Nonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains *

HAL Id: hal-02507092

https://hal.archives-ouvertes.fr/hal-02507092

Submitted on 12 Mar 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

semilinear inequalities in cone-like domains *

Abdallah El Hamidi, Gennady Laptev

To cite this version:

Abdallah El Hamidi, Gennady Laptev. Nonexistence of solutions to systems of higher-order semi-

linear inequalities in cone-like domains *. Electronic Journal of Differential Equations, Texas State

University, Department of Mathematics, 2002, 2002, pp.1 - 19. �hal-02507092�

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

u

i

∂t

^k

− ∆(a

i

(x, t)u

i

(x, t)) ≥ t

^γi+1

| x |

^σi+1

| u

i+1

(x, t) |

^pi+1

, 1 ≤ i ≤ n, u

n+1

= u

1

,

in cones and cone-like domains in R

^N

, t > 0. Our results apply to nonneg- ative 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 un- bounded cone-like domains. Nonexistence results concerning nonnegative solu- tions 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

u

t

− ∆( | 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.

2002 Southwest Texas State University.c

Submitted March 10, 2002. Published November 14, 2002.

1

In this paper, we obtain nonexistence results for systems of semilinear higher- order evolution differential inequalities in unbounded cones and cone-like do- mains. More precisely, for n ≥ 2, we study the problem

∂

^k

u

i

∂t

^k

− ∆(a

_i

u

_i

) ≥ t

^γi+1

| x |

^σi+1

| u

_i+1

|

^pi+1

, 1 ≤ i ≤ n, u

n+1

= u

1

,

(1.1) where x belongs to a cone (or a cone-like domain), t ∈ ]0, + ∞ [, k ≥ 1, p

n+1

= p

1

, γ

_n+1

= γ

₁

and σ

_n+1

= σ

₁

.

For n = 1, we study the problem

∂

^k

u

∂t

^k

− ∆(au) ≥ t

^γ

| x |

^σ

| u |

^p

, (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 sys- tems 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 & V´ eron [27] and Laptev [11, 13, 14].

Let Ω ⊂ S

^N−1

be a connected submanifold of the unit sphere S

^N−1

in R

^N

with smooth boundary ∂Ω ⊂ S

^N−1

and having positive N − 2 dimensional measure. By a cone in R

^N

with cross section Ω with vertex at the origin, we mean a set

K = { (r, ω) ∈ R

^N

; 0 < r < + ∞ and ω ∈ Ω } , where r = | x | , x ∈ R

^N

. The boundary of K is

∂K = { (r, ω); r = 0 or ω ∈ ∂Ω } . For ε > 0 fixed, the cone-like domain K

ε

is defined as

K

_ε

= { x ∈ K; | x | > ε } and its boundary as

∂K

ε

= { (r, ω); r = ε or ω ∈ ∂Ω } .

The outward normal vector to the boundary ∂Ω (resp. ∂K ) will be denoted by ν

_ω

(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 well- known that the laplacian operator in R

^N

can be written, in polar coordinates (r, ω), as

∆ = 1

r

^N−1

∂

∂r r

^N−1

∂

∂r + 1

r

²

∆

ω

= ∂

²

∂r

²

+ N − 1 r

∂

∂r + 1

r

²

∆

ω

.

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 γ

i

= 0, i ∈ { 1, 2, . . . , n } . Section 5 concerns the general system (1.1), with γ

i

≤ 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 p

⁰

such that 1/p + 1/p

⁰

= 1.

We define now the weak solutions of the problems that we will consider in the sequel. Let us consider the higher order inequality

∂

^k

u

∂t

^k

− ∆(au) ≥ | x |

^σ

| u |

^p

, x ∈ K

ε

, t ∈ ]0, + ∞ [, (2.1) where p > 1, σ > − 2, with the initial data

u(x, 0) = u

0

(x), in K

ε

,

∂

ⁱ

u

∂t

ⁱ

(x, 0) = u

i

(x), i ∈ { 1, 2, . . . , k − 1 } , in K

ε

. Definition 2.1 Let a be in L

^∞

K

ε

× ]0, + ∞ [

. A weak solution u of the in- equality (2.1) on K

ε

× ]0, + ∞ [ is continuous function on K

ε

× [0, + ∞ [ such that the traces

^∂_∂t^ju_j

(x, 0), j ∈ { 1, .., k − 1 } , are well defined and locally integrable on K

ε

and

Z

∞ 0

Z

Kε

a u∆ϕ − u( − 1)

^k

∂

^k

ϕ

∂t

^k

+ | x |

^σ

| u |

^p

ϕ dx dt

− Z

_∞

0

Z

∂K_ε

a u ∂ϕ

∂ν dx dt +

k−1

X

j=0

( − 1)

^j

Z

K_ε

∂

^k−1−j

u

∂t

^k−1−j

(x, 0) ∂

^j

ϕ

∂t

^j

(x, 0) dx ≤ 0,

(2.2)

holds for any nonnegative test function ϕ ∈ C

^2,k

(K

ε

× ]0, + ∞ [) with compact support, such that ϕ |

∂Kε×]0,+∞[

= 0.

Similarly, we define the weak solutions of the system

∂

^k

u

_i

∂t

^k

− ∆(a

_i

u

_i

) ≥ | x |

^σi+1

| u

_i+1

|

^pi+1

, x ∈ K

_ε

, t ∈ ]0, + ∞ [, 1 ≤ i ≤ n, u

n+1

= u

1

,

(2.3) where p

i

> 1, σ

i

> − 2, for 1 ≤ i ≤ n, p

n+1

= p

1

, σ

n+1

= σ

1

, and the initial data

u

⁽⁰⁾_i

, u

⁽¹⁾_i

, . . . , u

^(k_i ⁻¹⁾

∈

L

¹_loc

(K

_ε

)

^k

, 1 ≤ i ≤ n.

Definition 2.2 Let a

_i

∈ L

^∞

K

_ε

× ]0, + ∞ [

, i ∈ { 1, 2, . . . , n } . A weak so- lution (u

₁

, . . . , u

_n

) of the system (2.3) on K

_ε

× ]0, + ∞ [ is a vector of contin- uous functions (u

1

, . . . , u

n

) on K

ε

× [0, + ∞ [ such that the traces

^∂_∂t^jujⁱ

(x, 0), (i, j) ∈ { 1, .., n } × { 1, .., k − 1 } , are well defined and locally integrable on K

ε

and the n estimates Z

∞

0

Z

K_ε

a

i

u

i

∆ϕ − u

i

( − 1)

^k

∂

^k

ϕ

∂t

^k

+ | x |

^σi+1

| u

i+1

|

^pi+1

ϕ dx dt

− Z

_∞

0

Z

∂Kε

a

i

u

i

∂ϕ

∂ν dx dt +

k−1

X

j=0

( − 1)

^j

Z

Kε

∂

^k−1−j

u

i

∂t

^k−1−j

(x, 0) ∂

^j

ϕ

∂t

^j

(x, 0) dx ≤ 0, (2.4) for any i ∈ { 1, 2, . . . , n − 1 } , and

Z

∞ 0

Z

K_ε

a

_n

u

_n

∆ϕ − u

_n

( − 1)

^k

∂

^k

ϕ

∂t

^k

+ | x |

^σ1

| u

₁

|

^p1

ϕ dx dt

− Z

_∞

0

Z

∂Kε

a

n

u

n

∂ϕ

∂ν dx dt +

k−1

X

j=0

( − 1)

^j

Z

Kε

∂

^k−1−j

u

n

∂t

^k−1−j

(x, 0) ∂

^j

ϕ

∂t

^j

(x, 0) dx ≤ 0, (2.5) hold, for any nonnegative test function ϕ ∈ C

^2,k

(K

_ε

× ]0, + ∞ [) with compact support, such that ϕ |

∂K_ε×]0,+∞[

= 0.

We shall construct the test functions which will be used in our proofs. Let ζ be the standard cut-off function

ζ(y) =

( 1 if 0 ≤ y ≤ 1, 0 if y ≥ 2.

Let p

₀

≥ 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 ≤ p

0

,

| η

^(k)

(y) |

^p

≤ C(η)η

^p−1

(y). (2.6)

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/ρ

^θ

). Remark that

supp | η(t/ρ

^θ

) | = { t ∈ R

⁺

, 0 ≤ t ≤ 2ρ

^θ

} and

supp

d

^k

η dt

^k

(t/ρ

^θ

)

= { t ∈ R

⁺

, ρ

^θ

≤ t ≤ 2ρ

^θ

} , where “supp” denotes the support. It follows that

Z

supp|^{dk η}_dtk(t/ρ^θ)|

|

^d_dt^kk^η

(t/ρ

^θ

) |

^p

η

^p−1

(t/ρ

^θ

) dt ≤ c

η

ρ

^{−θ(kp−1)}

. (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

∆ (r

^s

Φ(ω)) = r

^s−2

Φ(ω) s

²

+ (N − 2)s − λ . Denote by s

₊

and s

₋

the two roots of the equation

s

²

+ (N − 2)s − λ = 0.

These roots are given by s

+

= − N − 2

2 +

r N − 2

2

²

+ λ and s

₋

= − N − 2

2 −

r N − 2

2

²

+ λ.

Consider the function ζ defined on K

ε

by ζ(x) ≡ ζ(r, ω) = r

ε

^s+

− r ε

^s−

Φ(ω).

It is interesting to note that the function ζ is harmonic in K

ε

and vanishes on the boundary ∂K

ε

. Moreover,

∂ζ

∂ν

_∂K

ε

≤ 0,

where ν is the outer normal to the full surface ∂K

ε

. Indeed, thanks to the Hopf lemma,

∂ζ

∂ν

_ω

= r ε

^s+

− r ε

^s−

∂Φ(ω)

∂ν

_ω

≤ 0.

Moreover, since s

+

> 0 and s

₋

< 0, then

− ∂ζ

∂r

_r=ε

= s

₋

− s

+

ε Φ(ω) ≤ 0.

Let us consider now the function of r, for r ≥ ε, ξ(r) = r

ε

^s+

− r ε

^s−

η r

ρ

.

Now, we give estimates of ∂ξ/∂r and ∂

²

ξ/∂r

²

. First, we have

∂ξ

∂r = s

+

ε r ε

^s+−1

− s

₋

ε

r ε

^s−−1

η r

ρ + 1

ρ r

ε

^s+

− r ε

^s−

η

⁰

r

ρ .

Whence, there is a positive constant C, independent of ρ and r, such that

∂ξ

∂r

p

≤ C n h s

₊

ε

r ε

s+−1

− s

₋

ε

r ε

s₋−1

i

^p

η

^p

r

ρ + 1

ρ

^p

h r

ε

s+

− r ε

s₋

i

^p

η

⁰

r

ρ

p

o . Consequently,

∂ξ

∂r

p

≤ C r

^p(s+−1)

η

^p−1

r ρ

1 + r

^p

ρ

^p

. (2.8)

Moreover,

∂

²

ξ

∂r

²

= s

+

(s

+

− 1) ε

²

r ε

^s+−2

− s

₋

(s

₋

− 1) ε

²

r ε

^s−−2

η r

ρ

+ 2 ρ

s

+

ε r ε

^s+−1

− s

₋

ε

r ε

^s−−1

η

⁰

r

ρ + 1

ρ

²

r

ε

^s+

− r ε

^s−

η

⁰⁰

r

ρ .

Similarly, there is C > 0, independent on ρ and r, such that

∂

²

ξ

∂r

²

p

≤ C r

^p(s+−2)

η

^p−1

r ρ

1 + r

^p

ρ

^p

+ r

^2p

ρ

^2p

. (2.9)

We introduce now the final test function of the space variable ψ

ρ

(x) = ζ(x)η | x |

ρ

= r ε

s+

− r ε

s₋

η r

ρ Φ(ω).

Then

∆ψ

ρ

(x) = ∂

²

ψ

_ρ

∂r

²

(x) + N − 1 r

∂ψ

_ρ

∂r (x) + 1

r

²

∆

ω

ψ

ρ

(x), where

∆

ω

ψ

ρ

= r ε

s+

− r ε

s₋

η r

ρ

( − λΦ) = − λψ

ρ

. Whence

| ∆ψ

_ρ

(x) |

^p

= Φ

^p

(ω)

n ∂

²

∂r

²

+ N − 1 r

∂

∂r − λ r

²

o (ξ(r))

p

,

≤ C Φ

^p

(ω)η

^p−1

r ρ

r

^p(s+−2)

1 + r

^p

ρ

^p

+ r

^2p

ρ

^2p

, (2.10)

≤ C ψ

_ρ^p−1

(x)r

^s+−2p

(r/ε)

^s+

(r/ε)

^s+

− (r/ε)

^s−

p−1

1 + r

^p

ρ

^p

+ r

^2p

ρ

^2p

,

where C is a positive constant, independent of ρ and r. Let us denote N = supp (∆ψ

ρ

). Since η(r/ρ) = 1 for r ≤ ρ and η(r/ρ) = 0 for r ≥ 2ρ, then N ⊂ { x ∈ K

ε

; ρ ≤ r ≤ 2ρ } . Moreover, since ρ > ε, then the expressions

1 + r

^p

ρ

^p

+ r

^2p

ρ

^2p

and (r/ε)

^s+

(r/ε)

^s+

− (r/ε)

^s−

are bounded on { x ∈ K

ε

; ρ ≤ r ≤ 2ρ } . We conclude that there is a positive constant C such that

| ∆ψ

ρ

(x) |

^p

≤ C ψ

_ρ^p−1

(x) r

^s+

ρ

^2p

∀ x ∈ N . Finally, for ρ sufficiently large, we have the estimate

Z

N

| ∆(ψ

_ρ

)(x) |

^p

ψ

^pρ⁻¹

(x) | x |

^σ(p−1)

dx

≤ C ρ

^2p

Z

2ρ ρ

Z

Ω

r

^s++N−1

r

^σ(p−1)

dθ dr (2.11)

≤ C



 

 

ρ

^{s++N−σ(p−1)−2p}

if s

+

+ N − σ(p − 1) > 0, ρ

^−2p

ln(ρ) if s

+

+ N − σ(p − 1) = 0, ρ

^−2p

if s

+

+ N − σ(p − 1) < 0.

Consider the final test function of the variables x and t:

ϕ

_ρ

(x, t) = η t ρ

^θ

ψ

_ρ

(x). (2.12)

On one hand, the same arguments used in (2.11) give the estimate Z

+∞

0

Z

N

| ∆(ϕ

_ρ

)(x, t) |

^p

ϕ

^pρ⁻¹

(x, t) | x |

^σ(p−1)

dxdt

≤ C ρ

^2p

Z

2ρ^θ 0

Z

2ρ ρ

Z

Ω

r

^s++N−1

r

^σ(p−1)

dθ dr dt (2.13)

≤ C



 

 

ρ

^{s++N−σ(p−1)+θ−2p}

if s

₊

+ N − σ(p − 1) > 0, ρ

^θ−2p

ln(ρ) if s

+

+ N − σ(p − 1) = 0, ρ

^θ−2p

if s

₊

+ N − σ(p − 1) < 0.

On the other hand, if we denote C (ε, ρ) := { x ∈ K

ε

; ε ≤ | x | ≤ 2ρ } , then

Z Z

supp

∂k ϕρ

∂tk

∂^kϕρ

∂t^k

p

ϕ

^pρ⁻¹

| x |

^(p−1)σ

dx dt

≤ Z

C(ε,ρ)

ψ

ρ

(x)

| x |

^(p−1)σ

dx Z

supp

dk η dtk(t/ρ^θ)

d^kη dt^k

(t/ρ

^θ

)

p

η

^p−1

(t/ρ

^θ

) dt. (2.14)

Furthermore, Z

C(ε,ρ)

ψ ˜

ρ

(x)

| x |

^(p−1)σ

dx = Z

Ω

Φ(ω) dθ Z

2ρ

ε

r ε

^s+

− r ε

^s−

η r

ρ r

^N−1

r

^(p−1)σ

dr

≤ | Ω | ε

^s+

Z

2ρ ε

r

^{s++N−1−(p−1)σ}

dr

≤ C



 

 

ρ

^{s++N−σ(p−1)}

if s

₊

+ N − σ(p − 1) > 0, 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

supp

∂k ϕρ

∂tk

∂^kϕρ

∂t^k

p

ϕ

^pρ⁻¹

| x |

^(p−1)σ

dx dt

≤ C



 

 

ρ

^{s++N−σ(p−1)−θ(kp−1)}

if s

+

+ N − σ(p − 1) > 0, ρ

^{−θ(kp−1)}

ln(ρ) if s

+

+ N − σ(p − 1) = 0, ρ

^{−θ(kp−1)}

if s

+

+ N − σ(p − 1) < 0.

(2.16)

In the following section, we consider the case n = 1.

3 Higher-Order Evolution Semilinear Inequali- ties

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) ∈ ∂K

ε

× [0, + ∞ [, a(x, t) ≥ 0 and u(x, t) ≥ 0. Also assume that

∀ x ∈ K

_ε

; ∂

^k−1

u

∂t

^k−1

(x, 0) ≥ 0.

Let σ + 2

p − 1 ≥ s

₊

+ N − 2 1 − 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

p − 1 ≥ s

+

+ N − 2 1 − 1 k

.

In definition 2.1, let us choose the test function ϕ(x, t) = ϕ

ρ

(x, t) defined in (2.12). Thanks to the Hopf lemma, we have

Z

_∞

0

Z

∂Kε

a u ∂ϕ

ρ

∂ν dx dt ≤ 0.

Moreover, the test function ϕ

ρ

satisfies the equalities

∂

^j

ϕ

ρ

∂t

^j

(x, 0) = 0, for j ∈ { 1, 2, . . . , k − 1 } . Finally, we have

Z

K_ε

∂

^k−1

u

∂t

^k−1

(x, 0)ϕ

_ρ

(x, 0) dx ≥ 0.

Then, inequality (2.2) implies that Z

∞

0

Z

K_ε

| x |

^σ

| u |

^p

ϕ

_ρ

dx dt ≤ Z

∞

0

Z

K_ε

u

− a∆ + ( − 1)

^k

∂

^k

∂t

^k

ϕ

_ρ

dxdt. (3.1) Let us introduce the notation

I(ρ) :=

Z

_∞

0

Z

K_ε

| x |

^σ

| u |

^p

ϕ

ρ

dx dt,

A (ρ) = Z

_∞

0

Z

Kε

| ∆ϕ

ρ

|

^p0

( | x |

^σ

ϕ

ρ

)

^p0−1

dx dt B (ρ) =

Z

_∞

0

Z

K_ε

|

^∂_∂t^kϕk^ρ

|

^p0

( | x |

^σ

ϕ

ρ

)

^p0−1

dx dt.

Applying H¨ older’s inequality to (3.1), we obtain I(ρ) ≤ max ( || a ||

_∞

, 1) I(ρ)

^1p

A (ρ)

^p10

+ B (ρ)

^p10

, (3.2)

or equivalently

I(ρ)

^1−1p

≤ max ( || a ||

∞

, 1)

A (ρ)

^p10

+ B (ρ)

^p10

. 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

Θ(ρ) =







ρ

^{s++N−σ(p0−1)−2(p0−1/k)}

if s

+

+ N − σ(p

⁰

− 1) > 0,

ρ

^{−2(p0−1/k)}

ln(ρ) if s

₊

+ N − σ(p

⁰

− 1) = 0,

ρ

^{−2(p0−1/k)}

if s

+

+ N − σ(p

⁰

− 1) < 0.

If s

+

+ N − σ(p

⁰

− 1) > 0, then explicit computation gives , for ρ sufficiently large,

I

^1−1p

≤ Cρ

^α

, where

α = p − 1 p

s

₊

+ N − 2 1 − 1 k

− σ + 2 p − 1

. Now, we require that α ≤ 0, which is equivalent to

σ + 2

p − 1 ≥ s

+

+ N − 2 1 − 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

(x, t) ≡ (r, ω, t) 7−→ | u(x, t) |

^p

| x |

^σ

r ε

s+

− r ε

s₋

Φ(ω) is in L

¹

(K

ε

× ]0, + ∞ [). Furthermore, note that

supp(∆ϕ

ρ

) ⊂ { t ∈ R

⁺

, 0 ≤ t ≤ 2ρ

^2/k

} × { x ∈ K

ε

, ρ ≤ | x | ≤ 2ρ } and

supp ∂

^k

ϕ

ρ

∂t

^k

⊂ { t ∈ R

⁺

, ρ

^2/k

≤ t ≤ 2ρ

^2/k

} × { x ∈ K

_ε

, ε ≤ | x | ≤ 2ρ } . Whence, instead of (3.2) we have more precisely

I(ρ) ≤ max ( || a ||

_∞

, 1) ˜ I(ρ)

^1p

A (ρ)

^p10

+ B (ρ)

^p10

, (3.3)

where ˜ I(ρ) = R

Cρ

| x |

^σ

| u |

^p

ϕ

ρ

dx dt and C

ρ

= supp(∆ϕ

ρ

) ∪ supp

^∂_∂t^kϕ_k^ρ

. Finally, using the dominated convergence theorem, we obtain

ρ→

lim

+∞

I(ρ) = 0.

This implies u ≡ 0, which contradicts the fact that u is assumed to be nontrivial weak solution.

Now, if s

+

+ N − σ(p

⁰

− 1) ≤ 0, then

ρ→

lim

+∞

ρ

^{−2(p0−1/k)}

ln(ρ) = 0 and lim

ρ→+∞

ρ

^{−2(p0−1/k)}

= 0.

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. In- deed, let us consider the higher order inequality

∂

^k

u

∂t

^k

− ∆(au) ≥ | x |

^σ

| u |

^p

, x ∈ K, t ∈ ]0, + ∞ [, (3.4)

where p > 1, σ > − 2, with the initial data

u(x, 0) = u

₀

(x), in K,

∂

ⁱ

u

∂t

ⁱ

(x, 0) = u

_i

(x), i ∈ { 1, 2, . . . , k − 1 } , in K.

Then we have the following statement.

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

∂t

^k−1

(x, 0) ≥ 0.

Let σ + 2

p − 1 ≥ s

+

+ N − 2 1 − 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 K

ε

for ε = 0. In this case, the test function ϕ

ρ

given by (2.12) is not well defined. We choose the new test function

˜

ϕ

_ρ

(x, t) ≡ ϕ ˜

_ρ

(r, ω, t) = r

^s+

Φ(ω) η r ρ

^θ

η t ρ

. The function

K −→ [0, + ∞ [ (r, ω) 7−→ r

^s+

Φ(ω),

is also harmonic. Following the different steps of the last proof with ˜ ϕ

ρ

(resp.

K) instead of ϕ

ρ

(resp. K

ε

), we obtain the result.

4 Higher Order Systems of Evolution Semilin- ear Inequalities

We establish here nonexistence results of global solutions to the system (S

ⁿ_k

).

The weak solutions of (S

ⁿ_k

) are defined in Definition 2.2. In this section, the initial conditions

^∂_∂t^jujⁱ

(x, 0) will be denoted by u

^(j)_i

(x, 0), for (i, j) ∈ { 1, .., n } × { 0, .., k − 1 } , and the vector (X

₁

, X

₂

, . . . , X

_n

) will denote the solution of the linear system



















− 1 p

₁

0 . . . 0 0 − 1 p

2

. . . .. . .. . . . . . . . . . . 0 0 . . . . . . p

n−1

p

n

0 . . . 0 − 1































 X

1

X

2

.. . X

n−1

X

n















=













 σ

1

+ 2 σ

2

+ 2

.. . σ

n−1

+ 2

σ

n

+ 2















(4.1)

We have the following non existence result.

Theorem 4.1 Assume that for all (x, t) ∈ ∂K

ε

× [0, + ∞ [ and i ∈ { 1, 2, . . . , n } , u

i

(x, t) ≥ 0 and a

i

(x, t) ≥ 0. Also assume that

∀ x ∈ K

_ε

, ∀ i ∈ { 1, 2, . . . , n } ; ∂

^k−1

u

i

∂t

^k−1

(x, 0) ≥ 0.

Let

max { X

1

, X

2

, . . . , X

n

} ≥ s

+

+ N − 2 1 − 1 k

,

where p

_i

> 1 and σ

_i

> − 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 (u

1

, u

2

, . . . , u

n

) with max { X

1

, X

2

, . . . , X

n

} ≥ s

+

+ N − 2(1 − 1/k). In definition 2.2, let us choose the test function ϕ(x, t) = ϕ

ρ

(x, t) defined in (2.12).

Thanks to the Hopf lemma, we have Z

∞

0

Z

∂K_ε

a

_i

u

_i

∂ϕ

_ρ

∂ν dx dt ≤ 0, for i ∈ { 1, 2, . . . , n } . Moreover, the test function ϕ

ρ

satisfies

∂

^j

ϕ

ρ

∂t

^j

(x, 0) = 0, for j ∈ { 1, 2, . . . , k − 1 } . Finally, we have

Z

Kε

∂

^k−1

u

i

∂t

^k−1

(x, 0)ϕ

ρ

(x, 0) dx ≥ 0, i ∈ { 1, 2, . . . , n } . Then, inequalities (2.4) and (2.5) imply

Z

_∞

0

Z

K_ε

| x |

^σ1

| u

1

|

^p1

ϕ

ρ

≤ Z

_∞

0

Z

K_ε

u

n

− a

n

∆ + ( − 1)

^k

∂

^k

∂t

^k

ϕ

ρ

, Z

_∞

0

Z

K_ε

| x |

^σi

| u

i

|

^pi

ϕ

ρ

≤ Z

_∞

0

Z

K_ε

u

i−1

− a

i−1

∆ + ( − 1)

^k

∂

^k

∂t

^k

ϕ

ρ

, 2 ≤ i ≤ n.

(4.2) For i ∈ { 1, 2, . . . , n } , we use the notation

I

_i

(ρ) :=

Z

∞ 0

Z

K_ε

| x |

^σi

| u

_i

|

^pi

ϕ

_ρ

dx dt,

A

i

(ρ) = Z

∞

0

Z

K_ε

| ∆ϕ

_ρ

|

^p0i

( | x |

^σi

ϕ

_ρ

)

^p0i−1

dx dt B

i

(ρ) =

Z

_∞

0

Z

Kε

|

^∂_∂t^kϕk^ρ

|

^p0i

( | x |

^σi

ϕ

_ρ

)

^p0i−1

dx dt.

Using the H¨ older’s inequality for system (4.2), we obtain I

1

(ρ) ≤ max( || a

n

||

_∞

, 1)I

n

(ρ)

^pn1

A

n

(ρ)

p01n

+ B

n

(ρ)

p01n

, I

i

(ρ) ≤ max ( || a

i−1

||

∞

, 1) I

i−1

(ρ)

1

pi−1

A

i−1

(ρ)

1 p0

i−1

+ B

i−1

(ρ)

1 p0

i−1

, 2 ≤ i ≤ n.

(4.3) Whence, there is a positive constant C, independent on ρ and r such that

I

¹⁻

1 p1p2...pn

i

≤ C

n

Y

j=1

A

^1/p

0 j

j

+ B

^1/p

0 j

j

_µij¹

, 1 ≤ i ≤ n, where

µ

_ij

= (Q

i−1

k=j+1

p

k

for i > j, Q

i−1

k=1

p

k

Q

n

k=j+1

p

k

for i ≤ j.

At this stage, we choose the real parameter θ = 2/k to obtain

A

i

(ρ) ≤ C Θ

i

(ρ) and B

i

(ρ) ≤ C Θ

i

(ρ), for i ∈ { 1, 2, . . . , n } , where

Θ

i

(ρ) =



 

 

ρ

^{s++N−σi(p0i−1)−2(p0i−1/k)}

if s

+

+ N − σ

i

(p

⁰_i

− 1) > 0, ρ

^{−2(p0i−1/k)}

ln(ρ) if s

₊

+ N − σ

_i

(p

⁰_i

− 1) = 0, ρ

^{−2(p0i−1/k)}

if s

+

+ N − σ

i

(p

⁰_i

− 1) < 0.

To estimate the expressions I

i

(ρ), i ∈ { 1, 2, . . . , n } , we consider two cases.

Case 1: s

+

+N − σ

i

(p

⁰_i

− 1) > 0, for any i ∈ { 1, 2, . . . , n } . Explicit computation gives , for ρ sufficiently large,

I

¹⁻

1 p1p2...pn

i

≤ Cρ

^αi

, 1 ≤ i ≤ n, where

α

i

=

1 − 1

p

₁

p

₂

. . . p

_n

s

+

+ N − 2 1 − 1 k

− P

n

j=1

(2 + σ

j

)τ

ij

p

₁

p

₂

. . . p

_n

− 1

, and

τ

_ij

= (Q

n

k=i

p

_k

Q

j−1

k=1

p

_k

for i > j, τ

ij

= Q

j−1

k=i

p

k

for i ≤ j.

Now, we require that, at least, one of α

i

, i ∈ { 1, 2, . . . , n } , be less than zero, which is equivalent to

max { X

1

, X

2

, . . . , X

n

} ≥ s

+

+ N − 2 1 − 1 k

,

where the vector (X

1

, X

2

, . . . , X

n

)

^T

is the solution of (4.1). In this case, there is

i

0

∈ { 1, 2, . . . , n } such that I

i₀

(ρ) is bounded uniformly with respect to the vari-

able ρ. Using the systems (4.2) and (4.3), we obtain that both I

i

(ρ), 1 ≤ i ≤ n,

are bounded uniformly with respect to the variable ρ. Moreover, the functions I

i

(ρ), i ∈ { 1, 2, . . . , n } , are increasing in ρ. Consequently, the monotone conver- gence theorem implies that the functions

(x, t) ≡ (r, ω, t) 7−→ | u

i

(x, t) |

^pi

| x |

^σi

r ε

s+

− r ε

s₋

Φ(ω).

is in L

¹

(K

ε

× ]0, + ∞ [). Precise that these functions correspond to

ρ→

lim

+∞

| u

i

(x, t) |

^pi

| x |

^σi

ϕ

ρ

(x, t).

Furthermore, note that

supp(∆ϕ

ρ

) ⊂ { t ∈ R

⁺

, 0 ≤ t ≤ 2ρ

^2/k

} × { x ∈ K

ε

, ρ ≤ | x | ≤ 2ρ } and

supp ∂

^k

ϕ

ρ

∂t

^k

⊂ { t ∈ R

⁺

, ρ

^2/k

≤ t ≤ 2ρ

^2/k

} × { x ∈ K

ε

, ε ≤ | x | ≤ 2ρ } . Whence, instead of (4.3) we have more precisely

I

1

(ρ) ≤ max ( || a

n

||

_∞

, 1) ˜ I

n

(ρ)

^pn1

A

n

(ρ)

p01n

+ B

n

(ρ)

p01n

, I

i

(ρ) ≤ max ( || a

i−1

||

_∞

, 1) ˜ I

i−1

(ρ)

1 pi−1

A

i−1

(ρ)

1 p0

i−1

+ B

i−1

(ρ)

1 p0

i−1

, 2 ≤ i ≤ n.

(4.4) where

I ˜

i

(ρ) = Z

Cρ

| x |

^σi

| u

i

|

^pi

ϕ

ρ

dx dt,

and C

ρ

= supp(∆ϕ

ρ

) ∪ supp

^∂_∂t^kϕk^ρ

. Finally, using the dominated convergence theorem, we obtain

ρ→

lim

+∞

I

i

(ρ) = 0, i ∈ { 1, 2, . . . , n } .

This implies that (u

₁

, u

₂

, . . . , u

_n

) ≡ (0, 0, . . . , 0), which contradicts the fact that (u

₁

, u

₂

, . . . , u

_n

) is assumed to be nontrivial weak solution. We complete the proof by treating the case

Case 2: There is i

₀

∈ { 1, 2, . . . , n } , such that s

₊

+ N − σ

_i0

(p

⁰_i

0

− 1) ≤ 0. The same arguments used in Case 1 give, for ρ sufficiently large,

I

¹⁻

1 p1p2...pn

i

≤ Cρ

^α˜i

, 1 ≤ i ≤ n,

where ˜ α

i

≤ α

i

, for i ∈ { 1, 2, . . . , n } . Then, if there is i

1

∈ { 1, 2, . . . , n } such that α

i₁

≤ 0 then ˜ α

i₁

≤ 0. This implies that I

i₁

(ρ) is bounded uniformly with respect to the variable ρ, which leads to the same conclusion as the Case 1. This

completes the proof.

When problem (2.3) is posed on the cone K instead of the cone-like domain K

ε

, our result is also valid and the proof changes very slightly. Indeed, consider

∂

^k

u

i

∂t

^k

− ∆(a

i

u

i

) ≥ | x |

^σi+1

| u

i+1

|

^pi+1

, x ∈ K, t ∈ ]0, + ∞ [, 1 ≤ i ≤ n, u

n+1

= u

1

,

(4.5)

where p

_i

> 1, σ

_i

> − 2, for 1 ≤ i ≤ n, p

_n+1

= p

₁

, σ

_n+1

= σ

₁

, and the initial data (u

⁽⁰⁾_i

, u

⁽¹⁾_i

, . . . , u

^(k_i ⁻¹⁾

) ∈ [L

¹_loc

(K)]

^k

, 1 ≤ i ≤ n.

The weak solutions of (4.5) are defined similarly as in Definition 2.1 with K instead of K

ε

. Then we have the following result.

Theorem 4.2 Assume that for all (x, t) ∈ ∂K × [0, + ∞ [ and i ∈ { 1, 2, . . . , n } , u

i

(x, t) ≥ 0 and a

i

(x, t) ≥ 0. Also assume that

∀ x ∈ K, , ∀ i ∈ { 1, 2, . . . , n } ; ∂

^k−1

u

i

∂t

^k−1

(x, 0) ≥ 0.

Let max { X

1

, X

2

, . . . , X

n

} ≥ s

+

+ N − 2 1 −

¹_k

, where p

i

> 1 and σ

i

> − 2, for 1 ≤ i ≤ n. Then problem ( S ˜

ⁿ_k

) has no nontrivial global weak solution.

For the Proof of this theorem, it suffices to use the same arguments of the last proof with ˜ ϕ

_ρ

(resp K) instead of ϕ

_ρ

(resp K

_ε

) to obtain the result.

5 General case

We consider the problem

∂^kui

∂t^k

− ∆(a

i

u

i

) ≥ t

^γi+1

| x |

^σi+1

| u

i+1

|

^pi+1

, x ∈ K

ε

, t ∈ ]0, + ∞ [, 1 ≤ i ≤ n, u

n+1

= u

1

,

(5.1) where p

i

> 1, σ

i

> − 2, for 1 ≤ i ≤ n, p

n+1

= p

1

, γ

n+1

= γ

1

, σ

n+1

= σ

1

, and the initial data (u

⁽⁰⁾_i

, u

⁽¹⁾_i

, . . . , u

^(k_i ⁻¹⁾

) ∈ [L

¹_loc

(K

_ε

)]

^k

, 1 ≤ i ≤ n. We will assume that γ

_i

≤ 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

N

| ∆(ϕ

ρ

)(x, t) |

^p

(t

^γ

| x |

^σ

ϕ

_ρ

)

^p−1

dxdt

≤ C ρ

^2p

Z

2ρ^θ 0

t

^−γ(p−1)

dt Z

2ρ

ρ

Z

Ω

r

^s++N−1

r

^σ(p−1)

dθ dr

≤ C



 

 

ρ

^{s++N−σ(p−1)+θ−2p−γθ(p−1)}

if s

+

+ N − σ(p − 1) > 0,

ρ

^{θ−2p−γθ(p−1)}

ln(ρ) if s

₊

+ N − σ(p − 1) = 0,

ρ

^{θ−2p−γθ(p−1)}

if s

+

+ N − σ(p − 1) < 0.

≤ C



 

 

ρ

^{s++N−(σ+γθ)(p−1)+θ−2p}

if s

+

+ N − σ(p − 1) > 0, ρ

^{θ−2p−γθ(p−1)}

ln(ρ) if s

₊

+ N − σ(p − 1) = 0, ρ

^{θ−2p−γθ(p−1)}

if s

+

+ N − σ(p − 1) < 0.

(5.2)

and Z Z

supp

∂k ϕρ

∂tk

∂^kϕ_ρ

∂t^k

p

(t

^γ

| x |

^σ

ϕ

ρ

)

^p−1

dx dt

≤ C



 

 

ρ

^{s++N−σ(p−1)−θ(kp−1)−γθ(p−1)}

if s

+

+ N − σ(p − 1) > 0, ρ

^{−θ(kp−1)−γθ(p−1)}

ln(ρ) if s

+

+ N − σ(p − 1) = 0, ρ

^{−θ(kp−1)−γθ(p−1)}

if s

+

+ N − σ(p − 1) < 0.

≤ C



 

 

ρ

^{s++N−(σ+γθ)(p−1)−θ(kp−1)}

if s

₊

+ N − σ(p − 1) > 0, ρ

^{−θ(kp−1)−γθ(p−1)}

ln(ρ) if s

+

+ N − σ(p − 1) = 0, ρ

^{−θ(kp−1)−γθ(p−1)}

if s

₊

+ N − σ(p − 1) < 0.

(5.3)

Let the vector (Y

₁

, Y

₂

, . . . , Y

_n

) be the solution of the linear system



















− 1 p

1

0 . . . 0 0 − 1 p

2

. . . .. . .. . . . . . . . . . . 0 0 . . . . . . p

_n−1

p

_n

0 . . . 0 − 1































 Y

1

Y

2

.. . Y

_n−1

Y

_n





























σ

1

+ 2 + 2γ

1

/k σ

2

+ 2 + 2γ

2

/k

.. .

σ

_n−1

+ 2 + 2γ

_n−1

/k σ

_n

+ 2 + 2γ

_n

/k















The we have the following non existence result.

Theorem 5.1 Assume that for all (x, t) ∈ ∂K

ε

× [0, + ∞ [ and i ∈ { 1, 2, . . . , n } , u

_i

(x, t) ≥ 0 and a

_i

(x, t) ≥ 0. Also assume that

∀ x ∈ K

_ε

, ∀ i ∈ { 1, 2, . . . , n } ; ∂

^k−1

u

_i

∂t

^k−1

(x, 0) ≥ 0.

Let

max { Y

1

, Y

2

, . . . , Y

n

} ≥ s

+

+ N − 2 1 − 1 k

,

where p

i

> 1 and σ

i

+ 2γ

i

/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 I

i

(ρ), A

i

(ρ) and B

i

(ρ) by

I

i

(ρ) :=

Z

∞ 0

Z

Kε

| x |

^σi

| u

i

|

^pi

ϕ

ρ

dx dt,

A

i

(ρ) = Z

_∞

0

Z

Kε

| ∆ϕ

ρ

|

^p0i

(t

^γi

| x |

^σi

ϕ

ρ

)

^p0i−1

dx dt

B

i

(ρ) = Z

∞

0

Z

K_ε

|

^∂_∂t^kϕk^ρ

|

^p0i

(t

^γi

| x |

^σi

ϕ

ρ

)

^p0i−1

dx dt,

respectively. By setting the parameter θ = 2/k, we conclude that A

i

(ρ) ≤ C Θ

i

(ρ) and B

i

(ρ) ≤ C Θ

i

(ρ), where

Θ

i

(ρ) =



 

 

ρ

^{s++N−(σi+2γi/k)(p0i−1)−2(p0i−1/k)}

if s

+

+ N − σ

i

(p

⁰_i

− 1) > 0, ρ

^{−2(p0i−1/k)−2γi(p0i−1)/k}

ln(ρ) if s

₊

+ N − σ

_i

(p

⁰_i

− 1) = 0, ρ

^{−2(p0i−1/k)−2γi(p0i−1)/k}

if s

+

+ N − σ(p

⁰_i

− 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 − (σ

i

+ 2γ

i

/k)(p

⁰_i

− 1) − 2(p

⁰_i

− 1/k).

In other words, the only difference with the last proof is that the parameter σ

i

must be replaced by σ

i

+ 2γ

i

/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 ref- erees 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.

References

[1] C. Bandle, H.A. Levine, On the existence and nonexistence of global so- lutions of reaction–diffusion equations in sectorial domains, Trans. Amer.

Math. Soc., Vol. 316 (1989), 595–622.

[2] C. Bandle, H.A. Levine, Fujita type results for convective-like reaction- diffusion equations in exterior domains, Z. Angew. Math. Phys., Vol. 40 (1989), 665–676.

[3] C. Bandle, M. Essen, On positive solutions of Emden equations in cone-like domains, Arch. Rational Mech. Anal., Vol. 112 (1990), 319–338.

[4] K. Deng, H.A. Levine, The role of critical exponents in blow-up theorems:

the sequel, J. Math. Anal. Appl., Vol. 243 (2000), 85–126.

[5] H. Egnell, Positive solutions of semilinear equations in cones, Trans. Amer.

Math. Soc., 330 no. 1 (1992), 191–201.

[6] A. El Hamidi, G. G. Laptev, Non existence de solutions d’inquations semili- naires dans des domaines coniques. (submitted to ”Canadian Journal of Mathematics”)

[7] M. Guedda, M. Kirane, Criticality for some evolution equations, Differ.

Equ., Vol. 37, no. 4 (2001), 540–550.

[8] N. Igbida, M. Kirane, Blowup for completely coupled Fujita type reaction- diffusion system, Colloq. Math., 92, no. 1 (2002), 87–96.

[9] M. Kirane, M. Qafsaoui, Fujita’s exponent for a semilinear wave equation with linear damping, Adv. Nonlinear Stud., 2 no. 1 (2002), 41–49.

[10] M. Kirane, E. Nabana, S. I. Pohozaev, Nonexistence results to elliptic sys- tems with dynamical boundary conditions, Differ. Equ., 38 no. 6 (2001), 1–8.

[11] G.G. Laptev, The absence of global positive solutions of systems of semi- linear elliptic inequalities in cones, Russian Acad. Sci. Izv. Math., Vol. 64 (2000), 108–124.

[12] G.G. Laptev, On the absence of solutions to a class of singular semilinear differential inequalities, Proc. Steklov Inst. Math., Vol. 232 (2001), 223–235.

[13] G.G. Laptev, Nonexistence of solutions to semilinear parabolic inequalities in cones, Mat. Sb., Vol. 192 (2001), 51–70.

[14] G.G. Laptev, Some nonexistence results for higher–order evolution inequal- ities in cone–like domains, Electron. Res. Announc. Amer. Math. Soc., Vol.

7 (2001), 87–93.

[15] G.G. Laptev, Nonexistence of solutions for parabolic inequalities in un- bounded cone–like domains via the test function method, J. Evolution Equa- tions (to appear).

[16] G.G. Laptev, Absence of solutions of evolution-type differential inequalities in the exterior of a ball, Russ. J. Math. Phys., Vol. 9 (2002), 180–187.

[17] H.A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., Vol. 32 (1990), 262–288.

[18] H.A. Levine, P. Meier, The value of the critical exponent for reaction- diffusion equations in cones, Arch. Rational Mech. Anal., Vol. 109 (1990), 73–80.

[19] E. Mitidieri, S.I. Pohozaev, The absence of global positive solutions of quasi-

linear elliptic inequalities, Dokl. Akad. Nauk, 359, No.4 (1998), 456–460.

[20] E. Mitidieri, S.I. Pohozaev, Nonexistence of positive solutions for a system of quasilinear elliptic equations and inequalities in R

^N

, Dokl. Akad. Nauk, 366, No. 1 (1999) 13–17.

[21] E. Mitidieri, S.I. Pohozaev, Nonexistence of positive solutions for quasilin- ear elliptic problems in R

^N

, Tr. Mat. Inst. Steklova 227 (1999), 192–222.

[22] E. Mitidieri, S.I. Pohozaev, Nonexistence of weak solutions for some degen- erate elliptic and parabolic problems on R

ⁿ

, J. Evolution Equations, Vol. 1 (2001), 189–220.

[23] E. Mitidieri, S.I. Pohozaev, A priori Estimates and Nonexistence of Solu- tions to Nonlinear Partial Differential Equations and Inequalities, Moscow, Nauka, 2001. (Proc. Steklov Inst. Math., Vol. 234).

[24] S.I. Pohozaev, Essential nonlinear capacities induced by differential opera- tors, Dokl. Russ. Acad. Sci., Vol. 357 (1997), 592–594.

[25] S.I. Pohozaev, A. Tesei, Blow-up of nonnegative solutions to quasilinear parabolic inequalities, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

Rend. Lincei (9) Mat. Appl., Vol. 11 (2000), 99–109.

[26] S.I. Pohozaev, A. Tesei, Critical exponents for the absence of solutions for systems of quasilinear parabolic inequalities, Differ. Uravn., Vol. 37 (2001), 521–528.

[27] S.I. Pohozaev, L. V´ eron, Blow-up results for nonlinear hyperbolic inequali- ties, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), Vol. 29 (2000), 393–420.

[28] J. Renclawowicz, Global existence and blow-up for a completely coupled Fujita type system, Appl. Math., 27, No.2 (2000), 203–218.

Abdallah El hamidi

Laboratoire de Math´ ematiques, Universit´ e de La Rochelle, Avenue Michel Cr´ epeau

17000 La Rochelle, France e-mail: [email protected] Gennady G. Laptev

Department of Function Theory, Steklov Mathematical Institute Gubkina 8, 117966 Moscow, Russia

e-mail: [email protected]