-hyperbolic equation;

systems of pseudo-hyperbolic equations; Heisenberg group.

c

2015 Texas State University - San Marcos.

Submitted January 5, 2015. Published April 22, 2015.

1

OnHit is natural to define a distance from η= (x, y, τ) =: (z, τ) to the origin by

|η|_H=

τ²+X^N

i=1

(x²_i +y_i²)21/4

= τ²+|z|^41/4 ,

wherex= (x₁,· · · , x_N) andy= (y₁,· · · , y_N).

The Laplacian ∆_H overHcan be defined from the vectors fields X_i=∂_xi+ 2y_i∂_τ and Y_i=∂_yi−2x_i∂_τ, fori= 1,· · ·, N, as follows

∆_H=

N

X

i=1

(X_i²+Y_i²).

A simple computation gives the expression

∆_Hu=

N

X

i=1

∂_x²_ixiu+∂²_yiyiu+ 4yi∂_x²_iτu−4xi∂_y²_iτu+ 4(x²_i +yi)²∂_{τ τ}² u .

The operator ∆_H satisfies the following properties:

• It is invariant with respect to the left multiplication in the group, i.e., for allη, η⁰ ∈H, we have

∆_H(u(η◦η⁰)) = ∆_Hu(η◦η⁰);

• It is homogeneous with respect to a dilatation. More precisely, forλ∈R and (x, y, τ)∈H, we have

∆_H(u(λx, λy, λ²τ)) =λ²(∆_Hu)(λx, λy, λ²τ);

• Ifu(η) =v(|η|_H), then

∆_Hv(ρ) =a(η)d²v

dρ² +Q−1 ρ

dv dρ

,

whereρ=|η|_H,a(η) =ρ⁻²PN

i=1(x²_i +y²_i) andQ= 2N+ 2 is the homoge- neous dimension ofH.

For more details on Heisenberg groups, we refer to [4, 7].

In this work, we first provide a sufficient condition for the nonexistence of weak solutions to the nonlinear problem (1.1)-(1.2), then we extend the result to the case of the 2×2 system

utt−∆_Hutt−∆_Hu=|v|^q, (η, t)∈H×(0,∞), v_tt−∆_Hv_tt−∆_Hv=|u|^p, (η, t)∈H×(0,∞),

u(η,0) =u0(η), u1(η,0) =u1(η), η∈H v(η,0) =v0(η), v(η,0) =v1(η), η∈H,

(1.3)

wherep, q >1 are real numbers, for which we provide a sufficient condition for the nonexistence of weak solutions.

2. Results and proofs LetH_T =H×(0, T),H=H×(0,∞). ForR >0, let

UR={(x, y, τ, t)∈ H: 0≤t⁴+|x|⁴+|y|⁴+τ²≤2R⁴}.

2.1. Case of a single equation. The definition of solutions we adopt for (1.1)- (1.2) is:

We say that u is a local weak solution to (1.1)-(1.2) on H with initial data u(0,·) =u₀∈L¹_loc(H), ifu∈L^p_loc(H) and satisfies

Z

H

|u|^pϕ dϑ dt+ Z

H

u₁(ϑ)ϕ(ϑ,0)dϑ+ Z

H

u₁(ϑ)∆_Hϕ(ϑ,0)dϑ

= Z

H

uϕ_ttdϑ dt+ Z

H

u∆_Hϕ_ttdt dϑ− Z

H

u∆_Hϕ dt dϑ,

for any test functionϕ,ϕ(·, t) = 0,ϕ_t(·, t) = 0, t≥T. The solutionuis said global if it exists on (0,∞).

Our first main result is given by the following theorem.

Theorem 2.1. Let u₁∈L¹(H). Suppose that Z

H

u₀dϑ >0. (2.1)

If

1< p≤1 + 2 Q−1,

then any weak solution to (1.1)-(1.2)blows-up in a finite time.

Proof. Suppose thatuis a weak solution to (1.1)-(1.2). Then for any regular test functionϕ, we have

Z

H

|u|^pϕ dϑ dt+ Z

H

u1(ϑ)ϕ(ϑ,0)dϑ

≤ Z

H

|u||ϕtt|dϑ dt+ Z

H

|u||∆_Hϕtt|dt dϑ

+ Z

H

|u||∆_Hϕ|dt dϑ+ Z

H

|u1(ϑ)||∆_Hϕ(ϑ,0)|dϑ.

(2.2)

Using theε-Young inequality

ab≤εa^p+Cεb^p0, a, b, ε, Cε>0, 1< p, p⁰, p+p⁰ =pp⁰, with parameterspandp⁰=p/(p−1), we obtain

Z

H

|u||ϕtt|dϑ dt= Z

H

|u|ϕ^1/pϕ^−1p|ϕtt|dϑ dt

≤ε Z

H

|u|^pϕ dϑ dt+c_ε Z

H

ϕ^−p−11 |ϕtt|^p−1p dϑ dt,

(2.3)

for some positive constantcε. Similarly, we have

Z

H

|u||∆_Hϕtt|dt dϑ≤ε Z

H

|u|^pϕ dϑ dt+cε

Z

H

ϕ^−p−11 |∆_Hϕtt|^p−1p dt dϑ, (2.4) Z

H

|u||∆_Hϕ|dt dϑ≤ε Z

H

|u|^pϕ dϑ dt+cε

Z

H

ϕ^−p−11 |∆_Hϕ|^P−1P dt dϑ. (2.5)

Using (2.2), (2.3), (2.4) and (2.5), forε >0 small enough, we obtain Z

H

|u|^pϕ dϑ dt+ Z

H

u₁(ϑ)ϕ(ϑ,0)dϑ

≤C

A_p(ϕ) +B_p(ϕ) +C_p(ϕ) + Z

H

|u₁(ϑ)||∆_Hϕ(ϑ,0)|dϑ ,

(2.6)

where

Ap(ϕ) = Z

H

ϕ^−p−11 |ϕtt|^p−1p dϑ dt, (2.7) B_p(ϕ) =

Z

H

ϕ^−p−11 |∆_Hϕ_tt|^p−1p dϑ dt, (2.8) C_p(ϕ) =

Z

H

ϕ^−p−11 |∆_Hϕ|^p−1p dϑ dt. (2.9) Now, let us consider the test function

ϕR(t, ϑ) =φ^ωt⁴+|x|⁴+|y|⁴+τ² R⁴

, R >0, ω1, (2.10) whereφ∈C₀^∞(R⁺) is a decreasing function satisfying

φ(r) =

(1 if 0≤r≤1, 0 ifr≥2.

Observe that supp(ϕR) is a subset of UR, while supp(ϕ_Rtt), supp(∆_HϕR) and supp(∆_H(ϕR)tt) are subsets of

ΘR={(t, x, y, τ)∈ H: R⁴≤t⁴+|x|⁴+|y|⁴+τ²≤2R⁴}.

Let

ρ= t⁴+|x|⁴+|y|⁴+τ²

R⁴ .

Then we have

∆_HϕR(t, ϑ)

= 4ω(N+ 4) R⁴

|x|²+|y|²

φ⁰(ρ)φ^ω−1(ρ) +16ω(ω−1)

R⁸

(|x|⁶+|y|⁶) + 2τ(|x|²− |y|²)x·y+τ²(|x|²+|y|²)

φ⁰²(ρ)φ^ω−2(ρ) +16ω

R⁸

(|x|⁶+|y|⁶) + 2τ(|x|²− |y|²)x·y+τ²(|x|²+|y|²)

φ⁰⁰(ρ)φ^ω−1(ρ) for example.

Observe first that (ϕR)t(ϑ,0) = 0 as required in the definition. It follows that there is a positive constantC >0, independent ofR, such that for all (t, ϑ)∈ ×R, we have

|∆_HϕR(t, ϑ)| ≤CR⁻²φ^ω−2(ρ)χ(ρ), (2.11) where

χ(ρ) =|φ⁰(ρ)|φ(ρ) +φ⁰²(ρ) +|φ⁰⁰(ρ)|φ(ρ), and

|(∆_Hϕ_R)_t(t, ϑ)| ≤CR⁻³, (2.12)

|(ϕ_R)_tt(t, ϑ)| ≤CR⁻⁴. (2.13)

Using (2.11) and (2.12), we obtain

A_p(ϕ_R)≤CR^{Q+1−p−12p} , (2.14) Bp(ϕR)≤CR^{Q+1−p−14p} , (2.15) Cp(ϕR)≤CR^{Q+1−p−12p} . (2.16) Let us consider now the change of variables

(t, x, y, τ) = (t, ϑ)7→(et,ev) = (et,x,e y,e eτ), (2.17) where

et=R⁻¹t, τe=R⁻²τ,xe=R⁻¹x, ye=R⁻¹y.

Let

ρe=et⁴+|x|e⁴+|y|e⁴+eτ², CfR={(et,ex,y,eτe)∈ H: 1≤ρe≤2},

CR={(x, y, τ)∈H:R⁴≤ |x|⁴+|y|⁴+τ²≤2R⁴}.

Using (2.6), (2.15) and (2.16), we obtain Z

H

|u|^pϕ_Rdϑ dt+ Z

H

u₁(ϑ)ϕ_R(ϑ,0)dϑ

≤C

R^ϑ1+R^ϑ2+ Z

C_R

|u₁(ϑ)||∆_Hϕ_R(ϑ,0)|dϑ ,

(2.18)

where

ϑ₁=Q+ 1− 2p

p−1 and ϑ₂=Q+ 1− 4p p−1. On the other hand, we have

lim inf

R→∞

Z

H

|u|^pϕ_Rdϑ dt+ Z

H

u₁(ϑ)ϕ_R(ϑ,0)dϑ

≥lim inf

R→∞

Z

H

|u|^pϕRdϑ dt+ lim inf

R→∞

Z

H

u1(ϑ)ϕR(ϑ,0)dϑ.

Using the monotone convergence theorem, we obtain lim inf

R→∞

Z

H

|u|^pϕ_Rdϑ dt= Z

H

|u|^pdϑ dt.

Sinceu1∈L¹(H), by the dominated convergence theorem, we have lim inf

R→∞

Z

H

u₁(ϑ)ϕ_R(ϑ,0)dϑ= Z

H

u₁(ϑ)dϑ.

Now, we have lim inf

R→∞

Z

H

|u|^pϕRdϑ dt+ Z

H

u1(ϑ)ϕR(ϑ,0)dϑ

≥ Z

H

|u|^pdϑ dt+`, where from (2.1),

`= Z

H

u1(ϑ)dϑ >0.

By the definition of the limit inferior, for everyε >0, there existsR0>0 such that Z

H

|u|^pϕRdϑ dt+ Z

H

u1(ϑ)ϕR(ϑ,0)dϑ

>lim inf

R→∞

Z

H

|u|^pϕRdϑ dt+ Z

H

u1(ϑ)ϕR(ϑ,0)dϑ

−ε

≥ Z

H

|u|^pdϑ dt+`−ε, for everyR≥R0. Takingε=`/2, we obtain

Z

H

|u|^pϕ_Rdϑ dt+ Z

H

u₁(ϑ)ϕ_R(ϑ,0)dϑ≥ Z

H

|u|^pdϑ dt+` 2, for everyR≥R0. Then from (2.18), we have

Z

H

|u|^pdϑ dt+` 2 ≤C

R^ϑ1+R^ϑ2+ Z

CR

|u0(ϑ)||∆_Hϕ_R(ϑ,0)|dϑ

, (2.19) forRlarge enough.

Now, we require that ϑ1 = max{ϑ1, ϑ2} ≤ 0, which is equivalent to 1 < p ≤ 1 + _Q−1² . We distinguish two cases.

Case 1: 1 < p <1 + _Q−1² . In this case, letting R → ∞ in (2.19) and using the dominated convergence theorem, we obtain

Z

H

|u|^pdϑ dt+` 2 ≤0, which is a contradiction as` >0.

Case 2: p= 1 +_Q−1² . From (2.19), we obtain Z

H

|u|^pdϑ dt≤C <∞ ⇒ lim

R→∞

Z

CR

|u|^pϕRdϑ dt= 0. (2.20) Using the H¨older inequality with parameterspandp/(p−1), from (2.2), we obtain

Z

H

|u|^pϕRdϑ dt+ `

2 ≤CZ

Θ_R

|u|^pϕRdϑ dt^1/p .

LettingR→ ∞in the above inequality and using (2.20), we obtain Z

H

|u|^pdϑ dt+` 2 = 0.

A contradiction; the proof of the theorem is complete.

2.1.1. The case of system (1.3). The definition of solutions we adopt for (1.3) is:

We say that the pair (u, v) is a local weak solution to (1.3) on H with initial data (u(0,·), v(0,·)) = (u0, v0)∈L¹_loc(H)×L¹_loc(H), if (u, v) ∈L^p_loc(H)×L^q_loc(H) and satisfies

Z

H

|v|^qϕ dϑ dt+ Z

H

u₁(ϑ)ϕ(ϑ,0)dϑ

= Z

H

uϕ_ttdϑ dt+ Z

H

u(∆_Hϕ)_ttdt dϑ− Z

H

u∆_Hϕ dt dϑ+ Z

H

u₁(ϑ)∆_Hϕ(ϑ,0)dϑ and

Z

H

|u|^pϕ dϑ dt+ Z

H

v1(ϑ)ϕ(ϑ,0)dϑ

= Z

H

vϕttdϑ dt+ Z

H

v(∆_Hϕ)ttdt dϑ− Z

H

v∆_Hϕ dt dϑ+ Z

H

v1(ϑ)∆_Hϕ(ϑ,0)dϑ,

for any test functionϕ,ϕ(·, t) = 0, ϕt(·, t) = 0, t≥T. The solution is said global if it exists forT = +∞.

Our second main result is given by the following theorem.

Theorem 2.2. Let (u₁, v₁)∈L¹(H)×L¹(H). Suppose that Z

H

u1dϑ >0 and Z

H

v1dϑ >0.

If 1< pq≤(pq)^∗, where

(pq)^∗= 1 + 2

Q−1max{p+ 1, q+ 1}, then there exists no nontrivial weak solution to (1.3).

Proof. Suppose that (u, v) is a nontrivial weak solution to (1.3). Then for any regular test functionϕ, we have

Z

H

|v|^qϕ dϑ dt+ Z

H

u1(ϑ)ϕ(ϑ,0)dϑ

≤ Z

H

|u||ϕtt|dϑ dt+ Z

H

|u||(∆_Hϕ)tt|dt dϑ

+ Z

H

|u||∆_Hϕ|dt dϑ+ Z

H

|u1(ϑ)||∆_Hϕ(ϑ,0)|dϑ and

Z

H

|u|^pϕ dϑ dt+ Z

H

v1(ϑ)ϕ(ϑ,0)dϑ

≤ Z

H

|v||ϕtt|dϑ dt+ Z

H

|v||(∆_Hϕ)tt|dt dϑ

+ Z

H

|v||∆_Hϕ|dt dϑ+ Z

H

|v1(ϑ)||∆_Hϕ(ϑ,0)|dϑ.

Takingϕ=ϕR, the test function given by (2.10), and using the H¨older inequality with parameterspandp/(p−1), we obtain

Z

H

|v|^qϕRdϑ dt+ Z

H

u1(ϑ)ϕR(ϑ,0)dϑ− Z

H

|u1(ϑ)||∆_HϕR(ϑ,0)|dϑ

≤

Ap(ϕR)^p−1p +Bp(ϕR)^p−1p +Cp(ϕR)^p−1p Z

H

|u|^pϕRdϑ dt^1/p ,

where A_p(ϕ), B_p(ϕ) and C_p(ϕ) are given respectively by (2.7), (2.8) and (2.9).

Similarly, by the H¨older inequality with parametersqand q/(q−1), we get Z

H

|u|^pϕRdϑ dt+ Z

H

v1(ϑ)ϕR(ϑ,0)dϑ− Z

H

|v1(ϑ)||∆_HϕR(ϑ,0)|dϑ

≤

Aq(ϕR)^q−1q +Bq(ϕR)^q−1q +Cq(ϕR)^q−1q Z

H

|v|^qϕRdϑ dt1/q

.

Without restriction of the generality, we may assume that forR large enough, we have

Z

H

u1(ϑ)ϕR(ϑ,0)dϑ− Z

H

|u1(ϑ)||∆_HϕR(ϑ,0)|dϑ≥0, Z

H

v1(ϑ)ϕR(ϑ,0)dϑ− Z

H

|v1(ϑ)||∆_HϕR(ϑ,0)|dϑ≥0.

Slight modifications yield the proof in the general case (see the proof of Theorem 2.1). Then forRlarge enough, we have

Z

H

|v|^qϕRdϑ dt

≤

A_p(ϕ_R)^p−1p +B_p(ϕ_R)^p−1p +C_p(ϕ_R)^p−1p Z

H

|u|^pϕ_Rdϑ dt^1/p

(2.21)

and Z

H

|u|^pϕ_Rdϑ dt

≤

A_q(ϕ_R)^q−1q +B_q(ϕ_R)^q−1q +C_q(ϕ_R)^q−1q Z

H

|v|^qϕ_Rdϑ dt^1/q .

(2.22)

Using the change of variables (2.17), from (2.21) and (2.22), we obtain Z

H

|v|^qϕRdϑ dt≤CR^{Q(p−1)−2p} Z

H

|u|^pϕRdϑ dt1/p

, (2.23)

Z

H

|u|^pϕRdϑ dt≤CR^{Q(q−1)−2q} Z

H

|v|^qϕRdϑ dt1/q

. (2.24)

Combining (2.23) with (2.24), we obtain Z

H

|u|^pϕ_Rdϑ dt1−_pq¹

≤CR^υ1, (2.25)

Z

H

|v|^qϕ_Rdϑ dt¹⁻_pq¹

≤CR^υ2, (2.26)

where

υ₁= Q(pq−1)−2(p+ 1)

pq−1 and υ₂= Q(pq−1)−2(q+ 1)

pq−1 .

We require thatυ1≤0 orυ2≤0 which is equivalent to 1< pq≤1 +_Q² max{p+

1, q+ 1}. We distinguish two cases.

Case 1: 1< pq <1 +_Q² max{p+ 1, q+ 1}. Without loss of the generality, we may suppose that 0< q≤p. In this case, lettingR→ ∞in (2.25), we obtain

Z

H

|u|^pdϑ dt= 0, which is a contradiction.

Case 2: pq= 1 +_Q² max{p+ 1, q+ 1}. This case can be treated in the same way

as in the proof of Theorem 2.1.

Remark 2.3. Ifp=qandu=vin Theorem 2.2, we obtain the result for a single equation given by Theorem 2.1.

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Mokhtar Kirane

Laboratoire de Math´ematiques, Image et Applications, Pˆole Sciences et Technologies, Universit´e de La Rochelle, Avenue M. Cr´epeau, 17042 La Rochelle, France.

NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdu- laziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

E-mail address:[email protected]

Lakhdar Ragoub

Al Yamamah University, College of Computers and Information Systems P.O. Box 45180, Riyadh 11512, Saudi Arabia

E-mail address:l [email protected]