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

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1 Electronic Journal of Differential Equations, Vol. 22(22, No. 97, pp ISSN: URL: or 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 u i t k (ai(x, tui(x, t tγ i+1 x σ i+1 u i+1(x, t p i+1, 1 i n, u n+1 = u 1, in cones and cone-like domains in R N, t >. 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 u t ( u m1 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 22 Southwest Texas State University. Submitted March 1, 22. Published November 14, 22. 1

2 2 Nonexistence of solutions to systems EJDE 22/97 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 u i t k (a iu 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 ], + [, 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 tγ x σ u p, (1.2 where x belongs to a cone (or a cone-like domain and t ], + [. Such systems were studied, in the whole space, by Renclawowicz [28], Guedda & Kirane [7], Igbida & Kirane [8] and Kirane, Nabana & Pohozaev [1]. 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 N1 be a connected submanifold of the unit sphere S N1 in R N with smooth boundary Ω S N1 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 = {, ω R N ; < r < + and ω Ω}, where r = x, x R N. The boundary of K is K = {, ω; r = or ω Ω}. For > fixed, the cone-like domain is defined as and its boundary as = {x K; x > } = {, ω; r = or ω Ω}. The outward normal vector to the boundary Ω esp. K will be denoted by ν ω esp. ν. The restriction of the laplacian operator to the unit sphere S N1 will be denoted by ω, which is the Laplace-Beltrami operator. It is wellknown that the laplacian operator in R N can be written, in polar coordinates, ω, as = 1 ( r N1 1 + r N1 r r r 2 ω = 2 r 2 + N 1 r r + 1 r 2 ω.

3 EJDE 22/97 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 = on Ω. Recall that λ > and Φ(ω >, for any ω Ω. We shall assume Φ is normalized so that < 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 ], + [, will be denoted by C 2,j (K ], + [. 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 =. In Section 4, we generalize the results of the Section 3 for n 2 and the parameters γ i =, i {1, 2,..., n}. Section 5 concerns the general system (1.1, with γ i, 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 such that 1/p + 1/p = 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, t ], + [, (2.1 where p > 1, σ > 2, with the initial data u(x, = u (x, in, i u t i (x, = u i(x, i {1, 2,..., k 1}, in. Definition 2.1 Let a be in L ( ], + [. A weak solution u of the inequality (2.1 on ], + [ is continuous function on [, + [ such that the traces j u t (x,, j {1,.., k 1}, are well defined and locally integrable on j and ( a u ϕ u(1 k k ϕ t k a u ϕ k1 ν dx dt + (1 j j= + x σ u p ϕ dx dt k1j u t k1j (x, ϕ j (x, dx, tj (2.2

4 4 Nonexistence of solutions to systems EJDE 22/97 holds for any nonnegative test function ϕ C 2,k ( ], + [ with compact support, such that ϕ K ],+ [ =. Similarly, we define the weak solutions of the system k u i t k (a iu i x σi+1 u i+1 pi+1, x, t ], + [, 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 ( i, u (1 i,..., u (k1 i [ L 1 loc ( ] k, 1 i n. Definition 2.2 Let a i L ( ], + [, i {1, 2,..., n}. A weak solution (u 1,..., u n of the system (2.3 on ], + [ is a vector of continuous functions (u 1,..., u n on [, + [ such that the traces j u i t (x,, j (i, j {1,.., n} {1,.., k 1}, are well defined and locally integrable on and the n estimates ( a i u i ϕ u i (1 k k ϕ t k + u x σi+1 i+1 pi+1 ϕ dx dt (1 j k1j u i k1 ϕ a i u i ν dx dt + for any i {1, 2,..., n 1}, and j= ( a n u n ϕ u n (1 k k ϕ t k + u x σ1 1 p1 ϕ dx dt (1 j k1 ϕ a n u n ν dx dt + j= t k1j (x, ϕ j (x, dx, tj k1j u n t k1j (2.4 (x, ϕ j (x, dx, tj (2.5 hold, for any nonnegative test function ϕ C 2,k ( ], + [ with compact support, such that ϕ K ],+ [ =. We shall construct the test functions which will be used in our proofs. Let ζ be the standard cut-off function { 1 if y 1, ζ(y = if y 2. Let p k + 1 and η(y = [ζ(y] p. Explicit computation shows that there is a positive constant C(η > such that, for y and 1 < p p, η (k (y p C(ηη p1 (y. (2.6

5 EJDE 22/97 Abdallah El Hamidi & Gennady G. Laptev 5 We introduce the term of the test function which depends on the variable t. Let the parameter >, the exponent θ > and the function t η(t/ θ. Remark that supp η(t/ θ = {t R +, t 2 θ } and supp dk η dt k (t/θ = {t R +, θ t 2 θ }, where supp denotes the support. It follows that supp dk η dt k (t/θ dk η dt k (t/ θ p η p1 (t/ θ dt c η θ(kp1. (2.7 Now, we construct the part of the test function in the space variable x, ω. Let s a real number, then s Φ(ω = r s2 Φ(ω ( s 2 + (N 2s λ. Denote by s + and s the two roots of the equation s 2 + (N 2s λ =. These roots are given by s + = N 2 (N λ and s = N Consider the function ζ defined on by ( ( r s+ s ζ(x ζ, ω = Φ(ω. (N λ. 2 It is interesting to note that the function ζ is harmonic in and vanishes on the boundary. Moreover, ζ, ν K where ν is the outer normal to the full surface. Indeed, thanks to the Hopf lemma, ζ ( ( r s+ s Φ(ω =. ν ω ν ω Moreover, since s + > and s <, then ζ = s s + Φ(ω. r r= Let us consider now the function of r, for r, ( ( r s+ s ξ = η ( r.

6 6 Nonexistence of solutions to systems EJDE 22/97 Now, we give estimates of ξ/ r and 2 ξ/ r 2. First, we have ξ r = ( s+ s+1 s s1 η ( r 1 + ( ( r s+ s η ( r. Whence, there is a positive constant C, independent of and r, such that ξ p r { [ s+ C s+1 s ] s1 p η p ( r 1 [ ( r + p s+ ] s p η ( r p}. Consequently, Moreover, ξ p C r p(s+1 η p1( r ( r p 1 +. (2.8 r p 2 ξ ( r 2 = s+ (s ( s+ s+2 s (s 1 2 s+1 s s1 η ( r s2 η ( r 1 ( ( r + 2 Similarly, there is C >, independent on and r, such that s+ s η ( r. 2 ξ p C r p(s +2 r 2 η p1( r ( 1 + rp p + r2p 2p. (2.9 We introduce now the final test function of the space variable Then where Whence ψ (x = ζ(xη ( x = ( ( r ψ (x = 2 ψ r 2 (x + N 1 r ω ψ = ( ( r s+ s+ s η ( r Φ(ω. ψ r (x + 1 r 2 ωψ (x, s η ( r (λφ = λψ. { ψ (x p = Φ p 2 (ω r 2 + N 1 r r λ } r 2 (ξ p, C Φ p (ωη p1( r r p(s +2 ( 1 + rp p + r2p 2p, (2.1 C ψ p1 (xr s+2p( / s+ / s+ / s p1( 1 + rp p + r2p 2p,

7 EJDE 22/97 Abdallah El Hamidi & Gennady G. Laptev 7 where C is a positive constant, independent of and r. Let us denote N = supp ( ψ. Since η/ = 1 for r and η/ = for r 2, then N {x ; r 2}. Moreover, since >, then the expressions 1 + rp p + r2p 2p and / s+ / s+ / s are bounded on {x ; constant C such that r 2}. We conclude that there is a positive ψ (x p C ψ p1 (x rs+ 2p x N. Finally, for sufficiently large, we have the estimate (ψ (x p N ψ p1 (x x C 2 2p σ(p1 dx Ω r s++n1 r σ(p1 dθ dr (2.11 s++nσ(p12p if s + + N σ(p 1 >, C 2p ln( if s + + N σ(p 1 =, 2p if s + + N σ(p 1 <. Consider the final test function of the variables x and t: ( t ϕ (x, t = η θ ψ (x. (2.12 On one hand, the same arguments used in (2.11 give the estimate + (ϕ (x, t p N C 2p ϕ p1 2 θ (x, t x 2 σ(p1 dxdt Ω r s++n1 r σ(p1 dθ dr dt (2.13 s++nσ(p1+θ2p if s + + N σ(p 1 >, C θ2p ln( if s + + N σ(p 1 =, θ2p if s + + N σ(p 1 <. On the other hand, if we denote C(, := {x ; x 2}, then supp k ϕ t k k ϕ t k ϕ p1 x C(, p (p1σ dx dt ψ (x dx x (p1σ supp dk η dt k (t/θ dk η (t/ θ dt p k η p1 (t/ θ dt. (2.14

8 8 Nonexistence of solutions to systems EJDE 22/97 Furthermore, C(, ψ (x dx = x (p1σ Ω Φ(ω dθ 2 2 ( ( r s+ s η ( r r N1 r (p1σ dr Ω r s++n1(p1σ dr s+ s++nσ(p1 if s + + N σ(p 1 >, C ln( if s + + N σ(p 1 =, 1 if s + + N σ(p 1 <. Combining the estimates (2.7, (2.14 and (2.15, we obtain supp k ϕ t k ϕ p1 k ϕ t k p dx dt x (p1σ s++nσ(p1θ(kp1 if s + + N σ(p 1 >, C θ(kp1 ln( if s + + N σ(p 1 =, θ(kp1 if s + + N σ(p 1 <. In the following section, we consider the case n = 1. (2.15 ( 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 [, + [, a(x, t and u(x, t. Also assume that Let x ; k1 u (x,. tk1 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

9 EJDE 22/97 Abdallah El Hamidi & Gennady G. Laptev 9 In definition 2.1, let us choose the test function ϕ(x, t = ϕ (x, t defined in (2.12. Thanks to the Hopf lemma, we have a u ϕ ν dx dt. Moreover, the test function ϕ satisfies the equalities Finally, we have j ϕ (x, =, for j {1, 2,..., k 1}. tj Then, inequality (2.2 implies that x σ u p ϕ dx dt Let us introduce the notation I( := A( = B( = k1 u t k1 (x, ϕ (x, dx. u (a + (1 k k x σ u p ϕ dx dt, ϕ K p dx dt ( x σ ϕ p 1 k ϕ p t k dx dt. ( x σ ϕ p 1 Applying Hölder s inequality to (3.1, we obtain I( max ( a, 1 I( 1 p t k ϕ dxdt. (3.1 (A( 1 p + B( 1 p, (3.2 or equivalently I( 1 1 p max ( a, 1 (A( 1 p + B( 1 p. At this stage, we choose the real parameter θ = 2/k and obtain A( C Θ( and B( C Θ(, where s++nσ(p 12(p 1/k if s + + N σ(p 1 >, Θ( = 2(p 1/k ln( if s + + N σ(p 1 =, 2(p 1/k if s + + N σ(p 1 <.

10 1 Nonexistence of solutions to systems EJDE 22/97 If s + + N σ(p 1 >, then explicit computation gives, for sufficiently large, I 1 1 p C α, where α = p 1 p ( s + + N 2 ( 1 1 σ + 2. k p 1 Now, we require that α, which is equivalent to σ + 2 p 1 s + + N 2 ( 1 1. k In this case, I( is bounded uniformly with respect to the variable. Moreover, the function I( is increasing in. Consequently, the monotone convergence theorem implies that the function (x, t, ω, t u(x, t p x σ ( is in L 1 ( ], + [. Furthermore, note that s+ s Φ(ω supp( ϕ {t R +, t 2 2/k } {x, x 2} and ( k ϕ supp t k {t R +, 2/k t 2 2/k } {x, x 2}. Whence, instead of (3.2 we have more precisely I( max ( a, 1 Ĩ( 1 p (A( 1 p + B( 1 p, (3.3 where Ĩ( = C x σ u p ϕ dx dt and C = supp( ϕ supp ( k ϕ t. Finally, k using the dominated convergence theorem, we obtain lim I( =. + This implies u, which contradicts the fact that u is assumed to be nontrivial weak solution. Now, if s + + N σ(p 1, then lim 1/k ln( = and lim 1/k =. + 2(p + 2(p Therefore the integral I( is bounded uniformly with respect to the variable. 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 k u t k (au x σ u p, x K, t ], + [, (3.4

11 EJDE 22/97 Abdallah El Hamidi & Gennady G. Laptev 11 where p > 1, σ > 2, with the initial data u(x, = u (x, in K, i u t i (x, = 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 [, + [, a(x, t and u(x, t. Also assume that k1 u x K; (x,. tk1 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 for =. In this case, the test function ϕ given by (2.12 is not well defined. We choose the new test function ϕ (x, t ϕ, ω, t = r s+ Φ(ω η ( r ( t η. θ The function K [, + [, ω r s+ Φ(ω, is also harmonic. Following the different steps of the last proof with ϕ esp. K instead of ϕ esp., we obtain the result. 4 Higher Order Systems of Evolution Semilinear 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 j u i t (x, will be denoted by u (j j i (x,, for (i, j {1,.., n} {,.., k 1}, and the vector (X 1, X 2,..., X n will denote the solution of the linear system 1 p p.. X 1 σ X 2 σ =. ( pn1 X n1 σ n1 + 2 X n σ n + 2 p n... 1 We have the following non existence result.

12 12 Nonexistence of solutions to systems EJDE 22/97 Theorem 4.1 Assume that for all (x, t [, + [ and i {1, 2,..., n}, u i (x, t and a i (x, t. Also assume that Let x, i {1, 2,..., n}; k1 u i (x,. tk1 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 (x, t defined in (2.12. Thanks to the Hopf lemma, we have ϕ a i u i dx dt, for i {1, 2,..., n}. ν Moreover, the test function ϕ satisfies Finally, we have j ϕ (x, =, for j {1, 2,..., k 1}. tj k1 u i t k1 (x, ϕ (x, dx, i {1, 2,..., n}. Then, inequalities (2.4 and (2.5 imply x σ1 u 1 p1 ϕ x σi u i pi ϕ u i1 u n (a n + (1 k k t k ϕ, (a i1 + (1 k k t k ϕ, 2 i n. For i {1, 2,..., n}, we use the notation I i ( := x σi u i pi ϕ dx dt, ϕ A i ( = K p i dx dt ( x σi ϕ p i1 k ϕ p t B i ( = k i dx dt. ( x σi ϕ p i1 (4.2

13 EJDE 22/97 Abdallah El Hamidi & Gennady G. Laptev 13 Using the Hölder s inequality for system (4.2, we obtain ( I 1 ( max( a n, 1I n ( 1 pn An ( 1 p n + B n ( 1 p n, I i ( max ( a i1, 1 I i1 ( 1 ( p i1 Ai1 ( 1 p i1 + Bi1 ( 1 p i1, 2 i n. (4.3 Whence, there is a positive constant C, independent on and r such that where I 1 1 p 1 p 2...pn i µ ij = C n j=1 ( A 1/p j j 1 + B 1/p µ j ij j, 1 i n, { i1 k=j+1 p k for i > j, i1 k=1 p k n k=j+1 p k for i j. At this stage, we choose the real parameter θ = 2/k to obtain A i ( C Θ i ( and B i ( C Θ i (, for i {1, 2,..., n}, where s++nσi(p i 12(p i 1/k if s + + N σ i (p i 1 >, Θ i ( = 2(p i 1/k ln( if s + + N σ i (p i 1 =, 2(p i 1/k if s + + N σ i (p i 1 <. To estimate the expressions I i (, i {1, 2,..., n}, we consider two cases. Case 1: s + +N σ i (p i 1 >, for any i {1, 2,..., n}. Explicit computation gives, for sufficiently large, where and α i = ( 1 I 1 1 p 1 p 2...pn i C αi, 1 i n, 1 p 1 p 2... p n (s + + N 2 ( 1 1 k τ ij = { n k=i p j1 k k=1 p k for i > j, τ ij = j1 k=i p k for i j. n j=1 (2 + σ jτ ij, p 1 p 2... p n 1 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 {1, 2,..., n} such that I i ( is bounded uniformly with respect to the variable. Using the systems (4.2 and (4.3, we obtain that both I i (, 1 i n,

14 14 Nonexistence of solutions to systems EJDE 22/97 are bounded uniformly with respect to the variable. Moreover, the functions I i (, i {1, 2,..., n}, are increasing in. Consequently, the monotone convergence theorem implies that the functions ( ( r (x, t, ω, t u i (x, t pi x σi s+ s Φ(ω. is in L 1 ( ], + [. Precise that these functions correspond to Furthermore, note that and lim u i(x, t pi x σi ϕ (x, t. + supp( ϕ {t R +, t 2 2/k } {x, x 2} supp ( k ϕ {t R + t k, 2/k t 2 2/k } {x, x 2}. Whence, instead of (4.3 we have more precisely I 1 ( max ( a n, 1 Ĩn( 1 pn (A n ( 1 p n + B n ( 1 p n, I i ( max ( a i1, 1 Ĩi1( 1 p i1 (A i1 ( where 1 p i1 Ĩ i ( = x σi u i pi ϕ dx dt, C + Bi1 ( 1 p i1, 2 i n. (4.4 and C = supp( ϕ supp ( k ϕ t. Finally, using the dominated convergence k theorem, we obtain lim I i( =, + i {1, 2,..., n}. This implies that (u 1, u 2,..., u n (,,...,, 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 {1, 2,..., n}, such that s + + N σ i (p i 1. The same arguments used in Case 1 give, for sufficiently large, I 1 1 p 1 p 2...pn i C αi, 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 then α i1. This implies that I i1 ( is bounded uniformly with respect to the variable, which leads to the same conclusion as the Case 1. This completes the proof.

15 EJDE 22/97 Abdallah El Hamidi & Gennady G. Laptev 15 When problem (2.3 is posed on the cone K instead of the cone-like domain, our result is also valid and the proof changes very slightly. Indeed, consider k u i t k (a iu i x σi+1 u i+1 pi+1, x K, t ], + [, 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 ( i, u (1 i,..., u (k1 i [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. Then we have the following result. Theorem 4.2 Assume that for all (x, t K [, + [ and i {1, 2,..., n}, u i (x, t and a i (x, t. Also assume that x K,, i {1, 2,..., n}; k1 u i (x,. tk1 Let max{x 1, X 2,..., X n } s + + N 2 ( 1 1 k, where pi > 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 esp K instead of ϕ esp to obtain the result. 5 General case We consider the problem k u i (a t k i u i t γi+1 x σi+1 u i+1 pi+1, x, t ], + [, 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 ( i, u (1 i,..., u (k1 i [L 1 loc (] k, 1 i n. We will assume that γ i 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 + (ϕ (x, t p N (t γ x σ p1 dxdt ϕ C 2 θ 2 2p t γ(p1 r s++n1 dt dθ dr Ω r σ(p1 s++nσ(p1+θ2pγθ(p1 if s + + N σ(p 1 >, C θ2pγθ(p1 ln( if s + + N σ(p 1 =, θ2pγθ(p1 if s + + N σ(p 1 <.

16 16 Nonexistence of solutions to systems EJDE 22/97 s++n(σ+γθ(p1+θ2p if s + + N σ(p 1 >, C θ2pγθ(p1 ln( if s + + N σ(p 1 =, θ2pγθ(p1 if s + + N σ(p 1 <. (5.2 and supp k ϕ t k k ϕ t k p (t γ x σ p1 dx dt ϕ s++nσ(p1θ(kp1γθ(p1 if s + + N σ(p 1 >, C θ(kp1γθ(p1 ln( if s + + N σ(p 1 =, θ(kp1γθ(p1 if s + + N σ(p 1 <. s++n(σ+γθ(p1θ(kp1 if s + + N σ(p 1 >, C θ(kp1γθ(p1 ln( if s + + N σ(p 1 =, θ(kp1γθ(p1 if s + + N σ(p 1 <. (5.3 Let the vector (Y 1, Y 2,..., Y n be the solution of the linear system 1 p p pn1 p n... 1 Y 1 Y 2. Y n1 Y n The we have the following non existence result. σ γ 1 /k σ γ 2 /k. σ n γ n1 /k σ n γ n /k Theorem 5.1 Assume that for all (x, t [, + [ and i {1, 2,..., n}, u i (x, t and a i (x, t. Also assume that Let x, i {1, 2,..., n}; k1 u i (x,. tk1 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 (, A i ( and B i ( by I i ( := x σi u i pi ϕ dx dt,

17 EJDE 22/97 Abdallah El Hamidi & Gennady G. Laptev 17 A i ( = B i ( = ϕ K p i dx dt (t γi x σi ϕ p i1 k ϕ p t k i dx dt, (t γi x σi ϕ p i1 respectively. By setting the parameter θ = 2/k, we conclude that A i ( C Θ i ( and B i ( C Θ i (, where s++n(σi+2γi/k(p i 12(p i 1/k if s + + N σ i (p i 1 >, Θ i ( = 2(p i 1/k2γi(p i 1/k ln( if s + + N σ i (p i 1 =, 2(p i 1/k2γi(p i 1/k if s + + N σ(p i 1 <, 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 i 1 2(p 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, 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. References [1] C. Bandle, H.A. Levine, On the existence and nonexistence of global solutions of reaction diffusion equations in sectorial domains, Trans. Amer. Math. Soc., Vol. 316 (1989, [2] C. Bandle, H.A. Levine, Fujita type results for convective-like reactiondiffusion equations in exterior domains, Z. Angew. Math. Phys., Vol. 4 (1989, [3] C. Bandle, M. Essen, On positive solutions of Emden equations in cone-like domains, Arch. Rational Mech. Anal., Vol. 112 (199, [4] K. Deng, H.A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., Vol. 243 (2,

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