Asymptotic and Quenching Behavior for a family of Parabolic System with General Singular Nonlinearities
aa r X i v : . [ m a t h . A P ] J u l On a Family of Parabolic System with General SingularNonlinearities and Applications to MEMS
Qi Wang ∗ , Yanyan Zhang † July 28, 2020
Abstract
In this paper, we study a family of parabolic system with general singular nonlinearities, which is ageneralization of MEMS system. We extend the classical results for single MEMS equation to coupledsystem. More precisely, the classification of global existence and finite time quenching according toparameters and initial data is given. Moreover, the convergence, convergence rate, quenching timeestimates are obtained. We point out that compared to single MEMS equation, some new ideas andtechniques are introduced in obtaining the convergence rate for system in our study. In fact, due tothe lack of variational characterization for the first eigenvalue of the linearized elliptic system, themethods in obtaining convergence rate for single equation cannot work completely here.
Keywords: semilinear parabolic system, singular nonlinearity, MEMS system, global existence, con-vergence rate, quenching, quenching time estimateMathematics Subject Classification (2010): 35B40, 35K51, 35K58, 35A01, 35B44
In this paper, we study the following coupled generalized singular parabolic system of the form u t − ∆ u = λα ( x ) f ( v ) , in Ω × (0 , T ) ,v t − ∆ v = µβ ( x ) g ( u ) , in Ω × (0 , T ) ,u = v = 0 , on ∂ Ω × (0 , T ) ,u ( x,
0) = u ( x ) , v ( x,
0) = v ( x ) , in Ω , (P)where Ω is a smooth bounded domain of R N , λ and µ are positive parameters, α ( x ) and β ( x ) arenonnegative nontrivial H¨older continuous functions in ¯Ω, f, g satisfy f, g ∈ C [0 ,
1) are positive, increasing and convex such that lim v → − f ( v ) = lim u → − g ( u ) = + ∞ , (H1)and the initial data satisfy u ( x ) , v ( x ) ∈ C ( ¯Ω) , ≤ u , v < , u = v = 0 on ∂ Ω . (H2) Remark 1.1.
In (H1), we fix the blow up level at u = 1 , v = 1 for simplicity. It is easy to see that withthe scaling, our approaches work for f, g blowing up at any positive values a and b , respectively. ∗ College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P.R. China. Email: [email protected] . † Corresponding author. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathe-matical Practice, East China Normal University, Shanghai 200241, P.R. China. Email: [email protected] . Y. Y.Zhang is sponsored by “Chenguang Program” supported by Shanghai Educational Development Foundation and ShanghaiMunicipal Education Commission [grant number: 13CG20]; NSFC [grant number: 11431005]; and STCSM [grant number:18dz2271000]. u t − ∆ u = λα ( x ) f ( u ) , in Ω × (0 , T ) ,u = 0 , on ∂ Ω × (0 , T ) ,u ( x,
0) = u ( x ) ∈ [0 ,
1) in Ω (1.1)as well as the associated stationary equation − ∆ u = λα ( x ) f ( u ) in Ω , u = 0 on ∂ Ω (1.2)with f satisfying (H1) have been studied in [18]. More precisely, it is showed in [18] that for any given α ≥ f satisfying (H1), there exists a critical value λ ∗ > λ ∈ (0 , λ ∗ ), problem (1.2) issolvable and the solution to (1.1) is global with u = 0; while for λ > λ ∗ , no solution of (1.2) exists, andthe solution to (1.1) will reach the value 1 at finite time T , i.e., the so called quenching or touchdownphenomenon occurs. In fact, besides [18], for the particular case f ( u ) = (1 − u ) − p , p >
0, especially for p = 2, as the mathematical model of micro-electromechanical systems (MEMS), (1.1) has been extensivelystudied by many authors in recent years(cf. [3, 5, 7] and references therein). MEMS device consists of anelastic membrane suspended over a rigid ground plate. For MEMS, u denotes the normalized distancebetween the membrane and the ground plate, α ( x ) represents the permitivity profile. When a voltage λ isapplied, the membrane deflects toward the ground plate and a snap-through may occur when it exceeds acertain critical value λ ∗ (pull-in voltage). This creates a so-called pull-in instability, which greatly affectsthe design of many devices (cf. [3, 14] for more details).As for system (P), if f ( v ) = (1 − v ) − p , g ( u ) = (1 − u ) − q , p, q > , due to the reason above system (P) iscalled general MEMS system(see [4]). For this parabolic general MEMS system, while λ = µ = 1 , α ( x ) = β ( x ) ≡ , some sufficient conditions related to domain for global existence and finite-time quenching ofsolutions, as well as the non-simultaneous quenching criteria for radial solutions are obtained in [19].Also, while λ = µ = 1 , α ( x ) = β ( x ) ≡ , for f, g being logarithmic singular (see [10]) and for general f, g satisfying (H1) (see [13]), some sufficient conditions related to domain Ω for finite-time quenching andglobal existence of the solutions, non-simultaneous quenching and the quenching rate are studied.In this paper, motivated by the above results related to single MEMS equation (cf. [3, 18]), for thecoupled parabolic system (P), one of our main purpose is to study the relationship between the existenceof global solution to (P) and existence of solution to the associated stationary problem − ∆ w = λα ( x ) f ( z ) , in Ω , − ∆ z = µβ ( x ) g ( w ) , in Ω ,w = z = 0 , on ∂ Ω . (E)Compared our paper to the research about coupled parabolic system as in [10, 13, 19], though weall care about the conditions for global existence and finite-time quenching of solutions, we turn tostudy conditions related to associated stationary problem, or we can say conditions related to λ, µ (seeTheorem 1.1, Theorem 1.2) rather than the conditions related to domain Ω. Therefore, in this paper,before studying the parabolic system (P), we will first consider the associated stationary problem (E).Recall that for (E) with f ( · ) = g ( · ) = (1 − · ) − , it has been proved in [2] that there exists a criticalcurve Γ splitting the positive quadrant of the ( λ, µ )-plane into two disjoint sets O and O such that theelliptic problem has a smooth minimal stable solution ( w λ,µ , z λ,µ ) for ( λ, µ ) ∈ O , while for ( λ, µ ) ∈ O there is no solution of any kind. In this paper we will first extend these results in [2] to elliptic problem(E) with general singular terms in Theorem A, which can be illustrated by Figure 1. Theorem A.
There exist < λ ∗ , µ ∗ < + ∞ , and a non-increasing continuous curve µ = Γ( λ ) connecting (0 , µ ∗ ) and ( λ ∗ , such that the positive quadrant R + × R + of the ( λ, µ ) -plane is separated into twoconnected components O and O . For ( λ, µ ) ∈ O , problem (E) has a positive classical minimal solution ( w λ,µ , z λ,µ ) . Otherwise, for ( λ, µ ) ∈ O , (E) admits no weak solution. Note that Theorem A can be established in a similar way to [2], we sketch the proof in Appendix Afor simplicity. 2 µ λ (0 , µ ∗ ) ( λ ∗ , O O : No weak solution for (E) andfinite time quenching for (P) µ = Γ ( λ ) Figure 1:
The critical curve Γ in ( λ, µ ) -plane The second fundamental problem is the local existence and uniqueness of solution to (P). In fact, onecan get the following Theorem B via the work of [15, Theorem 13, Chapter3] and C. V. Pao [12, Theorem2.2]. For the convenience of readers, we also sketch the proof in Appendix B.
Theorem B (Local existence and uniqueness) . Suppose (H1) and (H2) hold. Then for any λ > , µ > there exists T > such that problem (P) has a unique solution ( u, v ) . Moreover ( u, v ) ∈ C ((0 , T ) , C (Ω , R )) . Next, based on Theorem A and Theorem B, we will show in Theorem 1.1 that for ( λ, µ ) ∈ O (definedin Theorem A), there exist some initial data such that the solution to (P) exists globally and convergesto the unique minimal solution of (E) at the rate of (1.4). While for ( λ, µ ) ∈ O (also defined in TheoremA), we prove in Theorem 1.2 that for any initial data, the solution to (P) will quench at a finite time.Moreover, some estimates for quenching time are obtained. The solution ( u, v ) of (P) is called quenchingat time t = T < + ∞ if lim sup t → T − (max { max Ω u ( · , t ) , max Ω v ( · , t ) } ) = 1 . (1.3)The main results of this paper are listed as follows. Theorem 1.1 (Global existence, convergence and convergence rate) . Suppose (H1) and (H2) hold. Let O be the connected component defined in Theorem A as well as ( w λ,µ , z λ,µ ) be the minimal solution of (E) . Then there hold:(i) If ( λ, µ ) ∈ O , ( u ( x ) , v ( x )) is further a subsolution of (E) and satisfies u ≤ w λ,µ , v ≤ z λ,µ ,then the unique solution ( u ( x, t ) , v ( x, t )) to (P) exists globally and converges monotonically to the uniqueminimal solution ( w λ,µ , z λ,µ ) of (E) in C norm as t → + ∞ .(ii) Furthermore, for ≤ n ≤ , there exists T > such that k u ( t, x ) − w λ,µ ( x ) k + k v ( t, x ) − z λ,µ ( x ) k ≤ C exp (cid:18) − min (cid:26) λ , ν (cid:27) t (cid:19) , for t > T (1.4) with C = k w λ,µ − u k + k z λ,µ − v k + 2 k ψ ( w λ,µ − u ) + ϕ ( z λ,µ − v ) k . Here λ > is the firsteigenvalue of − ∆ on H (Ω) , ν > is the first eigenvalue of linearized elliptic system (2.10) , and ψ , ϕ are the corresponding strictly positive eigenfunction defined in Lemma 2.4. Remark 1.2.
In particular, Theorem 1.1 holds for zero initial data, by noting that (0 ,
0) is obviously asubsolution of (E).We also remark that to obtain the convergence rate (1.4), compared to single MEMS equation, somenew ideas and techniques are introduced in this paper (see Section 2.2). In fact, for single parabolic MEMSequation, the convergence rate of global solution has been obtained in [8], where the first eigenvalue ofthe linearized elliptic equation having a variational characterization plays an important role. However,no such analogous formulation is available for coupled system (P) considered in this paper (see [2]).3 heorem 1.2 (Quenching behavior) . Suppose (H1) holds. Let O , O be the connected componentdefined in Theorem A.(i) If ( λ, µ ) ∈ O , then for any ( u , v ) satisfying (H2) , the solution ( u, v ) to (P) will quench at afinite time T ∗ in the sense of (1.3) . Moreover, the quenching time T ∗ must verifies T ∗ ≥ min ( Z k v k ∞ d sµ k β k ∞ g (cid:18) G − (cid:16) λ k α k ∞ F ( s ) + µc µ k β k ∞ (cid:17)(cid:19) , Z k u k ∞ d sλ k α k ∞ f (cid:18) F − (cid:16) µ k β k ∞ G ( s ) − µc λ k α k ∞ (cid:17)(cid:19) ) , (1.5) where F ( s ) = Z s f ( τ )d τ , G ( s ) = Z s g ( τ )d τ , c := k β k ∞ G ( u ) − λµ k α k ∞ F ( v ) .(ii) For the particular case f = g ,there holds O ⊂ [0 , M ) × [0 , M ) (1.6) with M := λ { inf ¯Ω α, inf ¯Ω β } sup ≤ s< sf ( s ) . Furthermore, if min { λ, µ } > M , the quenching time T ∗ also satisfies T ∗ ≤ Z R Ω ( u + v ) φ d x − λ s + 2 min { λ, µ } min { inf ¯Ω α, inf ¯Ω β } f ( s ) d s < + ∞ , (1.7) where λ is the first eigenvalue of − ∆ in H (Ω) , and φ is the corresponding eigenfunction satisfying Z Ω φ d x = 1 . Corollary 1.3.
For the particular case f ( s ) = g ( s ) = 1(1 − s ) , i.e., the classical general MEMS system, (1.5) can be further calculated as T ∗ ≥ λ k α k ∞ µ k β k ∞ (cid:18) − c ln (cid:16) c µ k β k ∞ (1 − k v k ∞ ) + λ k α k ∞ λ k α k ∞ (cid:17) + µ k β k ∞ (1 − k v k ∞ ) c (cid:0) c µ k β k ∞ (1 − k v k ∞ ) + λ k α k ∞ (cid:1) + µ k β k ∞ (1 − k v k ∞ ) λ k α k ∞ (cid:19) > , (1.8) with c = 11 − k u k ∞ − λ k α k ∞ µ k β k ∞ (1 − k v k ∞ ) . In Theorem 1.2, the lower bound of quenching time is obtained for general f, g . Furthermore, theupper bound (1.7) is also obtained for the particular case f = g (including the MEMS system with f ( · ) = g ( · ) = (1 − · ) − ), by noting that Jensen’s inequality can be further applied.This paper is organized as follows. In Section 2 we will prove the global existence, convergence andconvergence rate of solutions to (P) in O , i.e. Theorem 1.1. In Section 3, we will prove that solutionsof (P) with ( λ, µ ) ∈ O must quench at a finite time and obtain some estimates for quenching time, i.e.Theorem 1.2 and Corollary 1.3. At last, we show the proof of Theorem A and B in Appendix A and B,respectively.In this paper, k · k p denotes always the standard norm of L p (Ω). ( λ, µ ) below the critical curve Γ In this section, our goal is to prove Theorem 1.1. More precisely, the global existence and convergencewill be proved in Subsection 2.1, and the convergence rate will be further obtained in Subsection 2.2.Here, we point out that compared to single MEMS equation, some new ideas are introduced to obtainthe convergence rate in Subsection 2.2. 4 .1 Global existence and convergence
In this subsection, we will apply the sub-super solution method to show the global existence andmaximum principle of parabolic system to demonstrate the convergence of the solution to (P).First, the global existence in O will be given in the following Proposition 2.1, which can be deduceddirectly by sub-super solution method in [11, Chapter 8]. Proposition 2.1.
Suppose (H1) and (H2) hold. Let O be the connected component defined in Theorem Aas well as ( w λ,µ , z λ,µ ) be the minimal solution of (E) . If ( λ, µ ) ∈ O , ( u ( x ) , v ( x )) is further a subsolutionof (E) and satisfies u ≤ w λ,µ , v ≤ z λ,µ , then there exists a unique global solution ( u ( x, t ) , v ( x, t )) for (P) . Secondly, we will show that the global solution ( u, v ) is monotonic increasing with respect to time t in Proposition 2.3. To verify Proposition 2.3, we need to borrow a comparison principle for the parabolicsystem below, which can be derived from [15, Theorem 13, Chapter3]. Lemma 2.2 (Comparison Principle) . Suppose that u = ( u , u , · · · , u k ) satisfies the following uniformlyparabolic system of inequalities in Ω × (0 , T ) . ∂u ∂t − ∆ u − k X i =1 h i u i ≤ ,∂u ∂t − ∆ u − k X i =1 h i u i ≤ , ... ∂u k ∂t − ∆ u k − k X i =1 h ki u i ≤ . (2.1) If u ≤ at t = 0 and on ∂ Ω × (0 , T ) and if h ij is bounded and satisfies h ji ≥ for i = j, i, j = 1 , , · · · , k, (2.2) then u ≤ in Ω × (0 , T ) . Moreover, if there exists i such that u i = 0 at an interior point ( x , t ) , then u i ≡ for t ≤ t . Here, we use the notation u ≤ to mean that every component u i , i = 1 , , · · · , k isnonpositive. Proposition 2.3.
Suppose ( u, v ) satisfies u t − ∆ u = f ( x, v ) > , in Q T = Ω × (0 , T ) ,v t − ∆ v = g ( x, u ) > , in Q T = Ω × (0 , T ) ,u = v = 0 , on ∂ Ω × (0 , T ) ,u ( x,
0) = u , v ( x,
0) = v , for x ∈ ¯Ω (2.3) with both ∂f∂v and ∂g∂u being positive and locally bounded. Then if ( u ( x ) , v ( x )) is a subsolution of thecorresponding stationary system to (2.3) , there holds u t ≥ , v t ≥ . Proof:
Differentiating system (2.3) with respect to t yields ( u t ) t − ∆ u t = ∂f∂v v t , in Q T = Ω × (0 , T ) , ( v t ) t − ∆ v t = ∂g∂u u t , in Q T = Ω × (0 , T ) ,u t = v t = 0 , on ∂ Ω × (0 , T ) ,u t ( x,
0) = ∆ u + f ( x, v ) ≥ , for x ∈ ¯Ω ,v t ( x,
0) = ∆ v + g ( x, u ) ≥ , for x ∈ ¯Ω . (2.4)5y the maximum principle for parabolic system stated in Lemma 2.2, we get that u t ≥
0. Similarly, v t ≥ (cid:3) At the last of this subsection, we conclude the proof of Theorem 1.1 as follows.
Proof of Theorem 1.1 (i):
Note that the unique global solution ( u ( x, t ) , v ( x, t )) obtained in Proposition2.1 for (P) is bounded by the unique minimal solution ( w λ,µ , z λ,µ ) of (E). By Proposition 2.3 andassumption (H1), we can conclude that u t ≥ v t ≥
0, which implies that ( u, v ) converges as t → + ∞ tosome functions ˜ u ( x ) , ˜ v ( x ) satisfying ˜ u ≤ w λ,µ <
1, ˜ v ≤ z λ,µ < ϕ ( x ) ∈ C ( ¯Ω) and ϕ | ∂ Ω = 0. Multiplying (P) by ϕ and integrating over Ω, we arrive at dd t Z Ω uϕ d x − Z Ω u ∆ ϕ d x = Z Ω λα ( x ) ϕf ( v )d x, dd t Z Ω vϕ d x − Z Ω v ∆ ϕ d x = Z Ω µβ ( x ) ϕg ( u )d x. (2.5)Operating on both sides with 1 T Z T , it follows that Z Ω u ( x, T ) − u ( x ) T ϕ d x + Z Ω ( − ∆ ϕ ) 1 T Z T u ( x, t )d t d x = Z Ω λα ( x ) ϕ T Z T f ( v )d t d x, Z Ω v ( x, T ) − v ( x ) T ϕ d x + Z Ω ( − ∆ ϕ ) 1 T Z T v ( x, t )d t d x = Z Ω µβ ( x ) ϕ T Z T g ( u )d t d x. (2.6)Note that lim T → + ∞ u ( x, T ) − u ( x ) T = 0 , lim T → + ∞ v ( x, T ) − v ( x ) T = 0 , lim T → + ∞ T Z T u ( x, t )d x = ˜ u ( x ) , lim T → + ∞ T Z T v ( x, t )d x = ˜ v ( x ) , lim T → + ∞ T Z T g ( u )d x = g (˜ u ) , lim T → + ∞ T Z T f ( v )d x = f (˜ v ) . (2.7)Therefore, by the Lebesgue dominated convergence theorem we get that as T → + ∞ Z Ω ˜ u ( − ∆ ϕ )d x = Z Ω λα ( x ) ϕf (˜ v )d x, Z Ω ˜ v ( − ∆ ϕ )d x = Z Ω µβ ( x ) ϕg (˜ u )d x, (2.8)which implies (˜ u, ˜ v ) is a weak solution of (E). By the L p estimates of Agmon, Douglis, Nirenberg [1], theSobolev embedding, and the classical Schauder estimate, we obtain that (˜ u, ˜ v ) is a classical solution of(E), and hence (˜ u, ˜ v ) = ( w λ,µ , z λ,µ ).Since u t ≥ , v t ≥ w λ,µ , z λ,µ are continuous, by [16, Theorem 7.13] the convergence of theunique global solution ( u ( x, t ) , v ( x, t )) to ( w λ,µ ( x ) , z λ,µ ( x )) is further uniform in x , i.e.,lim t →∞ (cid:0) k u ( t, x ) − w λ,µ ( x ) k ∞ + k v ( t, x ) − z λ,µ ( x ) k ∞ (cid:1) = 0 . (2.9)Therefore combined with Corollary 2.8, we complete the proof. (cid:3) To obtain the convergence rate of (P), we need to consider the stability of ( w λ,µ , z λ,µ ). For thispurpose, we first show a related lemma as follows. 6 emma 2.4. The problem − ∆ ϕ − λα ( x ) f ′ ( z λ,µ ) ψ = νϕ, in Ω , − ∆ ψ − µβ ( x ) g ′ ( w λ,µ ) ϕ = νψ, in Ω ,ϕ = ψ = 0 , on ∂ Ω . (2.10) has a first eigenvalue ν > (which means the minimal solution ( w λ,µ , z λ,µ ) is stable) with strictly positiveeigenfunction ( ϕ , ψ ) , that is, ϕ > , ψ > in Ω . Moreover ϕ and ψ are smooth. This result is standard. For the proof, see e.g., [9, Theorem 1.5] and [2, p10].Next, before verifying Theorem 1.1(ii), we shall introduce the following two useful lemmas and aproposition.
Lemma 2.5.
Given a smooth bounded domain Ω in R N . Suppose a ( x, t ) ∈ C ([0 , + ∞ ) , C ( ¯Ω)) , a ≥ in ¯Ω × [0 , + ∞ ) , b ( x ) ∈ C ( ¯Ω) , b > in Ω , a = b = 0 on ∂ Ω , ∂b∂~n < on ∂ Ω , lim t → + ∞ k a ( · , t ) k C = 0 . Thenthere exists T > such that a ( x, t ) ≤ b ( x ) in Ω for all t > T . Here ~n denotes the outward unit normalvector on ∂ Ω . Proof:
Since ∂b∂~n | ∂ Ω < b ( x ) ∈ C ( ¯Ω) , there exists a constant ε > x ∈ Ω ε := { x ∈ ¯Ω | dist ( x, ∂ Ω) ≤ ε } , there holds b ( x ) = b ( x ) − b ( x ) ≥ C | x − x | , where x ∈ ∂ Ω satisfying( x − x ) k ~n and C > x. On the other hand, for all x ∈ Ω ε , there alsoholds a ( x, t ) = a ( x, t ) − a ( x , t ) ≤ k a ( · , t ) k C | x − x | . Note that lim t → + ∞ k a ( · , t ) k C = 0 . Therefore, thereholds k a ( · , t ) k C ≤ C for t large enough and it follows that a ( x, t ) ≤ b ( x ) on Ω ε for t large enough. Atlast, it is obviously that for any given subset ˚Ω ⊂ Ω , a ( x, t ) ≤ b ( x ) on ˚Ω for t large enough. Hence, weconclude this lemma. (cid:3) Lemma 2.6.
For the solution ( u, v ) to Problem (P) with ( λ, µ ) ∈ O , if ( u ( x ) , v ( x )) is a subsolutionof (E) , then there exist c , c ∈ R + such that u t ≥ c v t ≥ , v t ≥ c u t ≥ . (2.11) Proof:
Let U = u t − c v t , V = v t − c u t . (2.12)Note by Proposition 2.3 that u t ≥ v t ≥
0. It can be deduced that U t − ∆ U + c µβ ( x ) g ′ ( u ) U = ( λαf ′ ( v ) − c µβg ′ ( u )) v t ,U | ∂ Ω = 0 ,U ( x,
0) = ∆ u + λα ( x ) f ( v ) − c (∆ v + µβ ( x ) g ( u )) . (2.13)Applying comparison principle, we have that u t − c v t = U ≥ c ≤ min (cid:26)s λµ inf Ω α ( x ) β ( x ) f ′ (0) g ′ ( k w λ,µ k ∞ ) , inf Ω ∆ u + λα ( x ) f ( v )∆ v + µβ ( x ) g ( u ) (cid:27) . (2.14)Here 0 ≤ u, v <
1, the nonnegativity of λ, µ, α, β, u t , v t , f ′ ( s ) , g ′ ( s ) for 0 ≤ s < f ′ ( s ) , g ′ ( s ) are used. Similarly, it can be proved that v t − c u t ≥ c ≤ min (cid:26)s µλ inf Ω β ( x ) α ( x ) g ′ (0) f ′ ( k z λ,µ k ∞ ) , inf Ω ∆ v + µβ ( x ) g ( u )∆ u + λα ( x ) f ( v ) (cid:27) . (2.15)This completes the proof of (2.11). (cid:3) Without causing confusion, for simplicity we use ( w, z ) instead of ( w λ,µ , z λ,µ ) to denote the minimalsolution of problem (E) in the rest part of this subsection.7 roposition 2.7. Suppose that the conditions in Proposition 2.1 are satisfied. Let ( u, v ) be the uniqueglobal solution of (P) , then we have lim t → + ∞ k u t k = lim t → + ∞ k v t k = 0 , (2.16) and k u k H + k v k H ≤ C ( δ ) , for all t ≥ δ > . (2.17) Proof:
First we claim that k∇ u k + k∇ v k ≤ C ( u , v , w, z ) , (2.18)where C is a constant independent of time t . To prove this claim, we denote ξ = u − w, η = v − z . Thenit follows from system (P) and (E) that ξ t − ∆ ξ = λα ( x )( f ( v ) − f ( z )) , in Q T = Ω × (0 , T ) ,η t − ∆ η = µβ ( x )( g ( u ) − g ( w )) , in Q T = Ω × (0 , T ) ,ξ = η = 0 , on ∂ Ω × (0 , T ) ,ξ ( x,
0) = u ( x ) − w ( x ) , η ( x,
0) = v ( x ) − z ( x ) , for x ∈ ¯Ω . (2.19)Multiplying the first equation of (2.19) by ξ t yields that12 ddt k∇ ξ k + k ξ t k = λ Z Ω α ( x )[ f ( v ) − f ( z )] ξ t dx ≤ , (2.20)where v ≤ z , assumption (H1) and Proposition 2.3 are used. The above inequality then implies that ddt k∇ ξ k ≤ k∇ ξ k ≤ k∇ ξ k ≤ C ( u , v , w, z ) . (2.22) k∇ η k ≤ C ( u , v , w, z ) can be obtained similarly. Then (2.18) follows by k∇ u k + k∇ v k ≤ k∇ ξ k + k∇ w k + k∇ η k + k∇ z k ≤ C ( u , v , w, z ) . (2.23)Next, we show that Z + ∞ ( k u t k + k v t k ) dt ≤ C. (2.24)After multiplying equations in (P) by v t and u t , respectively, adding them up and integrating over Ω, wecan see that Problem (P) admits a Lyapunov function E ( u, v ) = Z Ω (cid:0) ∇ u ∇ v − F ( x, v ) − G ( x, u ) (cid:1) dx, (2.25)where F ( x, v ) = λα ( x ) Z v f ( s )d s , G ( x, u ) = µβ ( x ) Z u g ( s )d s, and there holds ddt E ( u, v ) + 2 Z Ω u t v t dx = 0 . (2.26)Note that 0 < w < , < z <
1. By assumption (H1) and (2.18), integrating (2.26) with respect to t yields2 Z + ∞ Z Ω u t v t dxdτ ≤ E ( u , v ) + (cid:12)(cid:12)(cid:12)(cid:12) Z Ω ∇ u ∇ vdx (cid:12)(cid:12)(cid:12)(cid:12) + Z Ω ( F ( x, max Ω z ) + G ( x, max Ω w )) dx ≤ C. (2.27)8hen (2.24) can be concluded by (2.27), Proposition 2.3 and Lemma 2.6.Differentiating the first equation in (P) with respect to t yields u tt − ∆ u t = λα ( x ) f ′ ( v ) v t . (2.28)Multiplying (2.28) by u t and integrating over Ω, by Lemma 2.6 and assumption (H1) we have12 ddt k u t k + k∇ u t k = Z Ω λαf ′ ( v ) v t u t dx ≤ C k u t k . (2.29)By Young’s inequality we get ddt k u t k ≤ C k u t k + C . (2.30)Then by (2.24) and [20, Lemma 6.2.1], we get lim t → + ∞ k u t k = 0, while lim t → + ∞ k v t k = 0 can be obtainedsimilarly and (2.16) follows.Now integrating (2.29) with respect to t, by (2.24) we obtain12 k u t k + Z t k∇ u t k dτ ≤ k u t (0) k + C Z + ∞ k u t k dτ ≤ C, (2.31)which implies obviously Z t k∇ u t k dτ ≤ C. (2.32)Multiplying (2.28) by − ∆ u t and integrating over Ω, by Lemma 2.6 we have12 ddt k∇ u t k + k ∆ u t k = λ Z Ω αf ′ ( v ) v t ( − ∆ u t ) dx ≤ C k u t k k ∆ u t k ≤ C k u t k + 12 k ∆ u t k , (2.33)which yields ddt k∇ u t k + k ∆ u t k ≤ C k u t k . (2.34)Multiplying (2.34) by t , then integrating with respect to t in [0 , t ], by (2.24) and (2.32) there holds t k∇ u t k + Z t τ k ∆ u t k dτ ≤ Z t k∇ u t k dτ + Ct Z t k u t k dτ ≤ C + C t. (2.35)Thus, for t ≥ δ > , we have k∇ u t k ≤ C t + C ≤ C δ + C (2.36)and it follows k u t k H ≤ C ( δ ) for t ≥ δ. (2.37)Now we can deduce from the equation in (P) and the regularity theory for the elliptic problem (seee.g. [6, 20]) ( − ∆ u = λα ( x ) f ( v ) − u t , in Ω ,u = 0 , on ∂ Ω , (2.38)that k u ( · , t ) k H ≤ C ( k f ( v ) k H + k u t k H ) ≤ C ( δ )(1 + k f ′ ( v ) ∇ v k ) ≤ C ( δ )(1 + C k∇ v k ) ≤ C ( δ ) , (2.39)by (2.37) and (2.18). k v ( · , t ) k H can be treated similarly. In conclusion, we obtain (2.17). (cid:3) Corollary 2.8.
For ≤ n ≤ , there holds lim t →∞ k ξ ( · , t ) k C = lim t →∞ k u ( · , t ) − w k C = 0 , lim t →∞ k η ( t ) k C = lim t →∞ k v ( t ) − z k C = 0 . (2.40)9 roof: Note by (2.17) that ξ, η ∈ H (Ω), and H (Ω) ֒ → ֒ → C (Ω) for 1 ≤ n ≤ ξ ( t ) , η ( t ) in C andthe uniqueness of the limits. (cid:3) Now we give the proof of Theorem 1.1 (ii) as follows.
Proof of Theorem 1.1 (ii):
Multiplying equations in (2.19) by ξ and η, respectively, adding them up and integrating over Ω yields ddt Z Ω (cid:18) ξ + 12 η (cid:19) dx + k∇ ξ k + k∇ η k = Z Ω (cid:18) λα [ f ( z ) − f ( v )]( − ξ ) + µβ [ g ( w ) − g ( u )]( − η ) (cid:19) dx. (2.41)Rewrite equations in (2.19) as ( ξ t − ∆ ξ − λαf ′ ( z ) η = λα ( f ( v ) − f ( z ) − f ′ ( z ) η ) , in Q T = Ω × (0 , T ) ,η t − ∆ η − µβg ′ ( w ) ξ = µβ ( g ( u ) − g ( w ) − g ′ ( w ) ξ ) , in Q T = Ω × (0 , T ) . (2.42)Note that 0 ≤ v ≤ z < ≤ u ≤ w < . By the convexity of f and g , it is easy to deduce that f ( v ) − f ( z ) − f ′ ( z ) η ≥ g ( u ) − g ( w ) − g ′ ( w ) ξ ≥ . Thus it follows that ( ξ t − ∆ ξ − λαf ′ ( z ) η ≥ , in Q T = Ω × (0 , T ) ,η t − ∆ η − µβg ′ ( w ) ξ ≥ , in Q T = Ω × (0 , T ) . (2.43)Multiplying inequalities in (2.43) by ψ and ϕ , respectively, adding them up and integrating over Ωyields Z Ω ( ψ ξ + ϕ η ) t dx + ν Z Ω ( ψ ξ + ϕ η ) dx ≥ . (2.44)Here, ν is the principal eigenvalue of problem (2.10) and ( ϕ , ψ ) is the corresponding positive eigen-function. Multiplying (2.44) by −
1, then adding it and (2.41) together yields ddt Z Ω (cid:18) ξ + 12 η + (cid:2) ψ ( − ξ ) + ϕ ( − η ) (cid:3)(cid:19) dx + k∇ ξ k + k∇ η k + ν Z Ω (cid:2) ψ ( − ξ ) + ϕ ( − η ) (cid:3) d x ≤ Z Ω (cid:18) λα [ f ( z ) − f ( v )]( − ξ ) + µβ [ g ( w ) − g ( u )]( − η ) (cid:19) dx. (2.45)Now we claim that there exists T > t > T , there holds f ( z ) − f ( v ) ≤ ν λ k α k ∞ ψ , g ( w ) − g ( u ) ≤ ν µ k β k ∞ ϕ . (2.46)In fact, recalling (2.10), by Lemma 2.4 we have − ∆ ϕ = f ′ ( z ) ψ + ν ϕ ≥ , in Ω , − ∆ ψ = g ′ ( w ) ϕ + ν ψ ≥ , in Ω ,ϕ = ψ = 0 , on ∂ Ω . (2.47)Thus, by Hopf lemma there holds − ∂ϕ ∂~n ≥ ε , − ∂ψ ∂~n ≥ ε on ∂ Ω for some ε > . Then (2.46) follows byLemma 2.4, Lemma 2.5 and (2.40).Combining (2.45),(2.46) and the Poincar´ e inequality k u k ≤ λ k∇ u k for any u ∈ H (Ω) with λ > − ∆ on H (Ω) , we get ddt Z Ω (cid:18) ξ + 12 η + (cid:2) ψ ( − ξ ) + ϕ ( − η ) (cid:3)(cid:19) dx + λ k ξ k + λ k η k + ν Z Ω (cid:2) ψ ( − ξ ) + ϕ ( − η ) (cid:3) d x ≤ Y = Z Ω ( ξ + η + 2 (cid:2) ψ ( − ξ ) + ϕ ( − η ) (cid:3) ) dx. Note that Z Ω (cid:2) ψ ( − ξ ) + ϕ ( − η ) (cid:3) d x ≥ . (2.49)By (2.48) there holds dYdt + γY ≤ , γ = min (cid:26) λ , ν (cid:27) , (2.50)which yields Y ≤ Y (0) e − γt . Then by noting (2.49) again it follows that k u ( t, x ) − w ( x ) k + k v ( t, x ) − z ( x ) k ≤ Y ( t ) ≤ C exp (cid:18) − min (cid:26) λ , ν (cid:27) t (cid:19) , for t > T . (2.51)The proof of Theorem 1.1 (ii) is therefore completed. (cid:3) ( λ, µ ) above thecritical curve Γ In this section, our goal is to prove Theorem 1.2 and Corollary 1.3. More precisely, we will first provethe finite time quenching in the following Proposition 3.1, then obtain the quenching time estimates inthe rest part of this section.
Proposition 3.1.
Suppose (H1) holds. Let O be the connected component defined in Theorem A. If ( λ, µ ) ∈ O , then for any ( u , v ) satisfying (H2) , the solution ( u, v ) to (P) will quench at a finite time T ∗ in the sense of (1.3) . Proof:
We will only prove the case ( u , v ) ≡ (0 , λ, µ ) ∈ O . Suppose on the contrary that the local solution ( u, v ) (see Theorem B) exists globally,i.e. 0 ≤ u <
1, 0 ≤ v < t ≥
0. Take δ > a = λδ , b = µδ . Since U = uδ < u , V = vδ < v , it thenindicates that U ≤ δ < V ≤ δ <
1, and by the monotone increasing of f, g there holds U t − ∆ U = λα ( x ) f ( v ) δ ≥ aα ( x ) f ( V ) , in Q T = Ω × (0 , T ) ,V t − ∆ V = µβ ( x ) g ( u ) δ ≥ bβ ( x ) g ( U ) , in Q T = Ω × (0 , T ) ,U = V = 0 , on ∂ Ω × (0 , T ) ,U ( x,
0) = 0 , V ( x,
0) = 0 , for x ∈ ¯Ω . (3.1)Hence ( U, V ) is a supersolution of U t − ∆ U = aα ( x ) f ( V ) , in Q T = Ω × (0 , T ) , V t − ∆ V = bβ ( x ) g ( U ) , in Q T = Ω × (0 , T ) , U = V = 0 , on ∂ Ω × (0 , T ) , U ( x,
0) = 0 , V ( x,
0) = 0 , for x ∈ ¯Ω . (3.2)Therefore (3.2) has a global classical solution ( U ( x, t ) , V ( x, t )), since 0 ≤ U ≤ U ≤ δ <
1, 0 ≤ V ≤ V ≤ δ <
1. Note that there further holds lim t → + ∞ ( kU t k + kV t k ) = 0 , sup t> ( kUk H (Ω) + kVk H (Ω) ) < + ∞ , (3.3)11hich can be proved similarly to Proposition 2.7. By Sobolev embedding theorem, one can have thatthere exists a subsequence { t j } ∞ j =1 such that t j → + ∞ , ( U ( · , t j ) , V ( · , t j ) converges strongly in H (Ω) to( U ∞ , V ∞ ). Now take φ ∈ H (Ω). Multiplying (3.2) by φ and integrating by parts with respect to x yields, Z Ω φ U t ( · , t j )d x + Z Ω ∇U ( · , t j ) ∇ φ d x = Z Ω aα ( x ) φf ( V ( · , t j ))d x, Z Ω φ V t ( · , t j )d x + Z Ω ∇V ( · , t j ) ∇ φ d x = Z Ω bβ ( x ) φg ( U ( · , t j ))d x. (3.4)Passing to the limit t j → + ∞ , we obtain that ( U ∞ , V ∞ ) is a weak solution of − ∆ w = aα ( x ) f ( z ) , in Ω , − ∆ z = bβ ( x ) g ( w ) , in Ω ,u = v = 0 , on ∂ Ω . (3.5)Chose δ close to 1 such that ( a, b ) ∈ O . Then we get a contradiction with Theorem A. The proof of thisproposition is therefore completed. (cid:3) We now focus on estimates for quenching time T ∗ . Proof of Theorem 1.2 (i):
Note that the finite time quenching result has been proved in Proposition3.1. It is reduced to obtain (1.5) to complete the proof. Similar to the proof of Theorem B, let ( ζ, ρ ) bethe solution of the following ODE system: d ζ d t = λ k α k ∞ f ( ρ ) , in (0 , T ) , d ρ d t = µ k β k ∞ g ( ζ ) , in (0 , T ) ,ζ (0) = k u k ∞ , ρ (0) = k v k ∞ . (3.6)The local existence of (3.6) can be obtained by [17, Chapter III]. By the comparison principle forparabolic system (Lemma 2.2), it follows that ζ ≥ u , ρ ≥ v , and then the solution ( ζ, ρ ) of (3.6) will alsoquench at a finite time in the sense of (1.3). Denote the quenching time of ( ζ, ρ ) by T , which means ζ ( T ) = 1 , ρ ( T ) ≤ ζ ( T ) ≤ , ρ ( T ) = 1. Therefore T ∗ ≥ T .Obviously, we can see that d ζ d ρ = λ k α k ∞ f ( ρ ) µ k β k ∞ g ( ζ ) , which implies µ k β k ∞ g ( ζ )d ζ = λ k α k ∞ f ( ρ )d ρ . Then k β k ∞ G ( ζ ( t )) − λµ k α k ∞ F ( ρ ( t )) =: c is a constant for all t ≥
0. It hence follows that ζ ( t ) = G − (cid:16) λ k α k ∞ F ( ρ ( t )) + µc µ k β k ∞ (cid:17) , ρ ( t ) = F − (cid:16) µ k β k ∞ G ( ζ ( t )) − µc λ k α k ∞ (cid:17) . (3.7)Therefore by (3.6) there holdsd ρµ k β k ∞ g (cid:18) G − (cid:16) λ k α k ∞ F ( ρ ) + µc µ k β k ∞ (cid:17)(cid:19) = d t, d ζλ k α k ∞ f (cid:18) F − (cid:16) µ k β k ∞ G ( ζ ) − µc λ k α k ∞ (cid:17)(cid:19) = d t. (3.8)If ρ ( T ) = 1 and ζ ( T ) ≤
1, one can see that T = Z k v k ∞ d ρµ k β k ∞ g (cid:18) G − (cid:16) λ k α k ∞ F ( ρ )+ µc µ k β k ∞ (cid:17)(cid:19) = Z ζ ( T ) k u k ∞ d ζλ k α k ∞ f (cid:18) F − (cid:16) µ k β k ∞ G ( ζ ) − µc λ k α k ∞ (cid:17)(cid:19) ≤ Z k u k ∞ d ζλ k α k ∞ f (cid:18) F − (cid:16) µ k β k ∞ G ( ζ ) − µc λ k α k ∞ (cid:17)(cid:19) . (3.9)12imilarly, if ρ ( T ) ≤ ζ ( T ) = 1, we arrive at T = Z k u k ∞ d ζλ k α k ∞ f (cid:18) F − (cid:16) µ k β k ∞ G ( ζ ) − µc λ k α k ∞ (cid:17)(cid:19) = Z ρ ( T ) k v k ∞ d ρµ k β k ∞ g (cid:18) G − (cid:16) λ k α k ∞ F ( ρ )+ µc µ k β k ∞ (cid:17)(cid:19) ≤ Z k v k ∞ d ρµ k β k ∞ g (cid:18) G − (cid:16) λ k α k ∞ F ( ρ )+ µc µ k β k ∞ (cid:17)(cid:19) (3.10)In conclusion, (1.5) holds. (cid:3) As for the particular MEMS case f ( s ) = g ( s ) = 1(1 − s ) , Theorem 1.2 (i) can be rewritten asCorollary 1.3. Proof of Corollary 1.3:
Suppose that f ( s ) = g ( s ) = 1(1 − s ) . Then (3.6) can be rewritten as d ζ d t = λ k α k ∞ (1 − ρ ) , in (0 , T ) , d ρ d t = µ k β k ∞ (1 − ζ ) , in (0 , T ) ,ζ (0) = k u k ∞ , ρ (0) = k v k ∞ , (3.11)and there holds 11 − ζ − λ k α k ∞ µ k β k ∞ (1 − ρ ) =: c , (3.12)with c = 11 − k u k ∞ − λ k α k ∞ µ k β k ∞ (1 − k v k ∞ ) is a constant for all t ≥
0. Note that it follows from (3.12)that ζ , ρ will quench simultaneously. By (3.11) there holdsd ρ ( c + λ k α k ∞ µ k β k ∞ (1 − ρ ) ) = µ k β k ∞ d t. (3.13)Then by taking y = c + λ k α k ∞ µ k β k ∞ (1 − ρ ) , we obtain µ k β k ∞ λ k α k ∞ d t = d( 2 c ln (cid:12)(cid:12) yy − c (cid:12)(cid:12) − c y − y − c ). Nowone can see from lim ρ → y = + ∞ that T = λ k α k ∞ µ k β k ∞ (cid:0) − c ln (cid:12)(cid:12) c µ k β k ∞ (1 − k v k ∞ ) + λ k α k ∞ λ k α k ∞ (cid:12)(cid:12) + µ k β k ∞ (1 − k v k ∞ ) c ( c µ k β k ∞ (1 − k v k ∞ ) + λ k α k ∞ )+ µ k β k ∞ (1 − k v k ∞ ) λ k α k ∞ (cid:1) , (3.14)and (1.8) follows by T ∗ ≥ T . (cid:3) Next, we shall adapt and improve some of the arguments in [5] to further get an upper estimate ofquenching time T ∗ for the particular case f = g , i.e, Theorem 1.2 (ii). Proof of Theorem 1.2 (ii):
Let λ is the first eigenvalue of − ∆ in H (Ω), and φ is the correspondingeigenfunction satisfying Z Ω φ d x = 1. Introduce I ( t ) = Z Ω ( u + v ) φ d x. (3.15)13t is clear that I ( t ) is well defined on the existence interval of the solution ( u, v ). Using (P), we find that I ′ ( t ) = Z Ω φ ( u t + v t )d x = Z Ω ( u + v )∆ φ d x + Z Ω (cid:0) λαφf ( v ) + µβφf ( u ) (cid:1) d x ≥ − λ I ( t ) + min { λ, µ } min { inf ¯Ω α, inf ¯Ω β } Z Ω φ (cid:0) f ( v ) + f ( u ) (cid:1) d x. (3.16)By (H1) and Jensen’s inequality, Z Ω φf ( u )d x ≥ f ( Z Ω uφ d x ) , Z Ω φf ( v )d x ≥ f ( Z Ω vφ d x ) . (3.17)Substituting this into (3.16), using (H1) again we obtain that I ′ ( t ) + λ I ( t ) ≥ min { λ, µ } min { inf ¯Ω α, inf ¯Ω β } (cid:0) f ( R Ω vφ d x ) + f ( R Ω uφ d x ) (cid:1) ≥ { λ, µ } min { inf ¯Ω α, inf ¯Ω β } f ( I ) . (3.18)Therefore I ′ ( t ) ≥ − λ I ( t ) + 2 min { λ, µ } min { inf ¯Ω α, inf ¯Ω β } f ( I > , (3.19)when min { λ, µ } > λ { inf ¯Ω α, inf ¯Ω β } sup ≤ s< sf ( s ) . If max { u, v } remains smaller than 1 for all t , then I ( t ) is defined for all t . However, from the ODE theory, under the given assumptions, I ( t ) is only welldefined in (0 , ˜ T ), where˜ T = Z R Ω ( u + v ) φ d x − λ s + 2 min { λ, µ } min { inf ¯Ω α, inf ¯Ω β } f ( s ) d s < + ∞ . (3.20)That is to say, the solution ( u, v ) must touchdown at a finite time T ∗ ≤ ˜ T , i.e., (1.6) and (1.7) follows. (cid:3) Appendix A
We will show the proof of Theorem A in this Appendix. First we will prove that the elliptic problem(E) has a classical solution for λ and µ small enough, while (E) has no solution for λ or µ large enough.More precisely, we will prove that the setΛ := { ( λ, µ ) ∈ R + × R + : (E) has a classical minimal solution } (A.1)is nonempty and bounded. Lemma A.1. Λ is bounded, and there exist λ > , µ > such that (0 , λ ] × (0 , µ ] ⊆ Λ . Proof:
Let γ ∈ H (Ω) be the regular solution of − ∆ γ = 1 in Ω. It is then easy to verify that there exists α ∈ (0 , k γ k ∞ ) such that ( αγ, αγ ) is a supersolution of (E) if λ < x ∈ Ω α ( x ) sup λ ∈ (0 , λ ] and µ ∈ (0 , µ ].In fact, for these λ, µ , using (H2) and the monotone iteration for n ∈ N , w = z = 0 , − ∆ w n +1 = λα ( x ) f ( z n ) , in Ω , − ∆ z n +1 = µβ ( x ) g ( w n ) , in Ω ,w n +1 = z n +1 = 0 , on ∂ Ω , (A.2)14e get the minimal solution ( w λ,µ , z λ,µ ) = lim n → + ∞ ( w n , z n ). Therefore, Λ is nonempty.On the other hand, take a positive first eigenfunction ϕ of − ∆ in H (Ω) with the first eigenvalue λ such that Z Ω ϕ d x = 1. By (E) and w < z <
1, we arrive at λ ≥ λ Z Ω wϕ d x = Z Ω ϕ ( − ∆ w )d x = λ Z Ω α ( x ) f ( z ) ϕ d x ≥ λ Z Ω α ( x ) f (0) ϕ d x,λ ≥ λ Z Ω zϕ d x = Z Ω ϕ ( − ∆ z )d x = λ Z Ω β ( x ) g ( w ) ϕ d x ≥ µ Z Ω β ( x ) g (0) ϕ d x. (A.3)So Λ is bounded and Λ ⊆ (cid:0) , λ R Ω α ( x ) f (0) ϕ d x (cid:3) × (cid:0) , λ R Ω β ( x ) g (0) ϕ d x (cid:3) . (cid:3) Denote µ = Γ( λ ) as the critical curve such that if 0 ≤ µ < Γ( λ ) , then ( λ, µ ) ∈ Λ; if µ > Γ( λ ) , then( λ, µ ) ∈ ( R + × R + ) \ ¯Λ . By Lemma A.1, there further hold 0 < µ ∗ := Γ(0) < + ∞ and 0 < λ ∗ := Γ − (0) < + ∞ . Next we state that the critical curve µ = Γ( λ ) is non-increasing. More precisely, Lemma A.2. If ≤ λ ′ ≤ λ, ≤ µ ′ ≤ µ for some ( λ, µ ) ∈ Λ , then ( λ ′ , µ ′ ) ∈ Λ . Proof:
Indeed, the solution associated to ( λ, µ ) turns out to be a super-solution to ( E ) with ( λ ′ , µ ′ ) . (cid:3) Proof of Theorem A:
Define O = Λ \ Γ. For ( λ , µ ) , ( λ , µ ) ∈ O , there exist θ , θ > µ = θ λ and µ = θ λ . Using Lemma A.2, we can define a path linking ( λ , µ ) to (0 ,
0) and anotherpath linking (0 ,
0) to ( λ , µ ), which implies that O is connected. Now, define O = ( R + × R + ) \{ Λ S Γ } .Let ( λ , µ ) , ( λ , µ ) ∈ O . Then by Lemma 1.5 again that ( λ max , µ max ) ∈ O , where λ max = max { λ , λ } and µ max = max { µ , µ } . We can take a path linking ( λ , µ ) to ( λ max , µ max ) and another path linking( λ max , µ max ) to ( λ , µ ), which follows that O is connected.At last, it is reduced to prove that problem (E) admits no weak solution for ( λ, µ ) ∈ O . Suppose onthe contrary that ( w, z ) is a weak solution to (E). By the monotonicity of f, g , it is easy to verify thatfor any δ >
1, ( ˆ w, ˆ z ) = ( wδ , zδ ) is a weak super-solution for problem − ∆ w = λδ α ( x ) f ( z ) , in Ω , − ∆ z = µδ β ( x ) g ( w ) , in Ω ,w = z = 0 , on ∂ Ω , ( E δ )then the monotone iteration will enable us a weak solution ( ˜ w, ˜ z ) of ( E δ ) satisfying 0 ≤ ˜ w ≤ ˆ w ≤ δ < ≤ ˜ z ≤ ˆ z ≤ δ <
1. The regularity theory implies that ( ˜ w, ˜ z ) is a regular solution of ( E δ ). Thismeans that ( λδ , µδ ) ∈ O S Γ. Let δ tend to 1, we get ( λ, µ ) ∈ O S Γ, which contradicts with the as-sumption. Therefore, no weak solution exists for ( λ, µ ) ∈ O and the proof of Theorem A is completed. (cid:3) Appendix B
In this Appendix, we will show the proof of Theorem B.
Proof:
We first show the uniqueness of the solution to (P). For any given 0 < T < T, suppose(˜ u, ˜ v ), (ˆ u, ˆ v ) are two pair classical solutions of (P) on the interval [0 , T ] such that k ˜ u k L ∞ (Ω × [0 ,T ]) < , k ˜ v k L ∞ (Ω × [0 ,T ]) < , k ˆ u k L ∞ (Ω × [0 ,T ]) < , k ˆ v k L ∞ (Ω × [0 ,T ]) < U, V ) = (˜ u − ˆ u, ˜ v − ˆ v ) satisfies U t − ∆ U = λα ( x ) f ′ ( θ v ) V, in Q T = Ω × (0 , T ] ,V t − ∆ V = µβ ( x ) g ′ ( θ u ) U, in Q T = Ω × (0 , T ] ,U = V = 0 , on ∂ Ω × (0 , T ] ,U ( x,
0) = V ( x,
0) = 0 , for x ∈ ¯Ω (B.1)where θ v is between ˜ v and ˆ v , θ u is between ˜ u and ˆ u . The assumption on (˜ u, ˜ v ), (ˆ u, ˆ v ) implies that f ′ ( θ v ) , g ′ ( θ u ) ∈ L ∞ (Ω × [0 , T ]) for any T < T . By using the comparison principle stated in Lemma 2.2,we deduce that U = V ≡ × [0 , T ].To obtain Theorem B, it is reduced to show the existence. Let ( ζ, ρ ) be the solution of the ODEsystem (3.6). The local existence of (3.6) can be obtained by [17, Chapter III]. Obviously, ( ζ, ρ ) is a su-persolution of (P). Since (0 ,
0) is a subsolution of (P), it follows from [12, Theorem 2.2] that there existsa unique classical solution ( u, v ) to (P) between (0 ,
0) and ( ζ, ρ ). In conclusion, the proof of Theorem Bis completed. (cid:3)
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