We establish a Fujita-type theorem for the blow-up of nonnegative solutions to a certain class of parabolic inequalities in the Heisenberg group

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

BLOW-UP OF SOLUTIONS TO PARABOLIC INEQUALITIES IN THE HEISENBERG GROUP

IBTEHAL AZMAN, MOHAMED JLELI, BESSEM SAMET

Abstract. We establish a Fujita-type theorem for the blow-up of nonnegative solutions to a certain class of parabolic inequalities in the Heisenberg group.

Our proof is based on a duality argument.

1. Introduction

In this article, we establish a Fujita-type theorem for parabolic inequality u_t−div_HA(ϑ, u,∇_Hu) +f(ϑ, u,∇_Hu)≥u^q, inH,

u≥0, a.e. inH, u(ϑ,0) =u0(ϑ), inH.

(1.1)

Here, His the (2N+ 1)-dimensional Heisenberg group,H=H×(0,∞) and u0∈ L¹_loc(H). The operator A: H×R×R^2N+1 →R^2N+1 is a Carath´eodory function satisfying

(A(ϑ, ξ, v), v)≥cA|A(ϑ, ξ, v)|^m0, (1.2) where c_A > 0, (·,·) is the standard scalar product in R^2N+1, | · | = p

(·,·), and m⁰>1. The functionf :H×R×R^2N+1→Ris continuous and satisfies

f(ϑ, ξ, v)≤λ|A(ϑ, ξ, v)|^σ, (1.3) whereλ≥0,σ= ^m_q+1^0q, and q >max{1, m−1}, withm= _m^m0−1⁰ . The proof of our main result is based on a duality argument [4, 5, 6].

First, let us recall some background facts that will be used in this article. The (2N + 1)-dimensional Heisenberg group H is the space R^2N+1 endowed with the group operation

ϑϑ⁰= (x+x⁰, y+y⁰, τ+τ⁰+ 2(x·y⁰−x⁰·y)),

for allϑ= (x, y, τ), ϑ⁰ = (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, 35R03.

Key words and phrases. Nonexistence; global solution; differential inequality;

Heisenberg group.

c

2015 Texas State University - San Marcos.

Submitted May 15, 2015. Published June 17, 2015.

1

The distance from an elementϑ= (x, y, τ)∈Hto the origin is given by

|ϑ|_H=

τ²+X^N

i=1

x²_i +y²_i^21/4 , wherex= (x₁, . . . , x_N) andy= (y₁, . . . , y_N).

The Gradient∇_H overHis defined by

∇_H= (X₁, .., X_N, Y₁, .., Y_N), where fori= 1, . . . , N,

Xi=∂x_i+ 2yi∂τ and Yi=∂y_i−2xi∂τ. Let

M =

IN 0 2y

0 IN −2x

, whereIN is the identity matrix of sizeN. Then

∇_H=M∇_R2N+1. A simple computation gives the expression

|∇_Hu|²= 4(|x|²+|y|²)(∂τu)²+

N

X

i=1

(∂x_iu)²+ (∂y_iu)²+ 4∂τu(yi∂x_iu−xi∂y_iu) . The divergence operator inHis defined by

div_H(u) = div_R2N+1(M u).

For more details on Heisenberg groups and partial differential equations in Heisen- berg groups, we refer to [1, 2, 3, 8, 9] and references therein.

For the proof of our main result, the following inequality will be used several times.

Lemma 1.1. Let a, b, ε >0. Then

ab≤εa^p+cεb^p0,

where p > 1, p⁰ is its corresponding conjugate exponent, i.e., ¹_p + _p¹0 = 1; and cε= _εp^1p0/p1

p⁰.

2. Main result

Definition 2.1. Letu∈W_loc^1,m(H;R+)∩L^q_loc(H;R+) andu0 ∈ L¹_loc(H;R+). We say thatuis a global weak solution of (1.1) if the following conditions are satisfied:

(i) A(ϑ, u,∇_Hu)∈L^m_loc⁰(H;R^2N+1);

(ii) For anyϕ∈W_loc^1,m(H;R+) with a compact support, Z

H

u^qϕ dH ≤ Z

H

(A(ϑ, u,∇_Hu),∇_Hϕ)dH+ Z

H

f(ϑ, u,∇_Hu)ϕ dH

− Z

H

uϕtdH − Z

H

u0(ϑ)ϕ(ϑ,0)dϑ.

(2.1)

Observe that all the integrals in (2.1) are well defined. Our main result is given in the following theorem.

Theorem 2.2. Assume that conditions (1.2) and (1.3)are satisfied. Let us con- siderα∈(α0,0), whereα0= max{−1,1−m}<0. If

0≤λ < λ^∗= (q+ 1)|α|cA

q _q+1^q

(2.2) and

max{1, m−1}< q < m−1 + m

Q, (2.3)

where Q = 2N+ 2 is the homogeneous dimension of H, then (1.1) has no global nontrivial weak solutions.

The following lemma provides a preliminary estimate of solutions.

Lemma 2.3. Suppose that all the assumptions of Theorem 2.2 are satisfied. Let u be a global weak solution to (1.1). Then for any α ∈ (α₀,0) and any ϕ ∈ W^1,∞(H;R+), we have

Z

H

u^q+αϕ dH+ Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1ϕ dH+ Z

H

u₀(ϑ)^α+1ϕ(ϑ,0)dϑ

≤CZ

H

|ϕt|^r ϕ

_r−1¹ dH+

Z

H

ϕ^1−ms|∇_Hϕ|^msdH ,

(2.4)

for some constant C >0, wherer= ^q+α_1+α, ands=_q−m+1^q+α .

Proof. Letε >0 be fixed andα∈(α₀,0). Suppose thatuis a global weak solution to (1.1). Let

u_ε(ϑ, t) =u(ϑ, t) +ε, (ϑ, t)∈ H.

Defineϕ_εas

ϕ_ε(ϑ, t) =u^α_ε(ϑ, t)ϕ(ϑ, t),

where ϕ∈W^1,∞(H;R+) has a compact support. Observe thatϕε belongs to the set of admissible test functions in the sense of Definition 2.1. By (2.1), we have

Z

H

u^qu^α_εϕ dH

≤α Z

H

(A(ϑ, u,∇_Hu),∇_Hu)u^α−1_ε ϕ dH+ Z

H

(A(ϑ, u,∇_Hu),∇_Hϕ)u^α_εdH +

Z

H

f(ϑ, u,∇_Hu)u^α_εϕ dH − 1 α+ 1

Z

H

u^α+1_ε ϕ_tdH

− 1 α+ 1

Z

H

(u₀(ϑ) +ε)^α+1ϕ(ϑ,0)dϑ.

(2.5)

Using the condition (1.2), we obtain Z

H

(A(ϑ, u,∇_Hu),∇_Hu)u^α−1_ε ϕ dH ≥cA

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH.

Sinceα <0, we have α

Z

H

(A(ϑ, u,∇_Hu),∇_Hu)u^α−1_ε ϕ dH ≤cAα Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH. (2.6) The Cauchy-Schwarz inequality yields

Z

H

(A(ϑ, u,∇_Hu),∇_Hϕ)u^α_εdH ≤ Z

H

|A(ϑ, u,∇_Hu)||∇_Hϕ|u^α_εdH. (2.7)

Using the condition (1.3), we obtain Z

H

f(ϑ, u,∇_Hu)u^α_εϕ dH ≤λ Z

H

|A(ϑ, u,∇_Hu)|^σu^α_εϕ dH. (2.8) Now, (2.5), (2.6), (2.7) and (2.8) yield

Z

H

u^qu^α_εϕ dH+cA|α|

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH

+ 1

α+ 1 Z

H

(u0(ϑ) +ε)^α+1ϕ(ϑ,0)dϑ

≤ Z

H

|A(ϑ, u,∇_Hu)||∇_Hϕ|u^α_εdH+λ Z

H

|A(ϑ, u,∇_Hu)|^σu^α_εϕ dH

+ 1

α+ 1 Z

H

u^α+1_ε |ϕt|dH.

(2.9)

Now, using Lemma 1.1, we estimate the individual terms on the right hand side of (2.9).

•Estimation ofR

H|A(ϑ, u,∇_Hu)||∇_Hϕ|u^α_εdH. We have

|A(ϑ, u,∇_Hu)||∇_Hϕ|u^α_ε =

|A(ϑ, u,∇_Hu)|u

α−1 m0

ε ϕ^m10 u

α+m−1

ε m ϕ^−1m0|∇_Hϕ|

. Applying Lemma 1.1 with parametersm⁰ andm, we obtain

Z

H

|A(ϑ, u,∇_Hu)||∇_Hϕ|u^α_εdH

≤ε₁ Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH+c_ε1 Z

H

u^α+m−1_ε ϕ^1−m|∇_Hϕ|^mdH, for someε1>0. Again, writing

u^α+m−1_ε ϕ^1−m|∇_Hϕ|^m=

ϕ^1−mss |∇_Hϕ|^m ϕ^s−1s u^α+m−1_ε and using Lemma 1.1 with parameterssands⁰, we obtain

Z

H

u^α+m−1_ε ϕ^1−m|∇_Hϕ|^mdH ≤ε₂ Z

H

ϕ^1−ms|∇_Hϕ|^msdH+c_ε2 Z

H

ϕu^(α+m−1)s_ε ⁰dH, for someε2>0. As consequence, we have

Z

H

|A(ϑ, u,∇_Hu)||∇_Hϕ|u^α_εdH

≤ε1

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH +cε₁ε2

Z

H

ϕ^1−ms|∇_Hϕ|^msdH+cε₁cε₂

Z

H

ϕu^(α+m−1)s_ε ⁰dH.

(2.10)

•Estimation ofR

H|A(ϑ, u,∇_Hu)|^σu^α_εϕ dH. We write

|A(ϑ, u,∇_Hu)|^σu^α_εϕ=

|A(ϑ, u,∇_Hu)|^σu

(α−1)σ m0

ε ϕ^mσ0 ϕ^{m0 −σm0} u

αm0 −(α−1)σ m0

ε

. We apply Lemma 1.1 with parameters ^m_σ⁰ and _m^m0−σ⁰ to obtain

Z

H

|A(ϑ, u,∇_Hu)|^σu^α_εϕ dH

≤ε3

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH+cε₃

Z

H

ϕu^α−1+

m0 m0 −σ

ε dH,

for someε3>0. Sinceσ=m⁰q/(q+ 1), we obtain Z

H

|A(ϑ, u,∇_Hu)|^σu^α_εϕ dH

≤ε3

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH+cε₃

Z

H

ϕu^α+q_ε dH.

(2.11)

•Estimation ofR

Hu^α+1_ε |ϕt|dH. Similarly, we write u^α+1_ε |ϕt|=

u^α+1_ε ϕ^1r ϕ^−1r|ϕt| . Lemma 1.1 with parametersrandr⁰ yields

Z

H

u^α+1_ε |ϕt|dH ≤ε4

Z

H

u^(α+1)r_ε ϕ dH+cε₄

Z

H

|ϕt|^r ϕ

_r−1¹

dH, (2.12)

for someε4>0. Now, substituting (2.10), (2.11) and (2.12) in (2.9), we obtain Z

H

u^qu^α_εϕ dH+cA|α|

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH

+ 1

α+ 1 Z

H

(u0(ϑ) +ε)^α+1ϕ(ϑ,0)dϑ

≤ε1

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH+cε₁ε2

Z

H

ϕ^1−ms|∇_Hϕ|^msdH +cε₁cε₂

Z

H

ϕu^(α+m−1)s_ε ⁰dH+λε3

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1_ε ϕ dH +λcε₃

Z

H

ϕu^α+q_ε dH+ ε4

α+ 1 Z

H

u^(α+1)r_ε ϕ dH+ cε4

α+ 1 Z

H

|ϕt|^r ϕ

_r−1¹ dH.

Now, we let ε → 0 in the obtained inequality, we use Fatou’s lemma and the dominated convergence theorem to obtain

Z

H

u^q+αϕ dH+cA|α|

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1ϕ dH+ 1 α+ 1

Z

H

u^α+1₀ (ϑ)ϕ(ϑ,0)dϑ

≤ε1

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1ϕ dH+cε₁ε2

Z

H

ϕ^1−ms|∇_Hϕ|^msdH +cε₁cε₂

Z

H

ϕu^q+αdH+λε3

Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1ϕ dH +λcε₃

Z

H

ϕu^α+qdH+ ε4

α+ 1 Z

H

u^q+αϕ dH+ cε₄

α+ 1 Z

H

|ϕt|^r ϕ

_r−1¹ dH, i.e.,

A Z

H

u^q+αϕ dH+B Z

H

|A(ϑ, u,∇_Hu)|^m0u^α−1ϕ dH

+ 1

α+ 1 Z

H

u^α+1₀ (ϑ)ϕ(ϑ,0)dϑ

≤max

cε₁ε2, cε₄

α+ 1 Z

H

ϕ^1−ms|∇_Hϕ|^msdH+ Z

H

|ϕt|^r ϕ

_r−1¹ dH

,

(2.13)

where

A= 1−λcε₃− ε4

α+ 1−cε₁cε₂ and B=cA|α| −ε1−λε3. From Lemma 1.1, we have

cε₁ = 1 m

1 ε1m⁰

^m0/m

, cε₂ = 1 s⁰

1 ε2s

^s0/s , c_ε3 = 1

q+ 1 q

ε3(q+ 1) ^q

, c_ε4 = 1 r⁰

1 ε4r

^r0/r . Forε1>0 small enough, taking

0≤λ < cA|α|

ε₃ , (2.14)

we obtainB>0. Forεi>0 small enough (i= 1,2,4), taking 0≤λ < 1

cε₃

, (2.15)

we obtainA>0. Now, we chooseε3>0 such that cA|α|

ε₃ = 1 c_ε3, i.e.,

cA|α|

ε₃ = (q+ 1) q ε₃(q+ 1)

−q

. A simple computation yields

ε3= (cA|α|)^q+11 q q+ 1

_q+1^q 1 q+ 1

_q+1¹ . We substituteε₃into (2.14) (or (2.15)) to get

0≤λ < λ^∗= (q+ 1)|α|cA

q _q+1^q

.

Thus, for 0≤λ < λ^∗and εi>0 small enough (i= 1,2,4), we have

A>0 and B>0. (2.16)

Finally, the desired result follows from (2.13) and (2.16) with C= max{cε₁ε2,_α+1^cε4 }

min{A,B,_α+1¹ } .

The lemma is proved.

Proof of Theorem 2.2. Suppose thatuis a nontrivial global weak solution to (1.1).

Let us consider the test function

ϕR(ϑ, t) =ϕR(x, y, τ, t) =φ^ωt^2θ1+|x|^4θ2+|y|^4θ2+τ^2θ2 R^4θ2

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

φ(z) =

(1 if 0≤z≤1 0 ifz≥2,

andθj, j= 1,2 are positive parameters, whose exact values will be specified later.

Let

ρ=t^2θ1+|x|^4θ2+|y|^4θ2+τ^2θ2

R^4θ2 .

ClearlyϕRis supported on

Ω_R={(ϑ, t)∈ H: 0≤ρ≤2}, while (ϕR)tand ∇_HϕRare supported on

ΘR={(ϑ, t)∈ H: 1≤ρ≤2}.

A simple computation yields

∂tϕR(ϑ, t) = 2θ1ωt^2θ1−1R^−4θ2φ^ω−1(ρ)φ⁰(ρ), while

∇_HϕR(t, ϑ)|²= 16θ²₂ω²R^−8θ2(φ⁰(ρ))²φ^2ω−2(ρ)

(|x|²+|y|²)τ^4θ2−2 + (|x|^8θ2−2+|y|^8θ2−2) + 2τ^2θ2−1

N

X

i=1

xiyi(|x|^4θ2−2− |y|^4θ2−2) . Then, for all (ϑ, t)∈Ω_R, we have

R|∇_HϕR|+R^2θ2/θ1|∂tϕR| ≤C|φ⁰(ρ)|φ^ω−1(ρ). (2.17) For simplicity, in the sequel, we will write ϕ in the place of ϕR. Let us consider now the change of variables

(x, y, τ, t) = (ϑ, t)7→(ex,y,e eτ ,et) = (eϑ,et), where

et=R^−2θ2/θ1t, xe=R⁻¹x, ey=R⁻¹y, τe=R⁻²τ.

In the same way, let

ρe=et^2θ1+|ex|^4θ2+|y|e^4θ2+τe^2θ2, Ω =e {(x,e y,e eτ ,et)∈ H: 0≤ρe≤2}, Θ =e {(ex,y,eτ ,e et)∈ H: 1≤ρe≤2}.

Using the above change of variables together with (2.17), we obtained Z

H

|ϕ_t|^r ϕ

r−1¹

dH ≤CR^{Q+2θθ21(1−r−1r )} Z

H

φ^ω−r−1r |φ⁰|^r−1r dHe (2.18)

and Z

H

ϕ^1−ms|∇_Hϕ|^msdH ≤CR^Q+2

θ2 θ1−msZ

H

φ^ω−ms|φ⁰|^msdH.e (2.19) Setting

θ2

θ1

=ms(r−1) 2r , we have

Q+ 2θ2

θ₁(1− r

r−1) =Q+ 2θ2

θ₁ −ms=Q− m(q+α)

q−m+ 1 + m(q−1)

q−m+ 1. (2.20) Using (2.4), (2.18)-(2.20), we obtain

Z

H

u^q+αϕ dH ≤CR^{Q−m(q+α)q−m+1+m(q−1)kq−m+1}. (2.21)

Furthermore, noting that

Q− m(q+α)

q−m+ 1 + m(q−1) q−m+ 1 <0

forq < m−1 +^m_Q and someα∈(α0,0) small enough. Under the above condition, lettingR→ ∞in (2.21) and using the monotone convergence theorem, we obtain

Z

H

u^q+αdH ≤0,

which contradicts our assumption aboutu. This completes the proof.

Let us consider now some examples where Theorem 2.2 can be applied.

Corollary 2.4. If max{1, m−1}< q < m−1 +^m_Q, then the problem ut−div_H(|∇_Hu|^m−2∇_Hu)≥u^q inH,

u≥0, a.e. inH, u(ϑ,0) =u₀(ϑ), in H,

whereu0∈L¹_loc(H;R+), has no nontrivial global weak solution.

Proof. The result follows from Theorem 2.2 withλ= 0 and A(ϑ, u,∇_H) =|∇_Hu|^m−2∇_Hu.

Observe that condition (1.2) is satisfied withc_A= 1.

Takem= 2 in Corollary 2.4, we obtain the following Heisenberg version of Fujita theorem (see [7]).

Corollary 2.5. If 1< q <1 + _Q², then the problem ut−∆_Hu≥u^q inH,

u≥0, a.e. inH, u(ϑ,0) =u0(ϑ), in H,

whereu0∈L¹_loc(H;R+), has no nontrivial global weak solution.

Corollary 2.6. If 1< q <1 + _Q², then the problem ut−div_H |∇_Hu|

p1 +|∇_Hu|²

≥u^q inH, u≥0, a.e. inH,

u(ϑ,0) =u₀(ϑ), in H,

whereu₀∈L¹_loc(H;R+), has no nontrivial global weak solution.

Proof. The result follows from Theorem 2.2 withλ= 0 and A(ϑ, u,∇_H) = |∇_Hu|

p1 +|∇_Hu|².

Observe that condition (1.2) is satisfied withcA= 1.

Note that Corollary 2.6 is a Heisenberg version of [5, Corollary 33.3].

Acknowledgments. The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No RGP-1435-034.

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Ibtehal Azman

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

E-mail address:[email protected]

Mohamed Jleli

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

E-mail address:[email protected]

Bessem Samet

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

E-mail address:[email protected]