Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains
aa r X i v : . [ m a t h . A P ] A p r CRITICAL CRITERIA OF FUJITA TYPE FOR A SYSTEM OFINHOMOGENEOUS WAVE INEQUALITIES IN EXTERIOR DOMAINS
MOHAMED JLELI, BESSEM SAMET, AND DONG YE
Abstract.
We consider blow-up results for a system of inhomogeneous wave inequalities inexterior domains. We will handle three type boundary conditions: Dirichlet type, Neumanntype and mixed boundary conditions. We use a unified approach to show the optimal criteriaof Fujita type for each case. Our study yields naturally optimal nonexistence results for thecorresponding stationary wave system and equation. We provide many new results and closesome open questions. Introduction
This paper is concerned with the study of existence and nonexistence of global weak solu-tions to the system of wave inequalities (cid:3) u ≥ | x | a | v | p , (cid:3) v ≥ | x | b | u | q in (0 , ∞ ) × Ω c . (1.1)Here (cid:3) := ∂ tt − ∆ is the wave operator, Ω c denotes the complement of Ω, with Ω a boundedsmooth open set in R N containing the origin and N ≥
2. Let p, q > a, b ≥ − u ( t, x ) , v ( t, x )) (cid:23) ( f ( x ) , g ( x )) , on (0 , ∞ ) × ∂ Ω;the Neumann type condition:(1.3) (cid:18) ∂u∂ν ( t, x ) , ∂v∂ν ( t, x ) (cid:19) (cid:23) ( f ( x ) , g ( x )) , on (0 , ∞ ) × ∂ Ω;and the mixed boundary condition:(1.4) (cid:18) u ( t, x ) , ∂v∂ν ( t, x ) (cid:19) (cid:23) ( f ( x ) , g ( x )) , on (0 , ∞ ) × ∂ Ω , where f, g ∈ L ( ∂ Ω , R + ) are two fixed functions and ν is the outward unit normal vector on ∂ Ω, relative to Ω c . By the notation (cid:23) , we mean the partial order on R , that is( y , y ) (cid:23) ( z , z ) ⇐⇒ y i ≥ z i , i = 1 , . We write y ≻ z , for y, z ∈ R if y (cid:23) z and y = z .The large-time behavior of solutions to the wave equation (cid:3) u = | u | p in [0 , ∞ ) × R N (1.5)has been studied extensively since four decades. Inspired by the seminal work of John [7] in R , Strauss conjectured in [15] that for each N ≥
2, there exists a critical exponent p c ( N ) of Mathematics Subject Classification.
Key words and phrases.
Inhomogeneous wave inequalities; exterior domain; blow-up; critical criteria.
Fujita type for the global existence question to (1.5) with compactly supported data, and itshould be the positive root of the polynomial( N − p − ( N + 1) p − . (1.6)This conjecture is finally showed to be true for all dimensions N ≥ N ≥ p c ( N ) = N + 1 + √ N + 10 N − N − , then • for any ( u, ∂ t u ) | t =0 compactly supported with positive average, the solution to (1.5)blows-up in a finite time if 1 < p ≤ p c ( N ); • if p > p c ( N ), there are compactly supported initial conditions ( u, ∂ t u ) | t =0 ≻ (0 , (cid:3) u ≥ | u | p in [0 , ∞ ) × R N . (1.7)He found another critical exponent e p c ( N ) = N +1 N − . Pohozaev & Veron [11] generalized Kato’swork and pointed out the sharpness of e p c for (1.7). More precisely, they proved that, • for any N ≥ < p ≤ e p c ( N ), there is no global weak solution to (1.7), if Z R N ∂ t u (0 , x ) dx > • inversely, if p > e p c ( N ), there are positive global solutions satisfying (1.7) and (1.8).A natural question is to understand the wave equation or inequality on other unboundeddomains of R N . The study of blow-up for wave equation on exterior domains was initializedby Zhang in [18]. Among many other things, he considered the inhomogeneous equation (cid:3) u = | x | α | u | p in (0 , ∞ ) × Ω c , (1.9)where N ≥ α > − ⊂ R N is a smooth bounded set. Under the Neumann boundarycondition ∂u∂ν = f ≥ , ∞ ) × ∂ Ω, Zhang showed that the critical exponent becomes now N + αN − : • when 1 < p < N + αN − , (1.9) has no global solution if f • when p > N + αN − , problem (1.9) has global solutions for some f > α = 0 and Ω = B r was studied in [6]. Here and after, B r denotesthe ball centered at 0 with radius r >
0. Our study for (1.1) will yield an optimal answer for(1.9) under the Dirichlet boundary condition, see Corollary 1.9 below.Here we are interested to understand the blow-up of solutions to (1.1) under various bound-ary conditions (1.2), (1.3) and (1.4). We will determine the critical criteria of Fujita type for( p, q ) in each case, without any assumption on the initial data. As far as we know, we arenot aware of such results concerning system of wave equations or inequalities. The study for(1.1) yields natural consequences for the corresponding stationary system, which seem alsoto be new for the Neumann type condition and the mixed boundary condition, see Corollary
YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 3 Q = (0 , ∞ ) × Ω c and Γ = (0 , ∞ ) × ∂ Ω . We introduce the test function space D = (cid:26) ϕ ∈ C ( Q, R + ) : ϕ | Γ = 0 , ∂ϕ∂ν | Γ ≤ (cid:27) . Here, C ( Q, R + ) means the space of nonnegative C functions compactly supported in Q .Notice that Ω c is closed and Γ ⊂ Q . Definition 1.1.
A pair ( u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is a global weak solution to (1.1) - (1.2) , iffor any ϕ ∈ D , (1.10) Z Q | x | a | v | p ϕdxdt − Z Γ ∂ϕ∂ν f dσdt ≤ Z Q u (cid:3) ϕdxdt and (1.11) Z Q | x | b | u | q ϕdxdt − Z Γ ∂ϕ∂ν gdσdt ≤ Z Q v (cid:3) ϕdxdt. For Neumann boundary problem, we consider the test function space N = (cid:26) ϕ ∈ C ( Q, R + ) : ∂ϕ∂ν | Γ = 0 (cid:27) . Definition 1.2.
A pair ( u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is called a global weak solution to (1.1) – (1.3) , if for any ψ ∈ N , (1.12) Z Q | x | a | v | p ψdxdt + Z Γ ψf dσdt ≤ Z Q u (cid:3) ψdxdt and (1.13) Z Q | x | b | u | q ψdxdt + Z Γ ψgdσdt ≤ Z Q v (cid:3) ψdxdt. For the mixed boundary problem, the natural test function space is then
D × N . Definition 1.3.
A pair ( u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is a global weak solution to (1.1) – (1.4) , iffor any ( ϕ, ψ ) ∈ D × N , there holds (1.10) and (1.13) . Define I f = Z ∂ Ω f dσ, for any f ∈ L ( ∂ Ω) . Let sgn denote the standard sign function over R . Our main result is the following. Theorem 1.4.
Assume that ( a, b ) ≻ ( − , − , f, g ∈ L ( ∂ Ω) , ( I f , I g ) ≻ (0 , and p, q > .Let either N = 2 ; or N ≥ and max (cid:26) sgn( I f ) × p ( q + 1) + pb + apq − , sgn( I g ) × q ( p + 1) + qa + bpq − (cid:27) > N. (1.14) Then
MOHAMED JLELI, BESSEM SAMET, AND DONG YE (i) there exists no global weak solution to (1.1) – (1.2) if f, g ≥ ; (ii) there exists no global weak solution to (1.1) – (1.3) ; (iii) there exists no global weak solution to (1.1) – (1.4) if p > and f ≥ .Furthermore, if Ω = B r , the sign condition for f, g can be erased in (i) and (iii) . Remark 1.5.
The condition (1.14) is equivalent to I f > and δ > N − or I g > and γ > N − , where δ = a + 2 + p ( b + 2) pq − , γ = b + 2 + q ( a + 2) pq − . (1.15) Therefore, (1.14) always holds true when N = 2 , p, q > and ( a, b ) ≻ ( − , − . In fact, the constants δ , γ come from the scaling transform of the stationary problem − ∆ u = | x | a v p , − ∆ v = | x | b u q . (1.16)Let ( u, v ) be a solution to the system (1.16), then for any λ > u λ ( x ) = λ δ u ( λx ) , v λ ( x ) = λ γ v ( λx ) satisfy still (1.16). Remark 1.6.
Assume that N ≥ , p, q > , pq > , and < min( δ, γ ) ≤ max( δ, γ ) < N − . (1.17) Let ( u ∗ , v ∗ )( x ) = ( A u | x | − δ , A v | x | − γ ) with A u , A v > given by A pq − u = δ ( N − − δ )[ γ ( N − − γ )] p , A pq − v = γ ( N − − γ )[ δ ( N − − δ )] q . We can check that ( u ∗ , v ∗ ) is a positive solution to (1.16) in R N \{ } . If Ω is star-shapedwith respect to the origin, there holds ∂u ∗ ∂ν , ∂v ∗ ∂ν ≥ on ∂ Ω with respect to Ω c . So ( u ∗ , v ∗ ) isa stationary solution to (1.1) and satisfies all the boundary conditions (1.2) , (1.3) and (1.4) for suitable f, g ≥ . This means that the condition (1.14) is optimal for the nonexistence ofglobal solution to the wave system (1.1) . Remark 1.7.
Assume that B r ⊂ Ω with r > . Let a, b ≤ , p, q > , pq > . Similarly asabove, there are suitable A , A > such that u ( t, x ) = A ( t + 1) − p +1) pq − , v ( t, x ) = A ( t + 1) − q +1) pq − satisfy (cid:3) u = r a v p and (cid:3) v = r b u q in R + × R N . Therefore, ( u, v ) resolves (1.1) and satisfiesall the boundary conditions (1.2) , (1.3) and (1.4) with f = g = 0 . This means the necessityof the assumption ( I f , I g ) ≻ (0 , in Theorem 1.4 when a, b ≤ . Clearly, Theorems 1.4 yields nonexistence results for the corresponding stationary problem − ∆ u ≥ | x | a | v | p , − ∆ v ≥ | x | b | u | q in Ω c . (1.18) Corollary 1.8.
Let N ≥ , f, g ∈ L ( ∂ Ω) and ( a, b ) ≻ ( − , − . Assume that ( I f , I g ) ≻ (0 , and p, q > satisfy (1.14) . Then (1.18) has no weak solution if one of the followingconditions holds true: (i) f, g ∈ L ( ∂ Ω , R + ) , ( u, v ) (cid:23) ( f, g ) on ∂ Ω ; (ii) (cid:0) ∂u∂ν , ∂v∂ν (cid:1) (cid:23) ( f, g ) on ∂ Ω ; (iii) f ∈ L ( ∂ Ω , R + ) , p > and (cid:0) u, ∂v∂ν (cid:1) (cid:23) ( f, g ) on ∂ Ω . YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 5
We refind Corollary 1.3 in [16] for the Dirichlet boundary condition case, where a, b > − f, g can be erased if Ω = B r .Theorems 1.4 yields also new result for the following wave inequality in exterior domain (cid:3) u ≥ | x | a | u | p in (0 , ∞ ) × Ω c , u ( t, x ) ≥ f ( x ) on (0 , ∞ ) × ∂ Ω , (1.19)and answers an open question proposed in Remark 1.5 of [18]. Corollary 1.9.
Let a > − , f ∈ L ( ∂ Ω , R + ) and N ≥ . If (1.20) I f > and < p < N + aN − , there is no global weak solution in L ploc ( Q ) to (1.19) . In other words, p ∗ = N + aN − is the Fujitacritical exponent for (1.19) if N ≥ , a > − . Indeed, when b = a , q = p > N < p ( q + 1) + pb + apq − p + ap − ⇐⇒ p < N + aN − . Taking ( v, b, q, g ) = ( u, a, p, f ) in (1.1)–(1.2), we deduce the above nonexistence result frompart (i) of Theorem 1.4. Again the condition f ≥ B r . On the otherhand, (1.19) admits positive solution for a > − p > N + aN − , N ≥ f > k f k ∞ issufficiently small (see [18, Proposition 6.1]). Remark 1.10.
Similarly, for the exterior Neumann inequality (cid:3) u ≥ | x | a | u | p in (0 , ∞ ) × Ω c , ∂u∂ν ( t, x ) ≥ f ( x ) on (0 , ∞ ) × ∂ Ω , we refind the critical exponent p ∗ = N + aN − as indicated by [18, Theorem 1.4] . Let us say some words for our approach which is based on suitable test functions andintegral estimates. At first glance it looks like the method in [18, 16] or similar works for theblow-up study in exterior domains, however some key choices are completely different. • In most previous works, we use cut-off functions with fixed scaling for the time variable t , we obtain then integral estimates on cylinder type domain Q D = Σ t × Σ x where | Σ x | ∼ R N and Σ t is of length CR or CR . Here we consider a large scale for t bychoosing | Σ t | ∼ R θ with θ large enough. • In [18, 16], they often use test functions with support away from the boundary ∂ Ω,hence it’s more difficult to observe the effect of the Dirichlet boundary condition. Inthis work, we make use of harmonic function on Ω c with zero boundary condition,which permits to cut off only at infinity.These ideas make our method more transparent, for example we avoid the iterative step usedin [18, 16].The paper is organized as follows. In section 2, we establish some preliminary estimatesthat will be used in the proof of our main results. In Section 3, we prove Theorem 1.4 in twodimensional case. The proof of Theorem 1.4 for N ≥ MOHAMED JLELI, BESSEM SAMET, AND DONG YE
The symbols C or C i denote always generic positive constants, which are independent ofthe scaling parameter T and the solutions u, v . Their values could be changed from one lineto another. We will write B := B for the unit ball, and we will use the notation h ∼ k fortwo positive functions or quantities, which satisfy C h ≤ k ≤ C h .2. Preliminary estimates
Let N ≥
2. We introduce the following harmonic function in Ω c : − ∆ H Ω = 0 in Ω c , H Ω = 0 on ∂ Ω;and lim | x |→∞ H Ω ( x )ln | x | = 1 if N = 2; lim | x |→∞ H Ω ( x ) = 1 if N ≥ . Clearly H Ω is uniquely determined and H Ω > c .We need also two cut-off functions. Let ξ ∈ C ∞ ( R N ) satisfies0 ≤ ξ ≤ ξ ≡ B ; ξ ( x ) ≡ | x | ≥ . Fix also ϑ ∈ C ∞ ( R ) such that ϑ ≥ , ϑ , supp( ϑ ) ⊂ (0 , . For 0 < T < ∞ , let Ξ T ( x ) = H Ω ( x ) ξ (cid:16) xT (cid:17) k in Ω c and ϑ T ( t ) = ϑ (cid:18) tT θ (cid:19) k in (0 , ∞ ) . Here, k ≥ θ > D T ( t, x ) = ϑ T ( t )Ξ T ( x ) , ( t, x ) ∈ (0 , ∞ ) × Ω c and N T ( x ) = ϑ T ( t ) ξ (cid:16) xT (cid:17) k , ( t, x ) ∈ (0 , ∞ ) × Ω c . Obviously, for any
T > dist(0 , ∂
Ω) and θ > D T , N T ) ∈ D × N . Denote H := H B , i.e. H ( x ) = (cid:26) ln | x | if N = 2 , − | x | − N if N ≥ . In the following, we will give some integral estimates for D T and N T . Our approach usesonly the asymptotic behavior of H Ω and its derivatives at infinity. For simplicity, we willdetail our proof only for the unit open ball B . The readers can be convinced easily that thesame ideas work well for general smooth open sets Ω. More precisely, as B r ⊂ Ω ⊂ B r with r > r >
0, thanks to the maximum principle, we have H B r ≤ H Ω ≤ H B r in B cr . Thestandard elliptic theory yields that |∇ k H Ω | ( x ) ∼ |∇ k H | ( x ) as | x | → ∞ , for all k ≥ YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 7
Lemma 2.1.
Let N = 2 , α ∈ R and β > − . There holds, as T → + ∞ , Z < | x |
Let N ≥ , α ∈ R and β > − . There holds, as T → + ∞ . Z < | x |
Lemma 2.3. | ∆Ξ T ( x ) | ≤ C k (cid:18) H Ω ( x ) T + | x | − N T (cid:19) ξ (cid:16) xT (cid:17) k − χ { T < | x | < T } in Ω c . and | ϑ ′′ T ( t ) | ≤ C k T θ ϑ (cid:18) tT θ (cid:19) k − χ { Let τ ∈ R , θ > , m > , k > mm − and N = 2 . We have, as T → + ∞ , Z Q | x | − τm − D − m − T | ∂ tt D T | mm − dxdt = O (cid:16) T − τ +( m +1) θm − ln T (cid:17) if τ < m − ,O (cid:16) T − ( m +1) θm − (ln T ) (cid:17) if τ = 2( m − ,O (cid:16) T − ( m +1) θm − (cid:17) if τ > m − . MOHAMED JLELI, BESSEM SAMET, AND DONG YE Proof. Without loss of generality, let Ω = B . By the definition of D T and Lemma 2.3, we get Z Q | x | − τm − D − m − T | ∂ tt D T | mm − dxdt = Z ∞ ϑ T ( t ) − m − | ϑ ′′ T ( t ) | mm − dt × Z B c | x | − τm − Ξ T ( x ) dx ≤ CT − θmm − Z T θ ϑ k − mm − (cid:18) tT θ (cid:19) dt × Z < | x | < T | x | − τm − H ( x ) dx ≤ CT − ( m +1) θm − Z < | x | < T | x | − τm − H ( x ) dx. Using Lemma 2.1 with α = − τm − and β = 1, we obtain the claimed estimate. (cid:3) Similarly, we deduce from Lemma 2.2 that Lemma 2.5. Let τ ∈ R , θ > , m > , k > mm − and N ≥ . There holds, as T → + ∞ , Z Q | x | − τm − D − m − T | ∂ tt D T | mm − dxdt = O (cid:16) T N − τ +( m +1) θm − (cid:17) if τ < N ( m − ,O (cid:16) T − ( m +1) θm − ln T (cid:17) if τ = N ( m − ,O (cid:16) T − ( m +1) θm − (cid:17) if τ > N ( m − . Furthermore, there holds Lemma 2.6. Let τ ∈ R , θ > , m > , k > mm − and N = 2 . Then Z Q | x | − τm − D − m − T | ∆ D T | mm − dxdt = O (cid:16) T θ − τ +2 m − ln T (cid:17) , as T → + ∞ . Proof. Consider still Ω = B . By the definition of D T , Z Q | x | − τm − D − m − T | ∆ D T | mm − dxdt = Z ∞ ϑ T ( t ) dt × Z B c | x | − τm − H ( x ) − m − ξ (cid:16) xT (cid:17) − km − | ∆Ξ T ( x ) | mm − dx = CT θ Z B c | x | − τm − H ( x ) − m − ξ (cid:16) xT (cid:17) − km − | ∆Ξ T ( x ) | mm − dx. (2.1)Applying Lemma 2.3, there holds, for any | x | > H ( x ) − m − ξ (cid:16) xT (cid:17) − m − | ∆Ξ T ( x ) | mm − ≤ CH ( x ) − m − ξ k − mm − (cid:16) xT (cid:17) (cid:18) H ( x ) T + | x | − N T (cid:19) mm − χ { T < | x | < T } ≤ C h T − mm − H ( x ) + T − mm − H ( x ) − m − | x | (1 − N ) mm − i χ { T < | x | < T } . (2.2) YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 9 Combining (2.1)–(2.2), we obtain Z Q | x | − τm − D − m − T | ∆ D T | mm − dxdt ≤ CT θ − mm − Z T < | x | < T | x | − τm − H ( x ) dx + CT θ − mm − Z T < | x | < T H ( x ) − m − | x | − τ +(1 − N ) mm − dx = O (cid:16) T θ − τ +2 m − ln T (cid:17) , as T goes to + ∞ . The last line is given by N = 2 and Lemma 2.1. (cid:3) Very similarly, using the expression of H and Lemma 2.2, we have Lemma 2.7. Let τ ∈ R , θ > , m > , k > mm − and N ≥ , then Z Q | x | − τm − D − m − T | ∆ D T | mm − dxdt = O (cid:16) T N − θ − τ +2 m − (cid:17) , as T → + ∞ . Estimates involving N T .Lemma 2.8. Let τ ∈ R , θ > , m > , k > mm − and N ≥ . There holds, as T → + ∞ , Z Q | x | − τm − N − m − T | ∂ tt N T | mm − dxdt = O (cid:16) T − ( m +1) θm − ln T (cid:17) if τ ≥ N ( m − ,O (cid:16) T N − τ +( m +1) θm − (cid:17) if τ < N ( m − . Proof. Consider Ω = B . By the definition of N T and Lemma 2.3, we get Z Q | x | − τm − N − m − T | ∂ tt N T | mm − dxdt = Z ∞ ϑ T ( t ) − m − | ϑ ′′ T ( t ) | mm − dt × Z B c | x | − τm − ξ k (cid:16) xT (cid:17) dx ≤ CT − ( m +1) θm − Z < | x | < T | x | − τm − dx, The desired estimate follows directly from Lemmas 2.1–2.2 with α = − τm − and β = 0. (cid:3) Lemma 2.9. Let τ ∈ R , θ > , m > , k > mm − and N ≥ . Then Z Q | x | − τm − N − m − T | ∆ N T | mm − dx dt = O (cid:16) T N − θ − τ +2 m − (cid:17) , as T → + ∞ . Proof. As in Lemma 2.3, there holds (cid:12)(cid:12)(cid:12) ∆ h ξ k (cid:16) xT (cid:17)i(cid:12)(cid:12)(cid:12) ≤ C k T − ξ (cid:16) xT (cid:17) k − χ { T < | x | < T } . (2.3)We can claim the mentioned estimate similarly as for Lemmas 2.6. (cid:3) Estimates involving D T and N T . The following are some estimates necessary tohandle the mixed boundary problem (1.1)–(1.4). Lemma 2.10. Let τ ∈ R , θ > , m > , k > mm − and N ≥ . There holds, as T → + ∞ , Z Q | x | − τm − D − m − T | ∂ tt N T | mm − dxdt = O (cid:16) T − ( m +1) θm − ln T (cid:17) if τ ≥ N ( m − ,O (cid:16) T N − τ +( m +1) θm − (cid:17) if τ < N ( m − . Proof. Without loss of generality, consider Ω = B . By the definitions of D T and N T , thanksto Lemma 2.3, we get Z Q | x | − τm − D − m − T | ∂ tt N T | mm − dx dt = Z ∞ ϑ T ( t ) − m − | ϑ ′′ T ( t ) | mm − dt Z B c | x | − τm − H ( x ) − m − ξ k (cid:16) xT (cid:17) dx ≤ CT − ( m +1) θm − Z < | x | < T | x | − τm − H ( x ) − m − dx. Applying Lemmas 2.1–2.2 with α = − τm − and β = − m − ∈ ( − , 0) (here m > (cid:3) Similarly, we have Lemma 2.11. Let τ ∈ R , θ > , m > , k > mm − and N ≥ . Then Z Q | x | − τm − D − m − T | ∆ N T | mm − dx dt = O (cid:16) T N − θ − τ +2 m − (cid:17) , as T → + ∞ . Proof. Using (2.3), there holds, for large T , Z Q | x | − τm − D − m − T | ∆ N T | mm − dxdt = Z ∞ ϑ T ( t ) dt × Z Ω c | x | − τm − H ( x ) − m − ξ (cid:16) xT (cid:17) − km − (cid:12)(cid:12)(cid:12) ∆ h ξ k (cid:16) xT (cid:17)i(cid:12)(cid:12)(cid:12) mm − dx ≤ CT θ − mm − Z T < | x | < T | x | − τm − H ( x ) − m − ξ (cid:16) xT (cid:17) k − mm − dx ≤ CT θ − mm − Z T < | x | < T | x | − τm − H ( x ) − m − dx ≤ CT θ − mm − Z T < | x | < T | x | − τm − dx = CT N − θ − τ +2 m − . So we are done. (cid:3) Two dimensional situation In this section, we prove successively the parts (i), (ii) and (iii) of Theorem 1.4 for N = 2.We will detail the proof for (i). The proofs for parts (ii) and (iii) are similar, so we proceedmore quickly. Let p, q > k > max (cid:26) pp − , qq − (cid:27) . As mentioned above, we consider only Ω = B , and we explain in Remarks 3.1–3.2 how thesame ideas work for general case. YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 11 Proof of part (i). We argue by contradiction by assuming that the pair ( u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is a global weak solution to (1.1)–(1.2). For T > θ > 0, taking ϕ = D T in (1.10), then Z Q | x | a | v | p D T dxdt − Z Γ ∂D T ∂ν f dσ dt ≤ Z Q | u || (cid:3) D T | dxdt. Moreover, as ∂ ν H is constant on ∂B , − Z Γ ∂D T ∂ν f dσdt = C Z ∞ ϑ (cid:16) sT θ (cid:17) k ds × Z ∂B f ( x ) dσ = CI f T θ , (3.2)where C is a constant depending only on H and ϑ . This yields(3.3) Z Q | x | a | v | p D T dxdt + I f T θ ≤ C Z Q | u || (cid:3) D T | dxdt. Similarly, taking ϕ = D T in (1.11), we get(3.4) Z Q | x | b | u | q D T dxdt + I g T θ ≤ C Z Q | v || (cid:3) D T | dxdt. By H¨older’s inequality, there holds(3.5) Z Q | u || ∂ tt D T | dxdt ≤ (cid:18)Z Q | x | b | u | q D T dxdt (cid:19) q (cid:18)Z Q | x | − bq − D − q − T | ∂ tt D T | qq − dxdt (cid:19) q − q . Using Lemma 2.4 with τ = b and m = q , we obtain(3.6) Z Q | x | − bq − D − q − T | ∂ tt D T | qq − dxdt = O (cid:16) T − b +( q +1) θq − ln T (cid:17) if b < q − ,O (cid:16) T − ( q +1) θq − (ln T ) (cid:17) if b = 2( q − ,O (cid:16) T − ( q +1) θq − (cid:17) if b > q − , as T → + ∞ .On the other hand,(3.7) Z Q | u || ∆ D T | dxdt ≤ (cid:18)Z Q | x | b | u | q D T dxdt (cid:19) q (cid:18)Z Q | x | − bq − D − q − T | ∆ D T | qq − dxdt (cid:19) q − q . Applying Lemma 2.6 with τ = b and m = q , we have(3.8) (cid:18)Z Q | x | − bq − D − q − T | ∆ D T | qq − dxdt (cid:19) q − q = O (cid:16) T θ ( q − − b − q (ln T ) q − q (cid:17) , as T → + ∞ . Combining (3.3) with (3.5)–(3.8), for T large enough, there holds(3.9) J T ( a, v ) + I f T θ ≤ C [ J T ( b, u )] q α ( T ) , where J T ( a, v ) = Z Q | x | a | v | p D T dxdt, J T ( b, u ) = Z Q | x | b | u | q D T dxdt and(3.10) α ( T ) = T θ ( q − − b − q (ln T ) q − q + ( T − ( q +1) θq (ln T ) q − q if b ≥ q − ,T q − − b − ( q +1) θq (ln T ) q − q if b < q − . Exchanging now the roles of u and v , using (3.4), we have also(3.11) J T ( b, u ) + I g T θ ≤ C [ J T ( a, v )] p β ( T ) , where(3.12) β ( T ) = T θ ( p − − a − p (ln T ) p − p + ( T − ( p +1) θp (ln T ) p − p if a ≥ p − ,T p − − a − ( p +1) θp (ln T ) p − p if a < p − . Without loss of generality, we assume I f > 0, as ( I f , I g ) ≻ (0 , T , J T ( a, v ) + T θ ≤ CJ T ( a, v ) pq β ( T ) q α ( T ) . Using Young’s inequality, we get(3.13) T − θ α ( T ) pqpq − β ( T ) ppq − ≥ C > , for large T . However, we claim that with large θ > T → + ∞ T − θ α ( T ) pqpq − β ( T ) ppq − = 0 . By (3.10) and (3.12), for θ > α ( T ) ∼ T θ ( q − − b − q (ln T ) q − q , β ( T ) ∼ T θ ( p − − a − p (ln T ) p − p , as T → + ∞ . (3.15)Therefore T − θ α ( T ) pqpq − β ( T ) ppq − ∼ T − ( b +2) p +( a +2) pq − ln T, as T → + ∞ , (3.16)hence (3.14) holds true (with large but fixed θ ) since ( a, b ) ≻ ( − , − N = 2. Remark 3.1. For general smooth open sets Ω , we have no longer ∂ ν H Ω ≡ constant on ∂ Ω ,hence we have no longer the equality (3.2) for all f ∈ L ( ∂ Ω) . However, by Hopf ’s Lemma, ∂ ν H Ω ≤ − C Ω < on ∂ Ω . If now f ≥ and T > dist(0 , ∂ Ω) , there holds − Z Γ ∂D T ∂ν f dσdt ≥ C Ω Z ∞ ϑ (cid:16) sT θ (cid:17) k ds × Z ∂ Ω f ( x ) dσ ≥ CI f T θ . where C depends only on Ω and ϑ . It’s easy to see that all the arguments are still valid for f, g ≥ . Proof of part (ii). Assume that ( u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is a global weak solutionto (1.1)–(1.3). Let K T ( a, v ) = Z Q | x | a | v | p N T dxdt, K T ( b, u ) = Z Q | x | b | u | q N T dxdt. By H¨older’s inequality, Z Q | u || (cid:3) N T | dxdt ≤ CK T ( b, u ) q (cid:18)Z Q | x | − bq − N − q − T | (cid:3) N T | qq − dxdt (cid:19) q − q . (3.17) YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 13 Applying Lemmas 2.8–2.9 with τ = b , m = q and N = 2, remarking that the involvedestimates are exactly of the same order or better than those in Lemmas 2.4 and 2.6, wededuce that for T large, (cid:18)Z Q | x | − bq − N − q − T | (cid:3) N T | qq − dxdt (cid:19) q − q ≤ Cα ( T )(3.18)where α ( T ) is given by (3.10). Similarly, there holds, for T large, Z Q | v || (cid:3) N T | dxdt ≤ CK T ( a, v ) p (cid:18)Z Q | x | − ap − N − p − T | (cid:3) N T | pp − dxdt (cid:19) p − p ≤ CK T ( a, v ) p β ( T ) , (3.19)where β ( T ) is given by (3.12). Moreover, by the definition of N T , for T large, Z Γ f N T dσdt = Z Γ f ϑ T ( t ) dσdt = CI f T θ , Z Γ gN T dσdt = CI g T θ . (3.20)Take ψ = N T in (1.12)–(1.13), combining with (3.17)–(3.19), we get(3.21) K T ( a, v ) + I f T θ ≤ CK T ( b, u ) q α ( T ) , K T ( b, u ) + I g T θ ≤ CK T ( a, v ) p β ( T ) . Remark that (3.21) is just (3.9) and (3.11), if we replace K T by J T . Assuming withoutloss of generality I f > 0, repeating the previous arguments for part (i), (3.13) still holds true.However, we can always choose θ > Remark 3.2. To get (3.20) , we used only ξ (cid:0) xT (cid:1) ≡ on ∂ Ω when T is large, which is truefor general bounded open sets Ω . Proof of part (iii). We use again the method of contradiction. Assume that ( u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is a global weak solution to (1.1)–(1.4), with now p > 2. We take ( D T , N T )as a couple of test functions, and use the same notations J T , K T , α ( T ) and β ( T ) as before.Inserting ϕ = D T in (1.12), we obtain, for T large,(3.22) J T ( a, v ) + I f T θ ≤ C [ J T ( b, u )] q α ( T ) ≤ C [ K T ( b, u ) ln T ] q α ( T ) . The key point here is to estimate k v (cid:3) N T k L ( Q ) using J T ( a, v ). By H¨older’s inequality, Z Q | v || (cid:3) N T | dxdt ≤ J T ( a, v ) p (cid:18)Z Q | x | − ap − D − p − T | (cid:3) N T | pp − dx dt (cid:19) p − p ≤ CJ T ( a, v ) p β ( T ) . (3.23)The last inequality follows from Lemmas 2.10–2.11 with τ = a , m = p and N = 2. Moreover,let ψ = N T in (1.13), using (3.23), there holds(3.24) K T ( b, u ) + I g T θ ≤ C Z Q | v || (cid:3) N T | dxdt ≤ CJ T ( a, v ) p β ( T ) . • Assume first I f > 0, combining (3.22) and (3.24), we deduce that J T ( a, v ) + T θ ≤ CJ T ( a, v ) pq α ( T )[ β ( T ) ln T ] q . Applying Young’s inequality, there holds T − θ α ( T ) pqpq − [ β ( T ) ln T ] ppq − ≥ C > , for large T . However, fix θ > • Assume now I g > 0. Always using (3.22) and (3.24), there holds K T ( b, u ) + T θ ≤ CK T ( b, u ) pq β ( T )[ α ( T )(ln T ) q ] p , hence T − θ [ α ( T ) ln T ] qpq − β ( T ) pqpq − ≥ C > , for large T . Moreover, fixing a large θ such that (3.15) is valid, we get, as T → + ∞ , T − θ [ α ( T ) ln T ] qpq − β ( T ) pqpq − ∼ T − ( b +2)+( a +2) qpq − (ln T ) qpq − . This contradicts the previous inequality.To conclude, if ( I f , I g ) ≻ (0 , 0) and ( a, b ) ≻ ( − , − N = 2. (cid:3) Proof of Theorem 1.4 for N ≥ N ≥ p, q > k satisfy (3.1). As above, we can consider just Ω = B . The proofis very similar to the case N = 2.4.1. Proof of parts (i)–(ii). Without restriction of the generality, suppose I f > δ + 2 = 2 p ( q + 1) + pb + apq − > N, where δ is defined by (1.15). Assume that ( u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is a global weak solutionto (1.1)–(1.2). Proceeding as above, by Lemmas 2.5 and 2.7 with τ = b and m = q , H¨olderand Young’s inequalities, we obtain again (3.13) with now(4.2) α ( T ) = T ( N − θ )( q − − b − q + ( T − ( q +1) θq (ln T ) q − q if b ≥ N ( q − ,T N ( q − − b − ( q +1) θq if b < N ( q − β ( T ) = T ( N − θ )( p − − a − p + ( T − ( p +1) θp (ln T ) p − p if a ≥ N ( p − ,T N ( p − − a − ( p +1) θp if a < N ( p − . Taking θ large enough, when T → + ∞ , there holds α ( T ) ∼ T ( N − θ )( q − − b − q and β ( T ) ∼ T ( N − θ )( p − − a − p . (4.4)Hence T − θ α ( T ) pqpq − β ( T ) ppq − ∼ T N − − ( b +2) p +( a +2) pq − = T N − − δ . Thanks to (4.1), (3.14) follows by choosing a large θ .The contradiction between (3.13) and (3.14) means that no global weak solution exists for(1.1)–(1.2). The nonexistence result for (1.1)–(1.3) can be derived by similar arguments, sowe omit the proof. YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 15 Proof of part (iii). Let p > u, v ) ∈ L qloc ( Q ) × L ploc ( Q ) is a globalweak solution to (1.1)–(1.4). For T > 1, using ϕ = D T in (1.12), we can claim that(4.5) J T ( a, v ) + I f T θ ≤ C [ J T ( b, u )] q α ( T ) ≤ C [ K T ( b, u )] q α ( T ) . Here we used H ( x ) ≤ N ≥ N = 2, taking ψ = N T in (1.13), we get (3.24).Here α ( T ) and β ( T ) are given by (4.2) and (4.3). Assume first I f > θ large enough, which isimpossible.Assume now I g > γ + 2 = 2 q ( p + 1) + qa + bpq − > N, with γ given by (1.15). Combining (4.5) and (3.24), there holds K T ( b, u ) + T θ ≤ CK T ( b, u ) pq β ( T ) α ( T ) p , hence T − θ α ( T ) qpq − β ( T ) pqpq − ≥ C > , as T → ∞ . We can conclude if lim T → + ∞ T − θ α ( T ) qpq − β ( T ) pqpq − = 0 . (4.6)Taking θ large enough, by (4.4), there holds T − θ α ( T ) qpq − β ( T ) pqpq − ∼ T N − − γ for T large, so(4.6) holds true and the proof of part (iii) is completed. (cid:3) Further Remarks It’s worthy to mention that the system of wave equations in the whole space, i.e. (cid:3) u = | v | p , (cid:3) v = | u | q in (0 , ∞ ) × R N , p, q > , N ≥ ∂ t u (0 , x ), ∂ t v (0 , x ), there exists a criticalcurve for the global existence, which ismax (cid:26) p + 2 + q − pq − , q + 2 + p − pq − (cid:27) = N − . The corresponding system of inequalities was studied in [11], where Theorem 6 (see alsoApplication 2) proves the nonexistence of nontrivial global solution if1 < p, q < N + 1 N − , Z R N ∂ t u (0 , x ) dx ≥ Z R N ∂ t v (0 , x ) dx ≥ . We can see that the critical criteria in the above cases are quite different for our situation.This phenomenon is similar to comparing Strauss’s critical exponent p c ( N ) for (1.5), Kato’sexponent e p c ( N ) for (1.7) and Zhang’s exponent p ∗ for (1.9). In other words, the blow-upfor inequalities on exterior domains is of very different nature comparing to the whole spacesituation. The critical case N ≥ (cid:26) sgn( I f ) × p ( q + 1) + pb + apq − , sgn( I g ) × q ( p + 1) + qa + bpq − (cid:27) = N for the system (1.1) is not investigated here. It should be interesting to decide whether thiscritical curve in ( p, q )–plan belongs to the blow-up situation.For the mixed boundary condition case (1.4), we supposed that p > < p ≤ f = g = 0, is very special. We may have no critical criteria of Fujita type in general.However, the simple example there only works for a, b ≤ 0. It could be interesting tounderstand the long term behavior of solutions to (1.1) with a, b > f = g = 0.In the case of homogeneous constraints, another way to avoid the simple example in Remark1.7 is to add sign condition or nonnegative average constraint on ∂ t u (0 , x ), ∂ t v (0 , x ) as in[8, 11]. For example, consider the following problem: (cid:3) u ≥ | x | a | u | p in R + × B cr , u ≥ R + × ∂B r and ∂ t u (0 , x ) ≥ , where B r ⊂ R N , N ≥ a > − 2. Laptev [10] showed that the critical exponent for existenceof non trivial global solution is N +1+ aN − .The understanding for wave equation on exterior domains with homogeneous Dirichletboundary condition is more difficult. Consider (1.9) with N ≥ a = 0 and u = 0 on ∂ Ω.There are many works who suggest that the critical exponent of Fujita type could be thesame as for the whole space, i.e. p c ( N ) given by (1.6). • Let 1 < p ≤ p c ( N ), it’s showed that for special choice of ( u , u ) ≻ (0 , u, ∂ t u ) | t =0 = ( εu , εu ) will blow up for any ε > 0, see [9] and thereferences therein. However, the blow-up result for general ( u , u ) seems unknown. • For p > p c ( N ), there exist some global existence results for some p > p c ( N ) in lowdimensions N ≤ u , u > 0. See forinstance [2, 14].As far as we know, it seems that there is no general result for the global existence of waveequation on exterior domains (1.9) with homogeneous Neumann boundary condition. Acknowledgments . M.J. and B.S. extend their appreciation to the Deanship of ScientificResearch at King Saud University for funding this work through research group No. RGP-237.D.Y. is partially supported by Science and Technology Commission of Shanghai Municipality(STCSM), grant No. 18dz2271000. References [1] D. Del Santo, V. Georgiev, E. Mitidieri, Global existence of the solutions and formation of singularitiesfor a class of hyperbolic systems. Geometrical optics and related topics, Progr. Nonlinear Diff. Equ.Appl., 32, Birkh¨auser (1997), 117–140.[2] Y. Du, J. Metcalfe, CD. Sogge, Y Zhou, Concerning the Strauss conjecture and almost global existencefor nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Diff. Equ. 33 (2008), 1487–1506.[3] R.T. Glassey, Finite-time blow up for solutions of nonlinear wave equations, Math. Z. 177 (1981),323–340. YSTEM OF INHOMOGENEOUS WAVE INEQUALITIES 17 [4] R.T. Glassey, Existence in the large for (cid:3) u = F ( u ) in two space dimensions, Math. 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Zhou, Blow-up of solutions to semilinear wave equations with critical exponent in high dimensions,Chin. Ann. Math. 28B (2007), 205–212. Department of Mathematics, College of Science, King Saud University, Riyadh 11451,Saudi Arabia E-mail address : [email protected] Department of Mathematics, College of Science, King Saud University, Riyadh 11451,Saudi Arabia E-mail address : [email protected] Center for Partial Differential Equations, School of Mathematical Sciences and Shang-hai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, ChinaIECL, UMR 7502, D´epartement de Math´ematiques, Universit´e de Lorraine, 57073 Metz,France E-mail address ::