Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 167, pp. 1–9.
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
ut − divH A(ϑ, u, ∇H u) + f (ϑ, u, ∇H u) ≥ uq ,
u ≥ 0,
in H,
a.e. in H,
u(ϑ, 0) = u0 (ϑ),
(1.1)
in H.
Here, H is the (2N + 1)-dimensional Heisenberg group, H = H × (0, ∞) and u0 ∈
L1loc (H). The operator A : H × R × R2N +1 → R2N +1 is a Carathéodory function
satisfying
0
(A(ϑ, ξ, v), v) ≥ cA |A(ϑ, ξ, v)|m ,
(1.2)
p
2N +1
where cA > 0, (·, ·) is the standard scalar product in R
, | · | = (·, ·), and
m0 > 1. The function f : H × R × R2N +1 → R is continuous and satisfies
f (ϑ, ξ, v) ≤ λ|A(ϑ, ξ, v)|σ ,
m0 q
q+1 ,
(1.3)
0
where λ ≥ 0, σ =
and q > max{1, m − 1}, with m = mm
0 −1 . 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 R2N +1 endowed with the
group operation
ϑ ϑ0 = (x + x0 , y + y 0 , τ + τ 0 + 2(x · y 0 − x0 · y)),
for all ϑ = (x, y, τ ), ϑ0 = (x0 , y 0 , τ 0 ) ∈ RN × RN × R, where · denotes the standard
scalar product in RN . This group operation endows H with the structure of a Lie
group.
2010 Mathematics 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
2
I. AZMAN, M. JLELI, B. SAMET
EJDE-2015/167
The distance from an element ϑ = (x, y, τ ) ∈ H to the origin is given by
N
X
2 1/4
|ϑ|H = τ 2 +
x2i + yi2
,
i=1
where x = (x1 , . . . , xN ) and y = (y1 , . . . , yN ).
The Gradient ∇H over H is defined by
∇H = (X1 , .., XN , Y1 , .., YN ),
where for i = 1, . . . , N ,
and Yi = ∂yi − 2xi ∂τ .
Xi = ∂xi + 2yi ∂τ
Let
IN
0
2y
,
0
IN
−2x
is the identity matrix of size N . Then
M=
where IN
∇H = M ∇R2N +1 .
A simple computation gives the expression
N X
|∇H u| = 4(|x| + |y| )(∂τ u) +
(∂xi u)2 + (∂yi u)2 + 4∂τ u(yi ∂xi u − xi ∂yi u) .
2
2
2
2
i=1
The divergence operator in H is defined by
divH (u) = divR2N +1 (M u).
For more details on Heisenberg groups and partial differential equations in Heisenberg 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
0
ab ≤ εap + cε bp ,
where p > 1, p0 is its corresponding conjugate exponent, i.e.,
0
1 p /p 1
cε = εp
p0 .
1
p
+
1
p0
= 1; and
2. Main result
1,m
Wloc
(H; R+ )
Definition 2.1. Let u ∈
∩ Lqloc (H; R+ ) and u0 ∈ L1loc (H; R+ ). We
say that u is a global weak solution of (1.1) if the following conditions are satisfied:
0
2N +1
);
(i) A(ϑ, u, ∇H u) ∈ Lm
loc (H; R
1,m
(ii) For any ϕ ∈ Wloc (H; R+ ) with a compact support,
Z
Z
Z
uq ϕ dH ≤
(A(ϑ, u, ∇H u), ∇H ϕ) dH +
f (ϑ, u, ∇H u)ϕ dH
H
H
H
Z
Z
−
uϕt dH −
u0 (ϑ)ϕ(ϑ, 0) dϑ.
H
(2.1)
H
Observe that all the integrals in (2.1) are well defined. Our main result is given
in the following theorem.
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BLOW-UP OF SOLUTIONS
3
Theorem 2.2. Assume that conditions (1.2) and (1.3) are satisfied. Let us consider α ∈ (α0 , 0), where α0 = max{−1, 1 − m} < 0. If
q
|α|c q+1
A
0 ≤ λ < λ∗ = (q + 1)
(2.2)
q
and
m
(2.3)
max{1, m − 1} < q < m − 1 + ,
Q
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
Z
Z
0
α+1
uq+α ϕ dH +
|A(ϑ, u, ∇H u)|m uα−1 ϕ dH +
u0 (ϑ)
ϕ(ϑ, 0) dϑ
H
H
H
(2.4)
Z
1
Z |ϕ |r r−1
t
1−ms
ms
dH +
ϕ
|∇H ϕ| dH ,
≤C
ϕ
H
H
for some constant C > 0, where r =
q+α
1+α ,
and s =
q+α
q−m+1 .
Proof. Let ε > 0 be fixed and α ∈ (α0 , 0). Suppose that u is a global weak solution
to (1.1). Let
uε (ϑ, t) = u(ϑ, t) + ε, (ϑ, t) ∈ H.
Define ϕε as
ϕε (ϑ, t) = uα
ε (ϑ, t)ϕ(ϑ, t),
where ϕ ∈ W 1,∞ (H; R+ ) has a compact support. Observe that ϕε belongs to the
set of admissible test functions in the sense of Definition 2.1. By (2.1), we have
Z
uq uα
ε ϕ dH
H
Z
Z
α−1
≤ α (A(ϑ, u, ∇H u), ∇H u)uε ϕ dH +
(A(ϑ, u, ∇H u), ∇H ϕ)uα
ε dH
H
H
Z
Z
(2.5)
1
α+1
+
f (ϑ, u, ∇H u)uα
ϕ
dH
−
u
ϕ
dH
t
ε
α+1 H ε
H
Z
1
(u0 (ϑ) + ε)α+1 ϕ(ϑ, 0) dϑ.
−
α+1 H
Using the condition (1.2), we obtain
Z
Z
0
(A(ϑ, u, ∇H u), ∇H u)uεα−1 ϕ dH ≥ cA
|A(ϑ, u, ∇H u)|m uεα−1 ϕ dH.
H
H
Since α < 0, we have
Z
Z
0
α−1
α (A(ϑ, u, ∇H u), ∇H u)uε ϕ dH ≤ cA α
|A(ϑ, u, ∇H u)|m uεα−1 ϕ dH.
(2.6)
The Cauchy-Schwarz inequality yields
Z
Z
α
(A(ϑ, u, ∇H u), ∇H ϕ)uε dH ≤
|A(ϑ, u, ∇H u)||∇H ϕ|uα
ε dH.
(2.7)
H
H
H
H
4
I. AZMAN, M. JLELI, B. SAMET
EJDE-2015/167
Using the condition (1.3), we obtain
Z
Z
α
f (ϑ, u, ∇H u)uε ϕ dH ≤ λ
|A(ϑ, u, ∇H u)|σ uα
ε ϕ dH.
H
(2.8)
H
Now, (2.5), (2.6), (2.7) and (2.8) yield
Z
Z
0
uq uα
ϕ
dH
+
c
|α|
|A(ϑ, u, ∇H u)|m uεα−1 ϕ dH
A
ε
H
H
Z
1
(u0 (ϑ) + ε)α+1 ϕ(ϑ, 0) dϑ
+
α+1 H
Z
Z
≤
|A(ϑ, u, ∇H u)||∇H ϕ|uα
dH
+
λ
|A(ϑ, u, ∇H u)|σ uα
ε
ε ϕ dH
H
H
Z
1
uα+1 |ϕt | dH.
+
α+1 H ε
(2.9)
Now, using Lemma 1.1, we estimate the individual terms on the right hand side of
(2.9).
R
• Estimation of H |A(ϑ, u, ∇H u)||∇H ϕ|uα
ε dH. We have
α+m−1 −1
α−1
1
m0
m
m0
m0 |∇ ϕ| .
|A(ϑ, u, ∇H u)||∇H ϕ|uα
=
|A(ϑ,
u,
∇
u)|u
ϕ
u
ϕ
ε
ε
H
H
ε
Applying Lemma 1.1 with parameters m0 and m, we obtain
Z
|A(ϑ, u, ∇H u)||∇H ϕ|uα
ε dH
H
Z
Z
0
≤ ε1
|A(ϑ, u, ∇H u)|m uεα−1 ϕ dH + cε1
uεα+m−1 ϕ1−m |∇H ϕ|m dH,
H
H
for some ε1 > 0. Again, writing
1−ms
s−1
uεα+m−1 ϕ1−m |∇H ϕ|m = ϕ s |∇H ϕ|m ϕ s uα+m−1
ε
and using Lemma 1.1 with parameters s and s0 , we obtain
Z
Z
Z
0
1−m
m
1−ms
ms
uα+m−1
ϕ
|∇
ϕ|
dH
≤
ε
ϕ
|∇
ϕ|
dH
+
c
ϕuε(α+m−1)s dH,
H
2
H
ε2
ε
H
H
H
for some ε2 > 0. As consequence, we have
Z
|A(ϑ, u, ∇H u)||∇H ϕ|uα
ε dH
H
Z
0
≤ ε1
|A(ϑ, u, ∇H u)|m uεα−1 ϕ dH
H
Z
Z
0
1−ms
ms
+ cε1 ε2
ϕ
|∇H ϕ| dH + cε1 cε2
ϕuε(α+m−1)s dH.
H
• Estimation of
H
|A(ϑ, u, ∇H u)|σ uα
ε ϕ dH.
H
R
We write
(α−1)σ
m0
σ
|A(ϑ, u, ∇H u)|σ uα
ε ϕ = |A(ϑ, u, ∇H u)| uε
We apply Lemma 1.1 with parameters
Z
|A(ϑ, u, ∇H u)|σ uα
ε ϕ dH
H
(2.10)
m0
σ
and
σ
ϕ m0
m0
m0 −σ
ϕ
m0 −σ
m0
to obtain
αm0 −(α−1)σ
m0
uε
.
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BLOW-UP OF SOLUTIONS
Z
0
≤ ε3
H
|A(ϑ, u, ∇H u)|m uεα−1 ϕ dH + cε3
Z
5
0
α−1+ mm
0 −σ
ϕuε
dH,
H
for some ε3 > 0. Since σ = m0 q/(q + 1), we obtain
Z
|A(ϑ, u, ∇H u)|σ uα
ε ϕ dH
H
Z
Z
m0 α−1
≤ ε3
|A(ϑ, u, ∇H u)| uε ϕ dH + cε3
ϕuα+q
dH.
ε
H
• Estimation of
R
(2.11)
H
uα+1 |ϕt | dH.
H ε
Similarly, we write
1
1
ϕ− r |ϕt | .
uα+1
|ϕt | = uα+1
ϕr
ε
ε
Lemma 1.1 with parameters r and r0 yields
Z
Z
Z 1
|ϕt |r r−1
α+1
(α+1)r
dH,
uε |ϕt | dH ≤ ε4
uε
ϕ dH + cε4
ϕ
H
H
H
(2.12)
for some ε4 > 0. Now, substituting (2.10), (2.11) and (2.12) in (2.9), we obtain
Z
Z
0
q α
u uε ϕ dH + cA |α|
|A(ϑ, u, ∇H u)|m uα−1
ϕ dH
ε
H
H
Z
1
+
(u0 (ϑ) + ε)α+1 ϕ(ϑ, 0) dϑ
α+1 H
Z
Z
0
≤ ε1
|A(ϑ, u, ∇H u)|m uα−1
ϕ
dH
+
c
ε
ϕ1−ms |∇H ϕ|ms dH
ε
2
ε
1
H
H
Z
Z
0
(α+m−1)s0
+ cε1 cε2
ϕuε
dH + λε3
|A(ϑ, u, ∇H u)|m uεα−1 ϕ dH
H
H
Z
Z Z
1
cε4
|ϕt |r r−1
ε
4
(α+1)r
α+q
uε
ϕ dH +
dH.
+ λcε3
ϕuε dH +
α+1 H
α+1 H
ϕ
H
Now, we let ε → 0 in the obtained inequality, we use Fatou’s lemma and the
dominated convergence theorem to obtain
Z
Z
Z
0
1
uα+1
(ϑ)ϕ(ϑ, 0) dϑ
uq+α ϕ dH + cA |α|
|A(ϑ, u, ∇H u)|m uα−1 ϕ dH +
0
α
+
1
H
H
H
Z
Z
0
≤ ε1
|A(ϑ, u, ∇H u)|m uα−1 ϕ dH + cε1 ε2
ϕ1−ms |∇H ϕ|ms dH
H
H
Z
Z
0
q+α
+ cε1 cε2
ϕu
dH + λε3
|A(ϑ, u, ∇H u)|m uα−1 ϕ dH
H
H
Z
Z
Z 1
cε4
|ϕt |r r−1
ε
4
q+α
α+q
u
ϕ dH +
dH,
+ λcε3
ϕu
dH +
α+1 H
α+1 H
ϕ
H
i.e.,
Z
Z
0
uq+α ϕ dH + B
|A(ϑ, u, ∇H u)|m uα−1 ϕ dH
H
H
Z
1
+
uα+1 (ϑ)ϕ(ϑ, 0) dϑ
α+1 H 0
Z
Z 1
cε4 |ϕt |r r−1
1−ms
ms
≤ max cε1 ε2 ,
dH ,
ϕ
|∇H ϕ| dH +
α+1
ϕ
H
H
A
(2.13)
6
I. AZMAN, M. JLELI, B. SAMET
where
A = 1 − λcε3 −
ε4
− cε1 cε2
α+1
EJDE-2015/167
and B = cA |α| − ε1 − λε3 .
From Lemma 1.1, we have
0
0
1 1 m /m
1 1 s /s
,
c
=
,
ε
2
m ε1 m0
s0 ε2 s
0
q
q
1 1 r /r
1 , cε4 = 0
.
=
q + 1 ε3 (q + 1)
r ε4 r
cε1 =
cε3
For ε1 > 0 small enough, taking
0≤λ<
cA |α|
,
ε3
(2.14)
we obtain B > 0. For εi > 0 small enough (i = 1, 2, 4), taking
0≤λ<
1
,
cε3
(2.15)
we obtain A > 0. Now, we choose ε3 > 0 such that
cA |α|
1
,
=
ε3
cε3
i.e.,
−q
cA |α|
q
= (q + 1)
.
ε3
ε3 (q + 1)
A simple computation yields
q 1
q q+1
1
1 q+1
ε3 = (cA |α|) q+1
.
q+1
q+1
We substitute ε3 into (2.14) (or (2.15)) to get
q
|α|c q+1
A
0 ≤ λ < λ∗ = (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
C=
The lemma is proved.
ε4
max{cε1 ε2 , α+1
}
1
min{A, B, α+1
}
.
Proof of Theorem 2.2. Suppose that u is a nontrivial global weak solution to (1.1).
Let us consider the test function
t2θ1 + |x|4θ2 + |y|4θ2 + τ 2θ2 ϕR (ϑ, t) = ϕR (x, y, τ, t) = φω
, R > 0, ω 1,
R4θ2
where φ ∈ C0∞ (R+ ) is a decreasing function satisfying
(
1 if 0 ≤ z ≤ 1
φ(z) =
0 if z ≥ 2,
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BLOW-UP OF SOLUTIONS
7
and θj , j = 1, 2 are positive parameters, whose exact values will be specified later.
Let
t2θ1 + |x|4θ2 + |y|4θ2 + τ 2θ2
.
ρ=
R4θ2
Clearly ϕR is supported on
ΩR = {(ϑ, t) ∈ H : 0 ≤ ρ ≤ 2},
while (ϕR )t and ∇H ϕR are supported on
ΘR = {(ϑ, t) ∈ H : 1 ≤ ρ ≤ 2}.
A simple computation yields
∂t ϕR (ϑ, t) = 2θ1 ωt2θ1 −1 R−4θ2 φω−1 (ρ)φ0 (ρ),
while
∇H ϕR (t, ϑ)|2 = 16θ22 ω 2 R−8θ2 (φ0 (ρ))2 φ2ω−2 (ρ) (|x|2 + |y|2 )τ 4θ2 −2
+ (|x|8θ2 −2 + |y|8θ2 −2 ) + 2τ 2θ2 −1
N
X
xi yi (|x|4θ2 −2 − |y|4θ2 −2 ) .
i=1
Then, for all (ϑ, t) ∈ ΩR , we have
R|∇H ϕR | + R2θ2 /θ1 |∂t ϕR | ≤ C|φ0 (ρ)|φω−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
ee
(x, y, τ, t) = (ϑ, t) 7→ (e
x, ye, τe, e
t) = (ϑ,
t),
where
e
t = R−2θ2 /θ1 t,
In the same way, let
x
e = R−1 x,
ye = R−1 y,
τe = R−2 τ.
ρe = e
t2θ1 + |e
x|4θ2 + |e
y |4θ2 + τe2θ2 ,
e = {(e
Ω
x, ye, τe, e
t) ∈ H : 0 ≤ ρe ≤ 2},
e = {(e
Θ
x, ye, τe, e
t) ∈ H : 1 ≤ ρe ≤ 2}.
Using the above change of variables together with (2.17), we obtained
Z Z
1
θ2
r
r
r
|ϕt |r r−1
e
d H ≤ CRQ+2 θ1 (1− r−1 )
φω− r−1 |φ0 | r−1 dH
ϕ
H
H
and
Z
Z
θ2
e
ϕ1−ms |∇H ϕ|ms dH ≤ CRQ+2 θ1 −ms
φω−ms |φ0 |ms dH.
H
(2.18)
(2.19)
H
Setting
θ2
ms(r − 1)
=
,
θ1
2r
we have
θ2
r
θ2
m(q + α)
m(q − 1)
(1 −
) = Q + 2 − ms = Q −
+
.
θ1
r−1
θ1
q−m+1 q−m+1
Using (2.4), (2.18)-(2.20), we obtain
Z
m(q+α)
m(q−1)k
uq+α ϕ dH ≤ CRQ− q−m+1 + q−m+1 .
Q+2
H
(2.20)
(2.21)
8
I. AZMAN, M. JLELI, B. SAMET
EJDE-2015/167
Furthermore, noting that
Q−
m(q + α)
m(q − 1)
+
<0
q−m+1 q−m+1
for q < m − 1 + m
Q and some α ∈ (α0 , 0) small enough. Under the above condition,
letting R → ∞ in (2.21) and using the monotone convergence theorem, we obtain
Z
uq+α dH ≤ 0,
H
which contradicts our assumption about u. 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 − divH (|∇H u|m−2 ∇H u) ≥ uq
u ≥ 0,
in H,
a.e. in H,
u(ϑ, 0) = u0 (ϑ),
in H,
where u0 ∈ L1loc (H; R+ ), has no nontrivial global weak solution.
Proof. The result follows from Theorem 2.2 with λ = 0 and
A(ϑ, u, ∇H ) = |∇H u|m−2 ∇H u.
Observe that condition (1.2) is satisfied with cA = 1.
Take m = 2 in Corollary 2.4, we obtain the following Heisenberg version of Fujita
theorem (see [7]).
Corollary 2.5. If 1 < q < 1 +
2
Q,
then the problem
ut − ∆H u ≥ uq
u ≥ 0,
in H,
a.e. in H,
u(ϑ, 0) = u0 (ϑ),
in H,
where u0 ∈ L1loc (H; R+ ), has no nontrivial global weak solution.
Corollary 2.6. If 1 < q < 1 +
2
Q,
then the problem
|∇H u|
≥ uq
ut − divH p
1 + |∇H u|2
u ≥ 0, a.e. in H,
u(ϑ, 0) = u0 (ϑ),
where u0 ∈
L1loc (H; R+ ),
in H,
in H,
has no nontrivial global weak solution.
Proof. The result follows from Theorem 2.2 with λ = 0 and
A(ϑ, u, ∇H ) = p
|∇H u|
1 + |∇H u|2
.
Observe that condition (1.2) is satisfied with cA = 1.
Note that Corollary 2.6 is a Heisenberg version of [5, Corollary 33.3].
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BLOW-UP OF SOLUTIONS
9
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: ibtehalazman@yahoo.com
Mohamed Jleli
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455,
Riyadh 11451, Saudi Arabia
E-mail address: jleli@ksu.edu.sa
Bessem Samet
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455,
Riyadh 11451, Saudi Arabia
E-mail address: bsamet@ksu.edu.sa