Finite speed of propagation in 1-D degenerate Keller-Segel system
aa r X i v : . [ m a t h . A P ] F e b Finite speed of propagation in 1-D degenerateKeller-Segel system
Department of Mathematics, Tsuda University,2-1-1, Tsuda-chou, Kodaira-shi, Tokyo, 187-8577,
Japan , [email protected] Abstract
We consider the following Keller-Segel system of degenerate type:(KS) ∂u∂t = ∂∂x (cid:16) ∂u m ∂x − u q − · ∂v∂x (cid:17) , x ∈ IR , t > , ∂ v∂x − γv + u, x ∈ IR , t > ,u ( x,
0) = u ( x ) , x ∈ IR , where m > , γ > , q ≥ m . We shall first construct a weak solution u ( x, t ) of(KS) such that u m − is Lipschitz continuous and such that u m − δ for δ > C with respect to the space variable x . As a by-product, we prove theproperty of finite speed of propagation of a weak solution u ( x, t ) of (KS), i.e., thata weak solution u ( x, t ) of (KS) has a compact support in x for all t > u ( x ) has a compact support in IR. We also give both upper and lower boundsof the interface of the weak solution u of (KS). We consider the following Keller-Segel system of degenerate type:(KS) ∂u∂t = ∂∂x (cid:16) ∂u m ∂x − u q − · ∂v∂x (cid:17) , x ∈ IR , t > , ∂ v∂x − γv + u, x ∈ IR , t > ,u ( x,
0) = u ( x ) , x ∈ IR , m > , γ > , q ≥ m . The initial data u is a non-negative function and in L ∩ L ∞ (IR) with u m ∈ H (IR) . This equation is often called as the Keller-Segel modeldescribing the motion of the chemotaxis molds. (see e.g., [5].)The aim of this paper is to construct a weak solution u ( x, t ) of (KS) such that u m − is Lipschitz continuous and such that u m − δ for δ > C with respect to thespace variable x . The regularity property whether u m − is Lipschitz continuous or of class C plays an important role for the investigation of the behaviour of the interface to thesolution u of (KS). Our result shows that the power m − u exhibits the borderlinebehaviour between Lipschitz continuity and C -regularity. Indeed, as a by-product ofLipschitz continuity for u m − , we prove that a weak solution u ( x, t ) of (KS) possesses theproperty of finite speed of propagation i.e., that a weak solution u ( x, t ) of (KS) has acompact support in x for all t > u ( x ) has a compact support in IR.Similar results have been obtained for the porous medium equation:(PME) ∂U∂t = ∂ U m ∂x , x ∈ IR , t > ,U ( x,
0) = U ( x ) , x ∈ IR . It is known that the comparison principle gives both upper and lower bounds of allsolutions U to (PME) by means of the Barenblatt solution V B which is an exact solutionof (PME). Hence the property of finite speed of propagation of U is a direct consequenceof the explicit form of V B since supp V B ( · , t ) is compact in IR for all time t .Our purpose is to prove the property of finite speed of propagation for (KS) to whichthe comparison principle is not available. To this end, one makes use of the notion of thedomain of dependence which is useful for the proof of uniqueness of solutions to the linearwave equations. For instance, the half-cone like region D T defined by D T := n ( x, t ); − ct + a ≤ x ≤ ct + b, ≤ t < T o , a < b, c > c vanishes on D T for the initial data u such that u ( x ) ≡ I ≡ [ a, b ].To deal with (KS), we generalize such an idea, and consider the curved half-cone likeregion. Indeed, suppose that u ( x ) = 0 on I . Then our curved half-cone like region D T with respect to I can be expressed by D T := n ( x, t ); ξ ( t ) ≤ x ≤ Ξ( t ) , ≤ t < T o , (1.1)where ξ ( t ) and Ξ( t ) are the solutions of the following initial value problems:(IE) ξ ′ ( t ) = − ∂∂x (cid:16) mm − u m − (cid:17) ( ξ ( t ) , t ) + u q − · ∂v∂x ( ξ ( t ) , t ) , ξ (0) = a, Ξ ′ ( t ) = − ∂∂x (cid:16) mm − u m − (cid:17) (Ξ( t ) , t ) + u q − · ∂v∂x (Ξ( t ) , t ) , Ξ(0) = b. Unfortunately, Lipschitz continuity of u m − is too weak to ensure the existence of solutions { ξ ( t ) , Ξ( t ) } to (IE). Hence we need to regularize u by u ε with small parameter ε >
0, and2eal with the approximating solutions { ξ ε ( t ) , Ξ ε ( t ) } which correspond to (IE) with u replaced by u ε . It is shown that Lipschitz continuity of u m − guarantees the existenceof uniform limit { ξ ( t ) , Ξ( t ) } on 0 ≤ t ≤ T of { ξ ε ( t ) , Ξ ε ( t ) } as ε →
0. Then we see that u ( x, t ) = 0 on D T .Our definition of a weak solution to (KS) now reads: Definition 1
Let m, γ and q be constants as m > , γ > , q ≥ . Let u be a non-negative function in IR with u ∈ L ∩ L ∞ (IR) and u m ∈ H (IR) . A pair of non-negativefunctions ( u, v ) defined in IR × [0 , T ) is said to be a weak solution of (KS) on [0 , T ) if i) u ∈ L ∞ (0 , T ; L (IR)) , u m ∈ L (0 , T ; H (IR)) , ii) v ∈ L ∞ (0 , T ; H (IR)) , iii) ( u, v ) satisfies (KS) in the sense of distributions: i.e., Z T Z IR (cid:0) ∂ x u m · ∂ x ϕ − u q − ∂ x v · ∂ x ϕ − u · ∂ t ϕ (cid:1) dxdt = Z IR u ( x ) · ϕ ( x, dx, for all functions ϕ ∈ C ∞ (IR × [0 , T )) , − ∂ x v + γv − u = 0 for a . a . ( x, t ) in IR × (0 , T ) . Concerning the local-in-time existence of weak solutions to (KS), the following result canbe shown by a slight modification of argument developed by the author [15, Theorem 1.1].
Proposition 1.1 ( local existence of weak solution and its L ∞ uniform bound ) Let m > , γ > , q ≥ . Suppose that the initial data u is non-negative everywhere.Then, (KS) has a non-negative weak solution ( u, v ) on [0 , T ) with T = (cid:16) k u k L ∞ (IR) +2 (cid:17) − q . Moreover, u ( t ) satisfies the following a priori estimate k u ( t ) k L ∞ (IR) ≤ k u k L ∞ (IR) + 2 for all t ∈ [0 , T ) . (1.2) Remark 1.
Concerning the global-in-time existence of weak solutions to (KS), theauthor and Kunii [17] obtained the following result: Let m, γ, q and the initial data u be as in Definition 1. In the case q < m + 2, there exists a weak solution u of (KS) on[0 , ∞ ) . On the other hand, in the case q ≥ m + 2, the weak solution u of (KS) on [0 , ∞ )can be constructed provided k u k L q − m (IR) is sufficiently small.Now, we construct a weak solution u of (KS) with some additional regularity for thevelocity potential u m − . 3 heorem 1.2 Let m > , γ > and q ≥ m . Let the initial data u be as in Definition 1.In addition, we assume that u m − is Lipschitz continuous in IR . Then, the weak solution u of (KS) on [0 , T ) given by Proposition 1.1 has the following additional properties (i)and (ii):(i) u m − ( x, t ) is Lipschitz continuous with respect to x for all ≤ t < T with the estimate sup
Let m > , γ > and q ≥ m . Let the initial data u be as in Definition 1. In addition, we assume that u ( x ) = 0 on some interval I ≡ [ a, b ] and that u m − is Lipschitz continuous in IR . Suppose that u is the weak solution of (KS) on [0 , T ) given by Theorem 1.2. Then, there exists a pair { ξ ( t ) , Ξ( t ) } of continuous functions on [0 , T ) with the following properties (i) and (ii):(i) ξ, Ξ ∈ W , ∞ (0 , T ) with ξ (0) = a, Ξ(0) = b ;(ii) u ( x, t ) = 0 for ξ ( t ) ≤ x ≤ Ξ( t ) , ≤ t < T . emark 3. (i) Concerning (PME), the interface of U can be explicitly determined bythe solutions ˆ ξ ( t ) and ˆΞ( t ) of the following initial value problems: ˆ ξ ′ ( t ) = − ∂∂x (cid:16) mm − U m − (cid:17) ( ˆ ξ ( t ) , t ) , ˆ ξ (0) = a, ˆΞ ′ ( t ) = − ∂∂x (cid:16) mm − U m − (cid:17) (ˆΞ( t ) , t ) , ˆΞ(0) = b. Indeed, by the comparison principle Knerr [6] showed that if U ( x ) = 0 on some interval I = [ a, b ] and U ( x ) > I c = IR \ I , then it holds that U ( x, t ) = 0 for ˆ ξ ( t ) ≤ x ≤ ˆΞ( t )and U ( x, t ) > x < ˆ ξ ( t ) and x > ˆΞ( t ) for all 0 ≤ t < ∞ . We call such ˆ ξ ( t ) and ˆΞ( t )the interface of (PME).(ii) Compared with (PME), it is not clear whether (IE) determines the exact interface of(KS) to which the comparison principle is not available. However, if ξ ( t ) and Ξ ( t ) arethe interface of (KS), i.e., that ξ ( t ) and Ξ ( t ) have the property that u ( x, t ) = 0 in I t := [ ξ ( t ) , Ξ ( t )] and u ( x, t ) > I t for all 0 ≤ t < T , then we can see that ξ ( t ) and Ξ( t ) given by Theorem 1.3 satisfy theestimates ξ ( t ) ≤ ξ ( t ) , Ξ( t ) ≤ Ξ ( t ) for all 0 ≤ t < T . Hence our result may be regarded as an estimate of the maximum and the minimum ofthe interface of (KS). Other observations were done by Mimura-Nagai [13] and Bonami-Hilhorst-Logak-Mimura [4].This paper is organized as follows. In Section 2, we shall first recall the approximatingproblem (KS) ε of (KS) introduced by [17]. Our main purpose is devoted to the derivationof uniform gradient bound with respect to ε > w ε = mm − u m − ε , where u ε is the smooth solution of (KS) ε . Bernstein’s method plays animportant role to obtain our uniform estimate. (see e.g., [12].) Then in Section 3, by thestandard compactness argument, we shall prove the Lipschitz continuity of the velocitypotential w = mm − u m − for the weak solution u of (KS). It is expected that ∂ x w ( x, t )becomes a discontinuous function in x after some finite time t . However, we shall showthat for p > m − ∂ x u p ( x, t ) is, in fact, a continuous function in IR for all t ∈ [0 , T ).Section 4 is devoted to the construction of continuous curves ξ ( t ) and Ξ( t ) such that u ( x, t ) = 0 on D T defined by (1.1), which implies the property of the finite speed ofpropagation for (KS).We will use the simplified notations:1) ∂ t = ∂∂t , ∂ x = ∂∂x , ∂ x = ∂ ∂x , ∂ x = ∂ ∂x , k · k L r = k · k L r (IR) , (1 ≤ r ≤ ∞ ) , R · dx := R IR · dx, Q T := IR × (0 , T ),4) When the weak derivatives ∂ x u, ∂ x u and ∂ t u are in L p ( Q T ) for some p ≥
1, we say5hat u ∈ W , p ( Q T ), i.e. , W , p ( Q T ) := n u ∈ L p (0 , T ; W ,p (IR)) ∩ W ,p (0 , T ; L p (IR)); k u k W , p ( Q T ) := k u k L p ( Q T ) + k ∂ x u k L p ( Q T ) + k ∂ x u k L p ( Q T ) + k ∂ t u k L p ( Q T ) < ∞ o . In order to justify the formal arguments, we introduce the following approximatingequations of (KS):(KS) ε ∂ t u ε ( x, t ) = ∂ x (cid:16) ∂ x ( u ε + ε ) m − ( u ε + ε ) q − u ε · ∂ x v ε (cid:17) , ( x, t ) ∈ IR × (0 , T ) , ∂ x v ε − γv ε + u ε , ( x, t ) ∈ IR × (0 , T ) ,u ε ( x,
0) = u ε ( x ) , x ∈ IR , where ε > u ε with ε > (A.1) u ε ≥ x ∈ IR and u ε ∈ W ,p (IR) withsup <ε< k u ε k L p (IR) ≤ k u k L p (IR) for all p ∈ [1 , ∞ ] , k u ε − u k L p (IR) → ε → p ∈ [1 , ∞ ) . (A.2) u ε ∈ W , (IR) with sup <ε< k ∂ x u ε k L (IR) ≤ k ∂ x u k L (IR) . Definition 2
We call ( u ε , v ε ) a strong solution of (KS) ε if it belongs to W , p × W , p ( Q T )for some p ≥ ε is satisfied almost everywhere.For the strong solution, we consider the case p = 3 and introduce the space W ( Q T )defined by W ( Q T ) := W , × W , ( Q T ) . (2.1)In [15]–[17], the following proposition concerning the existence of the strong solutionwas proved : Proposition 2.1 ( local existence of approximating solution ) Let m ≥ , γ > , q ≥ . Wetake T := ( k u k L ∞ (IR) + 2) − q . Then, for every ε > and every initial data u ε satisfyingthe hypothesis (A.1) , (KS) ε has a unique non-negative strong solution ( u ε , v ε ) in W ( Q T ) .Moreover, u ε ( t ) satisfies the following a priori estimate k u ε ( t ) k L ∞ (IR) ≤ k u k L ∞ (IR) + 2 for all t ∈ [0 , T ) and all ε ∈ (0 , . (2.2) 6 emark 4. (i) It should be noted that the time interval [0 , T ) of the existence of thestrong solution ( u ε , v ε ) can be taken uniformly with respect to ε > u, v ) of (KS) on [0 , T ) given by Proposition 1.1 can be constructedas the weak limit of ( u ε , v ε ) as ε →
0, where ( u ε , v ε ) is the strong solution in Proposition2.1. More precisely, by choosing a subsequence of ( u ε , v ε ) which we denote by ( u ε , v ε )itself for simplicity, we have u ε ⇀ u weakly − star in L ∞ (0 , T ; L (IR)) ,u mε → u m weakly in L (0 , T ; H (IR)) and strongly in C ([0 , T ); L loc (IR)) ,v ε ⇀ v weakly − star in L ∞ (0 , T ; H (IR))as ε → . In what follows, we assume that the sequence of approximating solutions ( u ε , v ε )satisfies the above convergence.(iii) The strong solution ( u ε , v ε ) ∈ W ( Q T ) is more regular. Indeed, for every ε >
0, itcan be shown that u ε , v ε ∈ C ∞ (IR × (0 , T )) . The following lemma gives the gradient estimate for the velocity potential u m − . Lemma 2.2
Let m > , γ > and q ≥ m . Let the initial data u be as in Definition 1.For every ε > , we take u ε so that the hypothesis (A.1)–(A.2) are satisfied. In addition,we assume that u m − ε is Lipschitz continuous in IR . Then the strong solution u ε of (KS) ε on [0 , T ) given by Proposition 2.1 has the following property: sup <ε< (cid:16) sup In the proof of Lemma 2.2, we have used a sequence { η k ( x ) } ∞ k = −∞ of cut-offfunctions with properties (2.11)–(2.13). Taking η ( x ) as η ( x ) = x ≤ − x ) for − < x < − , − x + 1) for − < x ≤ − , − ≤ x ≤ , − x − for 1 < x ≤ , − x ) for < x < , x ≥ , and then defining η k by η k ( x ) := η ( x − k ) for k = 0 , ± , ± , · · · , we see that { η k ( x ) } ∞ k = −∞ has the desired properties (2.11)–(2.13). Let us first show that for every t ∈ [0 , T ), { u ε ( · , t ) } ε> is a sequence of uniformlybounded and equi-continuous functions in IR. Indeed, the uniform bound is a consequenceof (2.2). By (2.2), (2.3) and (1.4) with u replaced by u ε + ε , it holds | u ε ( x, t ) − u ε ( y, t ) | ≤ C ( k u k L ∞ + 2) | x − y | µ , µ = min { , m − } x, y ∈ IR , ≤ t < T , and all ε > 0, where C is the same constant as in (2.3). Thisimplies that { u ε ( · , t ) } ε> is a family of equi-continuous functions in IR for all 0 ≤ t < T .Hence by the Ascoli-Arzela theorem, there is a subsequence of { u ε ( · , t ) } ε> , which wedenoted by { u ε ( · , t ) } ε> itself such that u ε ( · , t ) −→ u ( · , t ) as ε → I ⊂ IR.On the other hand, by (2.3) and the weakly-star compactness of L ∞ ( Q T ), there existsa sequence of { u ε } ε> , which we denote by { u ε } ε> itself for simplicity, and a function˜ u ∈ L ∞ ( Q T ) such that ∂ x ( u ε + ε ) m − → ˜ u weakly − star in L ∞ ( Q T )with k ˜ u k L ∞ ( Q T ) ≤ lim inf ε → +0 k ∂ x ( u ε + ε ) m − k L ∞ ( Q T ) . By (3.1), it is easy to see that ˜ u = ∂ x u m − , which yields the desired estimate (1.3).Next, we shall show that ∂ x u m − δ ( · , t ) is a continuous function in IR for all 0 < t < T and for all δ > ∂ x u m − δ ( x, t ) = 0 at the point ( x, t )such as u ( x, t ) = 0. To this aim, we follow a similar argument employed in Aronson [3].Let u ( x , t ) > 0. Then we see by the standard argument that both ∂ x u and ∂ x u m − δ with δ > x , t ). Therefore, it sufficesto prove that ∂ x u m − δ ( · , t ) is a continuous function in a neighbourhood of x such as u ( x , t ) = 0 with the additional property that ∂ x u m − δ ( x , t ) = 0. By virtue of (3.1),for every t ∈ [0 , T ) and every compact interval I ⊂ IR, it holds that u ε ( · , t ) → u ( · , t )uniformly on I . Therefore, by Remark 2, there exists a > ≤ u ε ( x, t ) ≤ | u ε ( x, t ) − u ( x, t ) | + | u ( x, t ) − u ( x , t ) | + u ( x , t ) ≤ a µ (3.2)holds for all x ∈ I a ( x ) := { x ∈ IR; | x − x | < a } and for all 0 < a ≤ a and for all0 < ε < 1, where µ := min { , m − } .On the other hand, since we have u m − δε ( x, t ) − u m − δε ( x ′ , t ) = m − δm − Z xx ′ u δε ( x, t ) · ∂ x u m − ε ( x, t ) dx, (3.3)it follows from (3.2),(3.3) and Lemma 2.2 that | u m − δε ( x, t ) − u m − δε ( x ′ , t ) | ≤ C (2 a µ ) δ | x − x ′ | for all x, x ′ ∈ I a ( x )(3.4)and for all 0 < a ≤ a and for all 0 < ε < 1, where C depends on m, γ, q, u but not on ε . Letting ε → +0 in (3.4), we have by (3.1) that | u m − δ ( x, t ) − u m − δ ( x ′ , t ) |≤ C (2 a µ ) δ | x − x ′ | for all x, x ′ ∈ I a ( x ) and all 0 < a ≤ a . (3.5) 14aking x = x in (3.5) and then letting x ′ → x , we have | ∂ x u m − δ ( x , t ) | ≤ C (2 a µ ) δ , < a ≤ a . Hence we have by letting a → ∂ x u m − δ ( x , t ) = 0 . Similarly, letting x ′ → x in (3.5), we have | ∂ x u m − δ ( x, t ) | ≤ C (2 a µ ) δ for all 0 < a ≤ a , which implies that ∂ x u m − δ ( · , t ) is continuous at x . Since x can be taken arbitrary insuch a way that u ( x , t ) = 0, we conclude that ∂ x u m − δ ( · , t ) is a continuous function inIR for all t ∈ [0 , T ) with the additional property that ∂ x u m − δ ( x, t ) = 0 for the point( x, t ) such as u ( x, t ) = 0.The case of 1 < m < ∂ x u ( · , t ) is a continuous function in IR for all t ∈ [0 , T ) with the additional property that ∂ x u ( x, t ) = 0 for the point ( x, t ) such as u ( x, t ) = 0. This completes the proof of Theorem1.2. Let ( u ε , v ε ) be the unique strong solution of (KS) ε given by Proposition 2.1. For afixed R > 0, we take a, b > − R < a < b < R and consider the following ordinarydifferential equations:(IE) ξ : (cid:26) ξ ′ ε ( t ) = mm − ∂ x (cid:0) u ε + ε (cid:1) m − ( ξ ε ( t ) , t ) − (cid:0) u ε + ε (cid:1) q − u ε · ∂ x v ε ( ξ ε ( t ) , t ) , ≤ t < T ,ξ ε (0) = a, and(IE) Ξ : (cid:26) Ξ ′ ε ( t ) = mm − ∂ x (cid:0) u ε + ε (cid:1) m − (Ξ ε ( t ) , t ) − (cid:0) u ε + ε (cid:1) q − u ε · ∂ x v ε (Ξ ε ( t ) , t ) , ≤ t < T , Ξ ε (0) = b. By Remark 4 (iii), (2.2), (2.3) and (2.23), we have ∂ x ( u ε + ε ) m − ∈ C ([ − R, R ] × [0 , T ))(4.1)and sup <ε< (cid:16) sup 0, respectively.We consider the following domain: D τ := [ t ∈ [0 ,τ ] I t × { t } , I t := n x ∈ IR; ξ ε ( t ) ≤ x ≤ Ξ ε ( t ) o for 0 < τ < T . By the local uniqueness of the initial value problem (IE) ξ and (IE) Ξ , we obtain that ξ ε ( t ) < Ξ ε ( t ) for all 0 ≤ t < T . Let us define the gradient −→∇ and the vector F on ( x, t ) by −→∇ := ( ∂ x , ∂ t ) , F ( x, t ) := (cid:16) − ∂ x ( u ε + ε ) m + ( u ε + ε ) q − u ε · ∂ x v ε , u ε + ε (cid:17) . Then it follows from the first equation of (KS) ε that Z D τ −→∇ · F ( x, t ) dxdt = Z D τ ∂ t u ε − ∂ x (cid:16) ∂ x ( u ε + ε ) m − ( u ε + ε ) q − u ε · ∂ x v ε (cid:17) dxdt = 0(4.2)for all 0 < τ < T . Taking two curves C and C as C := { ( x, t ) = ( ξ ε ( t ) , t ); 0 < t < τ } , C := { ( x, t ) = (Ξ ε ( t ) , t ); 0 < t < τ } , we have ∂D τ = I ∪ C ∪ C ∪ I τ . Hence, the Stokes formula gives0 = Z D τ −→∇ · F ( x, t ) dxdt = Z ∂D τ F ( x, t ) · n dS = Z ba F ( x, · (0 , − dx − Z ξ ε ( τ )Ξ ε ( τ ) F ( x, t ) · (0 , dx + Z C F · n dS + Z C F · n dS = − Z ba ( u ε + ε ) dx + Z Ξ ε ( τ ) ξ ε ( τ ) ( u ε + ε ) dx + Z C F · n dS + Z C F · n dS, (4.3)where n and n denote the unit outer normals to C and C , respectively. Since n = (1 , ξ ′ ε ( t )) p ξ ′ ε ( t )) , n = (1 , Ξ ′ ε ( t )) p ′ ε ( t )) , 16e have by (IE) ξ and (IE) Ξ that F · n = 0 on C , F · n = 0 on C . (4.4)Combining (4.2)–(4.4), we have Z Ξ ε ( τ ) ξ ε ( τ ) ( u ε ( x, τ ) + ε ) dx = Z ba ( u ε ( x ) + ε ) dx, ≤ τ < T . (4.5)On the other hand, we obtain from (2.23), Proposition 2.1 and Lemma 2.2 thatsup <ε< k ξ ε k L ∞ (0 ,T ) ≤ a + (cid:16) mm − · C + 2( k u k L ∞ + 2) q − · k u k L (cid:17) · T , sup <ε< k ξ ′ ε k L ∞ (0 ,T ) ≤ mm − · C + 2( k u k L ∞ + 2) q − · k u k L . Hence it follows by the Ascoli-Arzela theorem that there exists a subsequence of { ξ ε ( t ) } ,still denoted by { ξ ε ( t ) } ε> , and a function ξ ∈ C , [0 , T ) such that ξ ε ( t ) → ξ ( t ) as ε → t ∈ [0 , T ) . (4.6)Obviously, a similar argument to Ξ ε ( t ) also holds, and there exist a subsequence of { Ξ ε ( t ) } ε> , still denoted by { Ξ ε ( t ) } , and Ξ ∈ C , [0 , T ) such thatΞ ε ( t ) → Ξ( t ) as ε → t ∈ [0 , T ) . (4.7)Since u ≡ a, b ], by letting ε → Z Ξ( t ) ξ ( t ) u ( x, t ) dx = Z ba u ( x ) dx = 0(4.8)for all 0 ≤ t < T . Indeed, we may assume − R < ξ ε ( t ) < Ξ ε ( t ) < R for all ε > , and all 0 ≤ t < T , where R > (cid:12)(cid:12)(cid:12) Z Ξ ε ( t ) ξ ε ( t ) ( u ε + ε ) dx − Z Ξ( t ) ξ ( t ) u dx (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z Ξ ε ( t ) ξ ε ( t ) ( u ε − u ) dx (cid:12)(cid:12)(cid:12) + ε Z Ξ ε ( t ) ξ ε ( t ) dx + (cid:12)(cid:12)(cid:12) Z Ξ ε ( t ) ξ ε ( t ) u dx − Z Ξ( t ) ξ ( t ) u dx (cid:12)(cid:12)(cid:12) ≤ (cid:16) sup − R ≤ x ≤ R | u ε ( x, t ) − u ( x, t ) | + ε (cid:17) (Ξ ε ( t ) − ξ ε ( t ))+ k u k L ∞ ( Q T ) (cid:16) | Ξ ε ( t ) − Ξ( t ) | + | ξ ε ( t ) − ξ ( t ) | (cid:17) ≤ R (cid:16) sup − R ≤ x ≤ R | u ε ( x, t ) − u ( x, t ) | + ε (cid:17) + (cid:16) k u k L ∞ + 2 (cid:17)(cid:16) | Ξ ε ( t ) − Ξ( t ) | + | ξ ε ( t ) − ξ ( t ) | (cid:17) → ε → +0 , u is non-negative in IR × [0 , T ), we conclude from (4.8) that u ( x, t ) = 0 for ξ ( t ) ≤ x ≤ Ξ( t ) , ≤ t < T . This proves Theorem 1.3. 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