Electronic Journal of Differential Equations, Vol. 2002(2002), No. 97, pp. 1–19.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Nonexistence of solutions to systems of
higher-order semilinear inequalities
in cone-like domains ∗
Abdallah El Hamidi & Gennady G. Laptev
Abstract
In this paper, we obtain nonexistence results for global solutions to
the system of higher-order semilinear partial differential inequalities
∂ k ui
− ∆(ai (x, t)ui (x, t)) ≥ tγi+1 |x|σi+1 |ui+1 (x, t)|pi+1 ,
∂tk
un+1 = u1 ,
1 ≤ i ≤ n,
in cones and cone-like domains in RN , t > 0. Our results apply to nonnegative solutions and to solutions which change sign. Moreover, we provide
a general formula of the critical exponent corresponding to this system.
Our proofs are based on the test function method, developed by Mitidieri
and Pohozaev.
1
Introduction
This paper is devoted to the study of nonexistence results for global solutions
to systems of semilinear higher-order evolution differential inequalities in unbounded cone-like domains. Nonexistence results concerning nonnegative solutions of parabolic equations in cones were obtained by Bandle & Levine [1] and
Levine & Meier [18]. Recently, new nonexistence results dealing with solutions
with arbitrary sign were established by Laptev [11, 13, 14] and by El Hamidi
& Laptev [6] when the domains are cones or product of cones. On the other
hand, for cone-like domains, only nonexistence results of nonnegative solutions
to semilinear evolution differential inequalities, were obtained in [1, 6, 11, 13, 14].
Recently, Laptev [15] obtained a nonexistence result for the semilinear parabolic
inequality
ut − ∆(|u|m−1 u) ≥ |x|σ |u|q
with 1 ≤ m < q and σ > −2, in cone-like domains. Which is the first result,
to our knowledge, dealing with solutions of evolution problems which are not
necessarily nonnegative in cone-like domains.
∗ Mathematics Subject Classifications: 35G25, 35R45.
Key words: nonexistence, blow-up, higher-order differential inequalities, critical exponent.
c
2002
Southwest Texas State University.
Submitted March 10, 2002. Published November 14, 2002.
1
2
Nonexistence of solutions to systems
EJDE–2002/97
In this paper, we obtain nonexistence results for systems of semilinear higherorder evolution differential inequalities in unbounded cones and cone-like domains. More precisely, for n ≥ 2, we study the problem
∂ k ui
− ∆(ai ui ) ≥ tγi+1 |x|σi+1 |ui+1 |pi+1 ,
∂tk
un+1 = u1 ,
1 ≤ i ≤ n,
(1.1)
where x belongs to a cone (or a cone-like domain), t ∈]0, +∞[, k ≥ 1, pn+1 = p1 ,
γn+1 = γ1 and σn+1 = σ1 .
For n = 1, we study the problem
∂ku
− ∆(au) ≥ tγ |x|σ |u|p ,
∂tk
(1.2)
where x belongs to a cone (or a cone-like domain) and t ∈]0, +∞[. Such systems
were studied, in the whole space, by Renclawowicz [28], Guedda & Kirane [7],
Igbida & Kirane [8] and Kirane, Nabana & Pohozaev [10].
Our results concern all weak solutions, specially nonnegative weak solutions.
We obtain general formulas of the critical exponents corresponding to the systems considered. These formulas are also valid in the scalar case of one inequality
(n = 1).
Our approach is based on the test function method developed by Mitidieri &
Pohozaev [19], Pohozaev & Tesei [25], Pohozaev & Véron [27] and Laptev [11,
13, 14].
Let Ω ⊂ S N −1 be a connected submanifold of the unit sphere S N −1 in RN
with smooth boundary ∂Ω ⊂ S N −1 and having positive N − 2 dimensional
measure. By a cone in RN with cross section Ω with vertex at the origin, we
mean a set
K = {(r, ω) ∈ RN ; 0 < r < +∞ and ω ∈ Ω},
where r = |x|, x ∈ RN . The boundary of K is
∂K = {(r, ω);
r = 0 or ω ∈ ∂Ω}.
For ε > 0 fixed, the cone-like domain Kε is defined as
Kε = {x ∈ K; |x| > ε}
and its boundary as
∂Kε = {(r, ω); r = ε or ω ∈ ∂Ω}.
The outward normal vector to the boundary ∂Ω (resp. ∂K) will be denoted
by νω (resp. ν). The restriction of the laplacian operator ∆ to the unit sphere
S N −1 will be denoted by ∆ω , which is the Laplace-Beltrami operator. It is wellknown that the laplacian operator in RN can be written, in polar coordinates
(r, ω), as
∆=
1
rN −1
∂ N −1 ∂ 1
∂2
N −1 ∂
1
r
+ 2 ∆ω = 2 +
+ ∆ω .
∂r
∂r
r
∂r
r ∂r r2
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
3
for the rest of this paper, λ denotes the first Dirichlet eigenvalue, and Φ the
corresponding eigenfunction, for the Laplace-Beltrami operator; namely,
−∆ω Φ = λΦ in Ω,
Φ = 0 on ∂Ω.
Recall that λ > 0 and Φ(ω) > 0, for any ω ∈ Ω. We shall assume Φ is normalized
so that
0 < Φ(ω) ≤ 1, ∀ω ∈ Ω.
The space of the C 2 functions with respect to the first variable and C j , j ∈
N∗ , with respect to the second variable, on K×]0, +∞[, will be denoted by
C 2,j (K×]0, +∞[).
This article is organized as follows. In Section 2, we introduce notation
and establish estimates which we shall use in the sequel. Section 3 is devoted
to nonexistence results to the inequality (1.2), where the parameter γ = 0. In
Section 4, we generalize the results of the Section 3 for n ≥ 2 and the parameters
γi = 0, i ∈ {1, 2, . . . , n}. Section 5 concerns the general system (1.1), with
γi ≤ 0, i ∈ {1, 2, . . . , n}.
2
Preliminary results
Throughout this paper, the letter C denotes a constant which may vary from
line to line but is independent of the terms which will take part in any limit
process. For a real number p > 1, we define p0 such that 1/p + 1/p0 = 1.
We define now the weak solutions of the problems that we will consider in
the sequel. Let us consider the higher order inequality
∂ku
− ∆(au) ≥ |x|σ |u|p ,
∂tk
x ∈ Kε , t ∈]0, +∞[,
(2.1)
where p > 1, σ > −2, with the initial data
u(x, 0) = u0 (x),
in Kε ,
i
∂u
(x, 0) = ui (x),
∂ti
i ∈ {1, 2, . . . , k − 1},
in Kε .
Definition 2.1 Let a be in L∞ K ε ×]0, +∞[ . A weak solution u of the inequality (2.1) on Kε ×]0, +∞[ is continuous function on K ε × [0, +∞[ such that
j
the traces ∂∂tuj (x, 0), j ∈ {1, .., k − 1}, are well defined and locally integrable on
Kε and
Z ∞Z ∂kϕ
a u∆ϕ − u(−1)k k + |x|σ |u|p ϕ dx dt
∂t
0
Kε
Z ∞Z
Z
k−1
X
∂ϕ
∂ k−1−j u
∂j ϕ
−
au
dx dt +
(−1)j
(x, 0) j (x, 0) dx ≤ 0,
k−1−j
∂ν
∂t
0
∂Kε
Kε ∂t
j=0
(2.2)
4
Nonexistence of solutions to systems
EJDE–2002/97
holds for any nonnegative test function ϕ ∈ C 2,k (Kε ×]0, +∞[) with compact
support, such that ϕ|∂Kε ×]0,+∞[ = 0.
Similarly, we define the weak solutions of the system
∂ k ui
− ∆(ai ui ) ≥ |x|σi+1 |ui+1 |pi+1 , x ∈ Kε , t ∈]0, +∞[, 1 ≤ i ≤ n,
∂tk
un+1 = u1 ,
(2.3)
where pi > 1, σi > −2, for
i ≤ n, pn+1 = p1 , σn+1 = σ1 , and the initial
1≤
1
k
(0)
(1)
(k−1)
data ui , ui , . . . , ui
∈ Lloc (Kε ) , 1 ≤ i ≤ n.
Definition 2.2 Let ai ∈ L∞ K ε ×]0, +∞[ , i ∈ {1, 2, . . . , n}. A weak solution (u1 , . . . , un ) of the system (2.3) on Kε ×]0, +∞[ is a vector of continj
uous functions (u1 , . . . , un ) on K ε × [0, +∞[ such that the traces ∂∂tuji (x, 0),
(i, j) ∈ {1, .., n} × {1, .., k − 1}, are well defined and locally integrable on Kε
and the n estimates
Z ∞Z ∂kϕ
ai ui ∆ϕ − ui (−1)k k + |x|σi+1 |ui+1 |pi+1 ϕ dx dt
∂t
0
Kε
Z ∞Z
Z
k−1
X
∂ϕ
∂ k−1−j ui
∂j ϕ
−
ai ui
dx dt +
(−1)j
(x, 0) j (x, 0) dx ≤ 0,
k−1−j
∂ν
∂t
0
∂Kε
Kε ∂t
j=0
(2.4)
for any i ∈ {1, 2, . . . , n − 1}, and
∞
∂kϕ
+ |x|σ1 |u1 |p1 ϕ dx dt
k
∂t
0
Kε
Z ∞Z
Z
k−1
X
∂ϕ
∂ k−1−j un
∂j ϕ
j
dx dt +
(−1)
(x,
0)
(x, 0) dx ≤ 0,
−
an un
k−1−j
∂ν
∂tj
0
∂Kε
Kε ∂t
j=0
Z
Z
an un ∆ϕ − un (−1)k
(2.5)
hold, for any nonnegative test function ϕ ∈ C 2,k (Kε ×]0, +∞[) with compact
support, such that ϕ|∂Kε ×]0,+∞[ = 0.
We shall construct the test functions which will be used in our proofs. Let
ζ be the standard cut-off function
(
1 if 0 ≤ y ≤ 1,
ζ(y) =
0 if y ≥ 2.
Let p0 ≥ k + 1 and η(y) = [ζ(y)]p0 .
Explicit computation shows that there is a positive constant C(η) > 0 such
that, for y ≥ 0 and 1 < p ≤ p0 ,
|η (k) (y)|p ≤ C(η)η p−1 (y).
(2.6)
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
5
We introduce the term of the test function which depends on the variable t. Let
the parameter ρ > ε, the exponent θ > 0 and the function t 7→ η(t/ρθ ). Remark
that
supp |η(t/ρθ )| = {t ∈ R+ , 0 ≤ t ≤ 2ρθ }
and
dk η
supp k (t/ρθ ) = {t ∈ R+ , ρθ ≤ t ≤ 2ρθ },
dt
where “supp” denotes the support. It follows that
Z
k
kη
(t/ρθ )|
dtk
supp | d
| ddtkη (t/ρθ )|p
dt ≤ cη ρ−θ(kp−1) .
η p−1 (t/ρθ )
(2.7)
Now, we construct the part of the test function in the space variable x ≡ (r, ω).
Let s 6= 0 a real number, then
∆ (rs Φ(ω)) = rs−2 Φ(ω) s2 + (N − 2)s − λ .
Denote by s+ and s− the two roots of the equation
s2 + (N − 2)s − λ = 0.
These roots are given by
r
N −2
N − 2 2
s+ = −
+
+λ
2
2
N −2
and s− = −
−
2
r
N − 2 2
+ λ.
2
Consider the function ζ defined on Kε by
r
r s− s+
ζ(x) ≡ ζ(r, ω) =
−
Φ(ω).
ε
ε
It is interesting to note that the function ζ is harmonic in Kε and vanishes on
the boundary ∂Kε . Moreover,
∂ζ ≤ 0,
∂ν ∂Kε
where ν is the outer normal to the full surface ∂Kε . Indeed, thanks to the Hopf
lemma,
r
∂ζ
r s− ∂Φ(ω)
s+
=
−
≤ 0.
∂νω
ε
ε
∂νω
Moreover, since s+ > 0 and s− < 0, then
∂ζ s− − s+
− =
Φ(ω) ≤ 0.
∂r r=ε
ε
Let us consider now the function of r, for r ≥ ε,
r
r s− r s+
ξ(r) =
−
η
.
ε
ε
ρ
6
Nonexistence of solutions to systems
EJDE–2002/97
Now, we give estimates of ∂ξ/∂r and ∂ 2 ξ/∂r2 . First, we have
s r
s− r s− −1 r 1 r s+
r s− 0 r ∂ξ
s+ −1
+
=
−
η
+
−
η
.
∂r
ε ε
ε ε
ρ
ρ ε
ε
ρ
Whence, there is a positive constant C, independent of ρ and r, such that
∂ξ p
∂r
n hs r
s− r s− −1 ip p r 1 h r s+
r s− ip 0 r p o
s+ −1
+
−
η
+ p
−
≤C
η
.
ε ε
ε ε
ρ
ρ
ε
ε
ρ
Consequently,
Moreover,
∂ξ p
r
rp 1+ p .
≤ C rp(s+ −1) η p−1
∂r
ρ
ρ
∂ 2 ξ s+ (s+ − 1) r s+ −2 s− (s− − 1)
=
−
∂r2
ε2
ε
ε2
2 s+ r s+ −1 s− r s− −1 0
+
−
η
ρ ε ε
ε ε
r s− −2 η
ε
r
1 + 2
ρ
ρ
(2.8)
r
ρ
r s+
r s− 00 r −
η
.
ε
ε
ρ
Similarly, there is C > 0, independent on ρ and r, such that
∂ 2 ξ p
r rp
r2p 1 + p + 2p .
2 ≤ C rp(s+ −2) η p−1
∂r
ρ
ρ
ρ
(2.9)
We introduce now the final test function of the space variable
ψρ (x) = ζ(x)η
|x| r s+
r s− r =
−
η
Φ(ω).
ρ
ε
ε
ρ
Then
∆ψρ (x) =
where
∆ω ψ ρ =
∂ 2 ψρ
N − 1 ∂ψρ
1
(x) +
(x) + 2 ∆ω ψρ (x),
∂r2
r
∂r
r
r
r s− r s+
−
η
(−λΦ) = −λψρ .
ε
ε
ρ
Whence
n ∂ 2
p
N −1 ∂
λo
|∆ψρ (x)|p = Φp (ω)
+
−
(ξ(r))
,
∂r2
r ∂r r2
r p(s+ −2) rp
r2p ≤ C Φp (ω)η p−1
r
1 + p + 2p ,
(2.10)
ρ
ρ
ρ
p−1 (r/ε)s+
rp
r2p ≤ C ψρp−1 (x)rs+ −2p
1
+
+
,
(r/ε)s+ − (r/ε)s−
ρp
ρ2p
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
7
where C is a positive constant, independent of ρ and r. Let us denote N =
supp (∆ψρ ). Since η(r/ρ) = 1 for r ≤ ρ and η(r/ρ) = 0 for r ≥ 2ρ, then
N ⊂ {x ∈ Kε ; ρ ≤ r ≤ 2ρ}. Moreover, since ρ > ε, then the expressions
1+
rp
r2p
+
ρp
ρ2p
(r/ε)s+
(r/ε)s+ − (r/ε)s−
and
are bounded on {x ∈ Kε ; ρ ≤ r ≤ 2ρ}. We conclude that there is a positive
constant C such that
|∆ψρ (x)|p ≤ C ψρp−1 (x)
r s+
ρ2p
∀x ∈ N .
Finally, for ρ sufficiently large, we have the estimate
Z
|∆(ψρ )(x)|p
dx
p−1
(x)|x|σ(p−1)
N ψρ
Z 2ρ Z s+ +N −1
C
r
≤
dθ dr
σ(p−1)
ρ2p ρ
Ω r

s+ +N −σ(p−1)−2p

if s+ + N − σ(p − 1) > 0,
ρ
−2p
≤ C ρ
ln(ρ)
if s+ + N − σ(p − 1) = 0,

 −2p
ρ
if s+ + N − σ(p − 1) < 0.
(2.11)
Consider the final test function of the variables x and t:
t ϕρ (x, t) = η θ ψρ (x).
ρ
(2.12)
On one hand, the same arguments used in (2.11) give the estimate
Z +∞ Z
|∆(ϕρ )(x, t)|p
dxdt
p−1
(x, t)|x|σ(p−1)
0
N ϕρ
Z 2ρθ Z 2ρ Z s+ +N −1
C
r
≤
dθ dr dt
2p
σ(p−1)
ρ
0
ρ
Ω r

s+ +N −σ(p−1)+θ−2p

if s+ + N − σ(p − 1) > 0,
ρ
θ−2p
≤ C ρ
ln(ρ)
if s+ + N − σ(p − 1) = 0,

 θ−2p
ρ
if s+ + N − σ(p − 1) < 0.
(2.13)
On the other hand, if we denote C(ε, ρ) := {x ∈ Kε ; ε ≤ |x| ≤ 2ρ}, then
Z Z
∂ k ϕρ supp ∂tk k p
∂ ϕρ ∂tk ϕp−1
|x|(p−1)σ
ρ
≤
Z
C(ε,ρ)
dx dt
ψρ (x)
dx
|x|(p−1)σ
Z
k
supp d kη (t/ρθ )
dt
k
p
d η
dtk (t/ρθ )
η p−1 (t/ρθ )
dt.
(2.14)
8
Nonexistence of solutions to systems
EJDE–2002/97
Furthermore,
Z
Z
Z 2ρ r s− r rN −1
ψ̃ρ (x)
r s+
dx
=
Φ(ω)
dθ
−
η
dr
(p−1)σ
ε
ε
ρ r(p−1)σ
C(ε,ρ) |x|
Ω
ε
Z
|Ω| 2ρ s+ +N −1−(p−1)σ
≤ s+
r
dr
ε
 ε
s+ +N −σ(p−1)

if s+ + N − σ(p − 1) > 0,
ρ
≤C ln(ρ)
if s+ + N − σ(p − 1) = 0,


1
if s+ + N − σ(p − 1) < 0.
(2.15)
Combining the estimates (2.7), (2.14) and (2.15), we obtain
Z Z
k p
∂ ϕρ ∂tk dx dt
ϕp−1
|x|(p−1)σ
ρ

s+ +N −σ(p−1)−θ(kp−1)

ρ
≤ C ρ−θ(kp−1) ln(ρ)

 −θ(kp−1)
ρ
∂ k ϕρ supp k
∂t
if s+ + N − σ(p − 1) > 0,
if s+ + N − σ(p − 1) = 0,
if s+ + N − σ(p − 1) < 0.
(2.16)
In the following section, we consider the case n = 1.
3
Higher-Order Evolution Semilinear Inequalities
In this section, we establish nonexistence results for global solutions to the
semilinear problem (2.1). The weak solutions of (2.1) are defined in Definition
2.1.
Theorem 3.1 Assume that for all (x, t) ∈ ∂Kε × [0, +∞[, a(x, t) ≥ 0 and
u(x, t) ≥ 0. Also assume that
∀x ∈ Kε ;
Let
∂ k−1 u
(x, 0) ≥ 0.
∂tk−1
σ+2
1
≥ s+ + N − 2 1 − ,
p−1
k
where p > 1 and σ > −2. Then there is no weak nontrivial solution u of the
inequality (2.1).
Proof.
Assume that (2.1) admits a nontrivial global weak solution u with
σ+2
1
≥ s+ + N − 2 1 − .
p−1
k
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
9
In definition 2.1, let us choose the test function ϕ(x, t) = ϕρ (x, t) defined in
(2.12). Thanks to the Hopf lemma, we have
Z ∞Z
∂ϕρ
au
dx dt ≤ 0.
∂ν
0
∂Kε
Moreover, the test function ϕρ satisfies the equalities
∂ j ϕρ
(x, 0) = 0,
∂tj
for j ∈ {1, 2, . . . , k − 1}.
Finally, we have
Z
Kε
∂ k−1 u
(x, 0)ϕρ (x, 0) dx ≥ 0.
∂tk−1
Then, inequality (2.2) implies that
Z
0
∞
Z
Z
|x|σ |u|p ϕρ dx dt ≤
Kε
∞
∂k
u −a∆ + (−1)k k ϕρ dxdt.
∂t
Kε
Z
0
(3.1)
Let us introduce the notation
Z
I(ρ) :=
A(ρ) =
∞
0
∞
Z
Z
B(ρ) =
∞
0
|∆ϕρ |p
Z
(|x|σ ϕρ )
Kε
∂ k ϕρ p0
|
∂tk
p0 −1
(|x|σ ϕρ )
|
Z
0
p0 −1
Kε
0
Z
|x|σ |u|p ϕρ dx dt,
Kε
dx dt
dx dt.
Applying Hölder’s inequality to (3.1), we obtain
1
1
1
I(ρ) ≤ max (||a||∞ , 1) I(ρ) p A(ρ) p0 + B(ρ) p0 ,
or equivalently
1
1
1
I(ρ)1− p ≤ max (||a||∞ , 1) A(ρ) p0 + B(ρ) p0 .
At this stage, we choose the real parameter θ = 2/k and obtain
A(ρ) ≤ C Θ(ρ)
and
B(ρ) ≤ C Θ(ρ),
where
 s +N −σ(p0 −1)−2(p0 −1/k)
 ρ+
0
Θ(ρ) =
ρ−2(p −1/k) ln(ρ)
 −2(p0 −1/k)
ρ
if s+ + N − σ(p0 − 1) > 0,
if s+ + N − σ(p0 − 1) = 0,
if s+ + N − σ(p0 − 1) < 0.
(3.2)
10
Nonexistence of solutions to systems
EJDE–2002/97
If s+ + N − σ(p0 − 1) > 0, then explicit computation gives , for ρ sufficiently
large,
1
I 1− p ≤ Cρα ,
where
p−1
α=
p
1 σ + 2
s+ + N − 2 1 −
−
k
p−1
.
Now, we require that α ≤ 0, which is equivalent to
1
σ+2
≥ s+ + N − 2 1 − .
p−1
k
In this case, I(ρ) is bounded uniformly with respect to the variable ρ. Moreover,
the function I(ρ) is increasing in ρ. Consequently, the monotone convergence
theorem implies that the function
r
r s− s+
(x, t) ≡ (r, ω, t) 7−→ |u(x, t)|p |x|σ
−
Φ(ω)
ε
ε
is in L1 (Kε ×]0, +∞[). Furthermore, note that
supp(∆ϕρ ) ⊂ {t ∈ R+ ,
0 ≤ t ≤ 2ρ2/k } × {x ∈ Kε ,
ρ ≤ |x| ≤ 2ρ}
and
supp
∂kϕ ρ
⊂ {t ∈ R+ ,
∂tk
ρ2/k ≤ t ≤ 2ρ2/k } × {x ∈ Kε ,
ε ≤ |x| ≤ 2ρ}.
Whence, instead of (3.2) we have more precisely
1
1
˜ p1 A(ρ) p0 + B(ρ) p0 ,
I(ρ) ≤ max (||a||∞ , 1) I(ρ)
R
˜
where I(ρ)
= Cρ |x|σ |u|p ϕρ dx dt and Cρ = supp(∆ϕρ ) ∪ supp
using the dominated convergence theorem, we obtain
(3.3)
∂ k ϕρ .
∂tk
Finally,
lim I(ρ) = 0.
ρ→+∞
This implies u ≡ 0, which contradicts the fact that u is assumed to be nontrivial
weak solution.
Now, if s+ + N − σ(p0 − 1) ≤ 0, then
0
lim ρ−2(p −1/k) ln(ρ) = 0 and
ρ→+∞
0
lim ρ−2(p −1/k) = 0.
ρ→+∞
Therefore the integral I(ρ) is bounded uniformly with respect to the variable ρ.
The same arguments used previously complete the proof.
The previous result is also valid for cones instead of cone-like domains. Indeed, let us consider the higher order inequality
∂ku
− ∆(au) ≥ |x|σ |u|p ,
∂tk
x ∈ K, t ∈]0, +∞[,
(3.4)
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
11
where p > 1, σ > −2, with the initial data
u(x, 0) = u0 (x),
in K,
i
∂u
(x, 0) = ui (x), i ∈ {1, 2, . . . , k − 1},
∂ti
Then we have the following statement.
in K.
Theorem 3.2 Assume that for all (x, t) ∈ ∂K × [0, +∞[, a(x, t) ≥ 0 and
u(x, t) ≥ 0. Also assume that
∀x ∈ K;
∂ k−1 u
(x, 0) ≥ 0.
∂tk−1
Let
σ+2
1
≥ s+ + N − 2 1 − ,
p−1
k
where p > 1 and σ > −2. Then there is no weak nontrivial solution u of the
system (3.4).
Proof. Note that the cone K coincides with Kε for ε = 0. In this case, the
test function ϕρ given by (2.12) is not well defined. We choose the new test
function
r t
ϕ̃ρ (x, t) ≡ ϕ̃ρ (r, ω, t) = rs+ Φ(ω) η θ η
.
ρ
ρ
The function
K
−→ [0, +∞[
(r, ω) 7−→ rs+ Φ(ω),
is also harmonic. Following the different steps of the last proof with ϕ̃ρ (resp.
K) instead of ϕρ (resp. Kε ), we obtain the result.
4
Higher Order Systems of Evolution Semilinear Inequalities
We establish here nonexistence results of global solutions to the system (Snk ).
The weak solutions of (Snk ) are defined in Definition 2.2. In this section, the
j
(j)
initial conditions ∂∂tuji (x, 0) will be denoted by ui (x, 0), for (i, j) ∈ {1, .., n} ×
{0, .., k − 1}, and the vector (X1 , X2 , . . . , Xn ) will denote the solution of the
linear system



 

−1 p1
0 ...
0
X1
σ1 + 2


.
.
..
..   X   σ + 2 
 0 −1 p2

 2   2

 ..
  ..  

..
.
.
.
..
.. ..
=
(4.1)
 .





0  .  
.








.. ..
Xn−1
σn−1 + 2
0
.
. pn−1 
Xn
σn + 2
pn
0 ... 0
−1
We have the following non existence result.
12
Nonexistence of solutions to systems
EJDE–2002/97
Theorem 4.1 Assume that for all (x, t) ∈ ∂Kε × [0, +∞[ and i ∈ {1, 2, . . . , n},
ui (x, t) ≥ 0 and ai (x, t) ≥ 0. Also assume that
∀x ∈ Kε , ∀i ∈ {1, 2, . . . , n};
∂ k−1 ui
(x, 0) ≥ 0.
∂tk−1
Let
max{X1 , X2 , . . . , Xn } ≥ s+ + N − 2 1 −
1
,
k
where pi > 1 and σi > −2, for 1 ≤ i ≤ n. Then the problem (2.3) has no
nontrivial global weak solution.
Proof. By contradiction, assume that (2.3) admits a nontrivial global weak
solution (u1 , u2 , . . . , un ) with max{X1 , X2 , . . . , Xn } ≥ s+ + N − 2(1 − 1/k). In
definition 2.2, let us choose the test function ϕ(x, t) = ϕρ (x, t) defined in (2.12).
Thanks to the Hopf lemma, we have
Z ∞Z
∂ϕρ
ai ui
dx dt ≤ 0, for i ∈ {1, 2, . . . , n}.
∂ν
0
∂Kε
Moreover, the test function ϕρ satisfies
∂ j ϕρ
(x, 0) = 0,
∂tj
Finally, we have
Z
Kε
for j ∈ {1, 2, . . . , k − 1}.
∂ k−1 ui
(x, 0)ϕρ (x, 0) dx ≥ 0, i ∈ {1, 2, . . . , n}.
∂tk−1
Then, inequalities (2.4) and (2.5) imply
∞
∞
∂k
|x| |u1 | ϕρ ≤
un −an ∆ + (−1)
ϕρ ,
∂tk
0
Kε
0
Kε
Z ∞Z
Z ∞Z
∂k
|x|σi |ui |pi ϕρ ≤
ui−1 −ai−1 ∆ + (−1)k k ϕρ , 2 ≤ i ≤ n.
∂t
0
Kε
0
Kε
(4.2)
For i ∈ {1, 2, . . . , n}, we use the notation
Z ∞Z
Ii (ρ) :=
|x|σi |ui |pi ϕρ dx dt,
Z
Z
σ1
Z
p1
Ai (ρ) =
Z
0
∞
Bi (ρ) =
0
∞
k
Kε
Z
0
|∆ϕρ |pi
p0i −1
Kε
(|x|σi ϕρ )
Kε
∂ k ϕρ p0i
|
∂tk
p0 −1
σ
(|x| i ϕρ ) i
0
Z
Z
Z
|
dx dt
dx dt.
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
13
Using the Hölder’s inequality for system (4.2), we obtain
1
1 1
I1 (ρ) ≤ max(||an ||∞ , 1)In (ρ) pn An (ρ) p0n + Bn (ρ) p0n ,
1
1
1
0
0
Ii (ρ) ≤ max (||ai−1 ||∞ , 1) Ii−1 (ρ) pi−1 Ai−1 (ρ) pi−1 + Bi−1 (ρ) pi−1 , 2 ≤ i ≤ n.
(4.3)
Whence, there is a positive constant C, independent on ρ and r such that
1
1 p2 ...pn
1− p
Ii
≤C
n Y
1/p0j
Aj
1/p0j
+ Bj
µ1
ij
, 1 ≤ i ≤ n,
j=1
where
µij =
(Qi−1
k=j+1
Qi−1
k=1
pk
pk
Qn
k=j+1
for i > j,
for i ≤ j.
pk
At this stage, we choose the real parameter θ = 2/k to obtain
Ai (ρ) ≤ C Θi (ρ)
and Bi (ρ) ≤ C Θi (ρ),
for i ∈ {1, 2, . . . , n},
where

s+ +N −σi (p0i −1)−2(p0i −1/k)

ρ
0
Θi (ρ) = ρ−2(pi −1/k) ln(ρ)

 −2(p0i −1/k)
ρ
if s+ + N − σi (p0i − 1) > 0,
if s+ + N − σi (p0i − 1) = 0,
if s+ + N − σi (p0i − 1) < 0.
To estimate the expressions Ii (ρ), i ∈ {1, 2, . . . , n}, we consider two cases.
Case 1: s+ + N − σi (p0i − 1) > 0, for any i ∈ {1, 2, . . . , n}. Explicit computation
gives , for ρ sufficiently large,
1
1 p2 ...pn
1− p
Ii
≤ Cραi ,
1 ≤ i ≤ n,
where
Pn
1
1
j=1 (2 + σj )τij
αi = 1 −
s+ + N − 2 1 −
−
,
p1 p2 . . . p n
k
p1 p 2 . . . p n − 1
and
τij =
(Qn
Qj−1
pk k=1 pk
Qj−1
τij = k=i pk
k=i
for i > j,
for i ≤ j.
Now, we require that, at least, one of αi , i ∈ {1, 2, . . . , n}, be less than zero,
which is equivalent to
max{X1 , X2 , . . . , Xn } ≥ s+ + N − 2 1 −
1
,
k
where the vector (X1 , X2 , . . . , Xn )T is the solution of (4.1). In this case, there is
i0 ∈ {1, 2, . . . , n} such that Ii0 (ρ) is bounded uniformly with respect to the variable ρ. Using the systems (4.2) and (4.3), we obtain that both Ii (ρ), 1 ≤ i ≤ n,
14
Nonexistence of solutions to systems
EJDE–2002/97
are bounded uniformly with respect to the variable ρ. Moreover, the functions
Ii (ρ), i ∈ {1, 2, . . . , n}, are increasing in ρ. Consequently, the monotone convergence theorem implies that the functions
r
r s− s+
(x, t) ≡ (r, ω, t) 7−→ |ui (x, t)|pi |x|σi
−
Φ(ω).
ε
ε
is in L1 (Kε ×]0, +∞[). Precise that these functions correspond to
lim |ui (x, t)|pi |x|σi ϕρ (x, t).
ρ→+∞
Furthermore, note that
supp(∆ϕρ ) ⊂ {t ∈ R+ , 0 ≤ t ≤ 2ρ2/k } × {x ∈ Kε , ρ ≤ |x| ≤ 2ρ}
and
supp
∂ k ϕρ ⊂ {t ∈ R+ , ρ2/k ≤ t ≤ 2ρ2/k } × {x ∈ Kε , ε ≤ |x| ≤ 2ρ}.
∂tk
Whence, instead of (4.3) we have more precisely
1
1 1
I1 (ρ) ≤ max (||an ||∞ , 1) I˜n (ρ) pn An (ρ) p0n + Bn (ρ) p0n ,
1
1
1
0
0
Ii (ρ) ≤ max (||ai−1 ||∞ , 1) I˜i−1 (ρ) pi−1 Ai−1 (ρ) pi−1 + Bi−1 (ρ) pi−1 , 2 ≤ i ≤ n.
(4.4)
where
Z
I˜i (ρ) =
|x|σi |ui |pi ϕρ dx dt,
Cρ
and Cρ = supp(∆ϕρ ) ∪ supp
theorem, we obtain
k
∂ ϕρ .
∂tk
Finally, using the dominated convergence
lim Ii (ρ) = 0,
ρ→+∞
i ∈ {1, 2, . . . , n}.
This implies that (u1 , u2 , . . . , un ) ≡ (0, 0, . . . , 0), which contradicts the fact that
(u1 , u2 , . . . , un ) is assumed to be nontrivial weak solution. We complete the
proof by treating the case
Case 2: There is i0 ∈ {1, 2, . . . , n}, such that s+ + N − σi0 (p0i0 − 1) ≤ 0. The
same arguments used in Case 1 give, for ρ sufficiently large,
1
1 p2 ...pn
1− p
Ii
≤ Cρα̃i ,
1 ≤ i ≤ n,
where α̃i ≤ αi , for i ∈ {1, 2, . . . , n}. Then, if there is i1 ∈ {1, 2, . . . , n} such
that αi1 ≤ 0 then α̃i1 ≤ 0. This implies that Ii1 (ρ) is bounded uniformly with
respect to the variable ρ, which leads to the same conclusion as the Case 1. This
completes the proof.
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
15
When problem (2.3) is posed on the cone K instead of the cone-like domain
Kε , our result is also valid and the proof changes very slightly. Indeed, consider
∂ k ui
− ∆(ai ui ) ≥ |x|σi+1 |ui+1 |pi+1 , x ∈ K, t ∈]0, +∞[, 1 ≤ i ≤ n,
∂tk
un+1 = u1 ,
(4.5)
where pi > 1, σi > −2, for 1 ≤ i ≤ n, pn+1 = p1 , σn+1 = σ1 , and the initial
(0)
(1)
(k−1)
data (ui , ui , . . . , ui
) ∈ [L1loc (K)]k , 1 ≤ i ≤ n.
The weak solutions of (4.5) are defined similarly as in Definition 2.1 with K
instead of Kε . Then we have the following result.
Theorem 4.2 Assume that for all (x, t) ∈ ∂K × [0, +∞[ and i ∈ {1, 2, . . . , n},
ui (x, t) ≥ 0 and ai (x, t) ≥ 0. Also assume that
∂ k−1 ui
(x, 0) ≥ 0.
∂tk−1
Let max{X1 , X2 , . . . , Xn } ≥ s+ + N − 2 1 − k1 , where pi > 1 and σi > −2, for
n
1 ≤ i ≤ n. Then problem (S̃k ) has no nontrivial global weak solution.
∀x ∈ K, , ∀i ∈ {1, 2, . . . , n};
For the Proof of this theorem, it suffices to use the same arguments of the
last proof with ϕ̃ρ (resp K) instead of ϕρ (resp Kε ) to obtain the result.
5
General case
We consider the problem
∂ k ui
− ∆(ai ui )
∂tk
un+1 = u1 ,
≥ tγi+1 |x|σi+1 |ui+1 |pi+1 ,
x ∈ Kε , t ∈]0, +∞[, 1 ≤ i ≤ n,
(5.1)
where pi > 1, σi > −2, for 1 ≤ i ≤ n, pn+1 = p1 , γn+1 = γ1 , σn+1 = σ1 , and
(0)
(1)
(k−1)
the initial data (ui , ui , . . . , ui
) ∈ [L1loc (Kε )]k , 1 ≤ i ≤ n. We will assume
that γi ≤ 0 for i ∈ {1, 2, . . . , n}.
We start by giving the new estimates corresponding to (2.13) and (2.16),
(for given p > 1 and γ ≤ 0)
Z
0
+∞
Z
|∆(ϕρ )(x, t)|p
p−1 dxdt
(tγ |x|σ ϕρ )
Z 2ρθ
Z 2ρ Z s+ +N −1
C
r
−γ(p−1)
dθ dr
t
dt
≤
σ(p−1)
ρ2p 0
ρ
Ω r

s+ +N −σ(p−1)+θ−2p−γθ(p−1)

if s+ + N − σ(p − 1) > 0,
ρ
≤ C ρθ−2p−γθ(p−1) ln(ρ)
if s+ + N − σ(p − 1) = 0,

 θ−2p−γθ(p−1)
ρ
if s+ + N − σ(p − 1) < 0.
N
16
Nonexistence of solutions to systems

s+ +N −(σ+γθ)(p−1)+θ−2p

ρ
≤ C ρθ−2p−γθ(p−1) ln(ρ)

 θ−2p−γθ(p−1)
ρ
EJDE–2002/97
if s+ + N − σ(p − 1) > 0,
if s+ + N − σ(p − 1) = 0,
if s+ + N − σ(p − 1) < 0.
(5.2)
and
Z Z
∂ k ϕρ supp k
∂t
k p
∂ ϕρ ∂tk p−1
(tγ |x|σ ϕρ )
dx dt

s+ +N −σ(p−1)−θ(kp−1)−γθ(p−1)

ρ
≤ C ρ−θ(kp−1)−γθ(p−1) ln(ρ)

 −θ(kp−1)−γθ(p−1)
ρ

s+ +N −(σ+γθ)(p−1)−θ(kp−1)

if
ρ
−θ(kp−1)−γθ(p−1)
≤ C ρ
ln(ρ)
if

 −θ(kp−1)−γθ(p−1)
ρ
if
if s+ + N − σ(p − 1) > 0,
if s+ + N − σ(p − 1) = 0,
if s+ + N − σ(p − 1) < 0.
s+ + N − σ(p − 1) > 0,
s+ + N − σ(p − 1) = 0,
s+ + N − σ(p − 1) < 0.
(5.3)
Let the vector (Y1 , Y2 , . . . , Yn ) be the solution of the linear system









−1
0
..
.
0
pn




...
0
Y1
σ1 + 2 + 2γ1 /k
.. 
..



.
−1 p2
. 
  Y2   σ2 + 2 + 2γ2 /k 





.
..
..
.. ..

 . 
.
.
.
0 
.
 . 






.. ..
Y
σ
+
2
+
2γ
/k
n−1
n−1
n−1
.
. pn−1 
Yn
σn + 2 + 2γn /k
0 ... 0
−1
p1
0
The we have the following non existence result.
Theorem 5.1 Assume that for all (x, t) ∈ ∂Kε × [0, +∞[ and i ∈ {1, 2, . . . , n},
ui (x, t) ≥ 0 and ai (x, t) ≥ 0. Also assume that
∀x ∈ Kε , ∀i ∈ {1, 2, . . . , n};
∂ k−1 ui
(x, 0) ≥ 0.
∂tk−1
Let
max{Y1 , Y2 , . . . , Yn } ≥ s+ + N − 2 1 −
1
,
k
where pi > 1 and σi + 2γi /k > −2, for 1 ≤ i ≤ n. Then the problem 1.1 has no
nontrivial global weak solution.
Proof. We follow the previous proof and replace the expressions Ii (ρ), Ai (ρ)
and Bi (ρ) by
Z ∞Z
I i (ρ) :=
|x|σi |ui |pi ϕρ dx dt,
0
Kε
EJDE–2002/97
Abdallah El Hamidi & Gennady G. Laptev
Ai (ρ) =
∞
Z
Z
B i (ρ) =
Z
0
∞
0
|∆ϕρ |pi
p0i −1
Kε
(tγi |x|σi ϕρ )
Kε
∂ k ϕρ p0i
|
∂tk
p0 −1
γ
σ
i
i
(t |x| ϕρ ) i
0
Z
17
|
dx dt
dx dt,
respectively. By setting the parameter θ = 2/k, we conclude that
Ai (ρ) ≤ C Θi (ρ) and B i (ρ) ≤ C Θi (ρ),
where

s+ +N −(σi +2γi /k)(p0i −1)−2(p0i −1/k)

ρ
0
0
Θi (ρ) = ρ−2(pi −1/k)−2γi (pi −1)/k ln(ρ)

0
 −2(pi −1/k)−2γi (p0i −1)/k
ρ
if s+ + N − σi (p0i − 1) > 0,
if s+ + N − σi (p0i − 1) = 0,
if s+ + N − σ(p0i − 1) < 0,
for any i ∈ {1, 2, . . . , N }. As in the previous proof, note that the leading
exponent in the previous estimate is
s+ + N − (σi + 2γi /k)(p0i − 1) − 2(p0i − 1/k).
In other words, the only difference with the last proof is that the parameter σi
must be replaced by σi + 2γi /k. Which achieves the proof.
Remark. If we consider the semilinear problem (1.2) instead of (2.1) with
γ ≤ 0, then σ has to be replaced by σ + 2γ/k in Theorem 3.1 and Theorem 3.2.
Acknowledgements. The authors would like to thank the anonymous referees for their interesting remarks and suggestions. G.G. Laptev is grateful
to L.M.C.A. of the University of La Rochelle for hospitality and nice working
atmosphere during his visits.
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Abdallah El hamidi
Laboratoire de Mathématiques, Université de La Rochelle,
Avenue Michel Crépeau
17000 La Rochelle, France
e-mail: aelhamid@univ-lr.fr
Gennady G. Laptev
Department of Function Theory, Steklov Mathematical Institute
Gubkina 8, 117966 Moscow, Russia
e-mail: laptev@home.tula.net