Improved conditions for single-point blow-up in reaction-diffusion systems
aa r X i v : . [ m a t h . A P ] J a n IMPROVED CONDITIONS FOR SINGLE-POINT BLOW-UP INREACTION-DIFFUSION SYSTEMS
NEJIB MAHMOUDI, PHILIPPE SOUPLET, AND SLIM TAYACHI
Abstract.
We study positive blowing-up solutions of the system: u t − δ ∆ u = v p , v t − ∆ v = u q , as well as of some more general systems. For any p, q >
1, we prove single-point blow-upfor any radially decreasing, positive and classical solution in a ball. This improves onpreviously known results in 3 directions:(i) no type I blow-up assumption is made (and it is known that this property mayfail);(ii) no equidiffusivity is assumed, i.e. any δ > F ( u, v ), G ( u, v ) can be handled, which need notfollow a precise power behavior.As side result, we also obtain lower pointwise estimates for the final blow-up profiles. Introduction
Problem and main results.
In this paper, we consider nonnegative solutions ofthe following reaction-diffusion system: u t − δ ∆ u = v p x ∈ Ω , t > ,v t − ∆ v = u q , x ∈ Ω , t > ,u = v = 0 , x ∈ ∂ Ω , t > ,u (0 , x ) = u ( x ) , x ∈ Ω ,v (0 , x ) = v ( x ) , x ∈ Ω , (1.1) Mathematics Subject Classification.
Primary: 35B40; 35B44; 35B50. Secondary: 35K61; 35K40;35K57.
Key words and phrases.
Nonlinear initial-boundary value problems, nonlinear parabolic equations,reaction-diffusion systems, asymptotic behavior of solutions, single-point blow-up, blow-up profile. as well as of the more general system u t − δ ∆ u = F ( u, v ) , x ∈ Ω , t > ,v t − ∆ v = G ( u, v ) , x ∈ Ω , t > ,u = v = 0 , x ∈ ∂ Ω , t > ,u (0 , x ) = u ( x ) , x ∈ Ω ,v (0 , x ) = v ( x ) , x ∈ Ω . (1.2)Here p, q > δ > , Ω = B (0 , R ) = { x ∈ R n ; | x | < R } with R > u , v ∈ L ∞ (Ω) , u , v ≥
0, radially symmetric, radially nonincreasing. (1.3)As for the functions F and G , we assume that F, G ∈ C ( R ) (1.4)and that system (1.2) is cooperative, i.e.: F v ( u, v ) , G u ( u, v ) ≥ , for all u, v ≥
0. (1.5)Additional assumptions on
F, G will be made below.Under assumptions (1.3)–(1.5), system (1.2) has a unique nonnegative, radially sym-metric and radially nonincreasing maximal solution ( u, v ), classical for t >
0. This factfollows by standard contraction mapping and maximum principle arguments. The maxi-mal existence time of ( u, v ) is denoted by T ∗ ∈ (0 , ∞ ]. If, moreover, T ∗ < ∞ , thenlim sup t → T ∗ ( k u ( t ) k + k v ( t ) k ∞ ) = ∞ , and we say that the solution blows up in finite time with blow-up time T ∗ . Also, withoutrisk of confusion, we shall denote ρ = | x | , u = u ( t, ρ ), v = v ( t, ρ ). So we have u ρ , v ρ ≤ , T ∗ ) × Ω. (1.6)Problem (1.1) is a basic model case for reaction-diffusion systems and, as such, it hasbeen the subject of intensive investigation for more than 20 years (see e.g. [16, Chapter 32]and the references therein). We are here mainly interested in proving single-point blow-upfor systems (1.1) and (1.2).For system (1.1), the blow-up set was first studied in [6]. In that work, Friedman andGiga proved that blow-up occurs only at the origin for symmetric nonincreasing initial datain dimension n = 1, under the very restrictive conditions p = q and δ = 1. Note that theseassumptions are essential in [6] in order to apply the maximum principle to suitable linearcombination of the components u and v , so as to derive comparison estimates betweenthem. INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 3
Let us recall that, for scalar equations, the first result on single-point blow-up wasobtained by Weissler [21], and that different methods were subsequently developed in[7, 14]. In turn, the method of Friedman and Giga for systems is based on an extensionof that in [7] for a single equation. More recently, the restriction p = q was removed bythe second author [17], who proved single-point blow-up for radial nonincreasing solutionsof (1.1) for any p, q > n ≥
1. However, the equidiffusivity assumption δ = 1 is stillneeded in [17] and, in addition, it is required that the solution satisfies the upper type Iblow-up rate estimatessup
1, is to further remove the previouslymade extra assumptions. More precisely, we shall improve the known results in threedirections, by proving single-point blow-up:(i) without assuming the type I blow-up rate estimate (1.7);(ii) without assuming equidiffusivity , i.e. for any δ > general problem such as (1.2).Direction (i) seems the more important and challenging one, since estimate (1.7) is notknown in general and need not even be true. It usually requires either the hypothesis that p or q are not too large (see e.g. [3, 5]), or that the solution is monotone in time. Indeed, forlarge p , even in the particular case of the scalar problem, there exist radial nonincreasing,single-point blow-up solutions of type II (i.e., such that (1.7) fails); see [10, 11, 13]. As forthe case of monotone in time solutions, it seems that the known proofs of (1.7) for systems(see e.g. [4]) usually require δ = 1. Also we recall that non-equidiffusive parabolic systemsare often much more involved, both in terms of behavior of solutions and at the technicallevel (cf. [15] and [16, Chapter 33]). As for the general problem (1.2), we shall be able tohandle a large class of nonlinearities which need not follow a precise power behavior. Thefeatures (i)-(iii) will require a number of nontrivial new ideas, building on the approachin [17], which is here improved and made more flexible. See Section 1.2 below for details.The main results of this paper are the following. N. MAHMOUDI, PH. SOUPLET, AND S. TAYACHI
Theorem 1.1.
Let
Ω = B (0 , R ) , p, q > and δ > . Assume (1.3) and let the solution ( u, v ) of (1.1) satisfy T ∗ < ∞ . Then blow-up occurs only at the origin, i.e. sup
0, there exist µ, A, κ , κ > κ κ <
1, such that(1 + µ ) F ≤ uF u + κ vF v and (1 + µ ) G ≤ vG v + κ uG u on D := n ( u, v ) ∈ [ A, ∞ ) ; C ≤ u q +1 v p +1 ≤ C o . (1.13) Theorem 1.2.
Let
Ω = B (0 , R ) , p, q > , δ > . Assume (1.3)–(1.5) and (1.10)–(1.13).Let the solution ( u, v ) of (1.2) satisfy T ∗ < ∞ . Then blow-up occurs only at the origin,i.e. (1.9) holds. We immediately give examples of nonlinearities to which Theorem 1.2 applies.
Examples 1.1. (i) The result of Theorem 1.2 is valid for system (1.2) with F ( u, v ) = λv p + m X i =1 λ i u r i v s i and G ( u, v ) = λu q + m X i =1 λ i u r i v s i , (1.14) where p, q > , m ≥ and for all ≤ i ≤ m, r i , s i , r i , s i , λ i , λ i ≥ , r i p + 1 q + 1 + s i ≤ p and r i + s i p + 1 q + 1 ≤ q. (1.15) We note that the requirement that
F, G be of class C imposes r i , s i , r i , s i ∈ { } ∪ [1 , ∞ ) .However, in case some of these numbers belong to (0 , , Theorem 1.2 still applies if F, G only coincide with the expressions in (1.14) for u and v sufficiently large. We stress that INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 5
F, G in (1.14) are not mere perturbations of v p , u q . Indeed, when we have equality in(1.15), the additional terms are critical in the sense of scaling.(ii) The result of Theorem 1.2 is also valid for system (1.2) with F ( u, v ) = v p (cid:2) λ sin (cid:0) k log(1+ v ) (cid:1)(cid:3) and G ( u, v ) = u q (cid:2) λ sin ( k log(1+ u ) (cid:1)(cid:3) (1.16) where p, q > λ, λ > , < k < ( p − √ λλ < k < ( q − p λλ . (1.17) Note that Theorem 1.2 thus allows nonlinearities
F, G with oscillations of arbitrarily largeamplitude around v p , u q (since λ, λ can be arbitrarily large in (1.17)). Finally, in the case of monotone in time solutions, we extend to system (1.2) the lowerpointwise estimates from [17] on the final blow-up profiles.
Theorem 1.3.
Let
Ω = B (0 , R ) , p, q ≥ and δ > . Assume (1.3)–(1.5), (1.10)–(1.12)and let the solution ( u, v ) of (1.2) satisfy T ∗ < ∞ . Assume in addition that u t , v t ≥ .Then there exist constants ε , ε > , such that | x | α u ( T ∗ , x ) ≥ ε , < | x | < ε and | x | β v ( T ∗ , x ) ≥ ε , < | x | < ε , where α and β are given by (1.8) . Remarks 1.1. ( i ) The results of Theorems 1.2 and 1.3 remain true for the Cauchy problem (that is, (1.2)with R = ∞ and ∂ Ω = ∅ ) provided u , v are not both constant. These follow from simplemodifications of the proofs.( ii ) Concerning Theorem 1.3, we note that the existence of a positive, radially symmetric,radially nonincreasing and classical solution of (1.2) such that T ∗ < ∞ and u t , v t ≥ λu , λv ) with λ > u , v satisfy (1.3) and ( u , v ∈ C (Ω) ∩ C (Ω) , u = v = 0 on ∂ Ω ,δ ∆ u + F ( u , v ) ≥ , ∆ v + G ( u , v ) ≥ . See [20].
N. MAHMOUDI, PH. SOUPLET, AND S. TAYACHI
Outline of proof.
As in [6, 17] (and cf. [7, 2]), the basic idea for proving single-pointblow-up is to consider auxiliary functions
J, J , either of the form (cf. [6]): J ( t, ρ ) = u ρ + εc ( ρ ) u γ , J ( t, ρ ) = v ρ + εc ( ρ ) v γ , (1.18)or (cf. [17]): J ( t, ρ ) = u ρ + εc ( ρ ) v γ , J ( t, ρ ) = v ρ + εc ( ρ ) u γ , (1.19)with suitable constants γ, γ > ε > c ( ρ ) , c ( ρ ). The couple ( J, J ) satisfiesa system of parabolic inequalities to which one aims at applying the maximum principle,so as to deduce that
J, J ≤
0. By integrating these inequalities in space, one then obtainsupper bounds on u and v which guarantee single-point blowup at the origin.However, in the case of systems, such a procedure turns out to require good comparisonproperties between u and v . Due to the global comparison properties employed in [6], theresult there for system (1.1) imposed the severe restriction p = q (as well as δ = 1, becausethis comparison was shown by applying the maximum principle to a linear combinationof u and v ). For type I blowup, radially decreasing solutions of (1.1) with δ = 1 and any p, q >
1, this was overcome in [17] by applying a different strategy. Instead of lookingfor comparison properties valid everywhere, one assumed for contradiction that (type I)single-point blow-up fails and then established sharp asymptotic estimates near blowuppoints. Namely, it was shown that, if ρ > t → T ∗ ( T ∗ − t ) α u ( t, ρ ) = A , lim t → T ∗ ( T ∗ − t ) β u ( t, ρ ) = B (1.20)uniformly on compact subsets of [0 , ρ ), for some uniquely determined constants A , B >
0, hence in particular the comparison propertylim t → T ∗ h u p +1 v q +1 i ( t, ρ ) = A p +10 B − ( q +1)0 . These estimates turned out to be sufficient to handle the system satisfied by suitablefunctions of the form
J, J in (1.19). As for estimate (1.20), its proof in [17] was longand technical, using similarity variables, delayed smoothing effects for rescaled solutions,monotonicity arguments and a precise classification of entire solutions of a related ODEsystem.Although we here follow the same basic strategy as in [17], we have been able to makethe method much more flexible, leading to the improvements mentioned above, owing toa number of new ideas, which we now describe.
INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 7 (i) An important observation, improving on [17], is that the proof that
J, J ≤ ( C ≤ ( T ∗ − t ) α u ( t, ρ ) ≤ C C ≤ ( T ∗ − t ) β v ( t, ρ ) ≤ C in [ T ∗ / , T ∗ ) × [ ρ , ρ ], (1.21)for some 0 < ρ < ρ < ρ and some (unrestricted) constants C , C >
0. Defining
J, J by (1.18) instead of (1.19), and localizing the function c ( ρ ), this can be achieved by choosing γ, ¯ γ > suitably close to u ( t, ρ ) ≤ C ρ − n ( T ∗ − t ) − α and v ( t, ρ ) ≤ C ρ − n ( T ∗ − t ) − β . (1.22)See Proposition 3.1. This is a rather easy consequence of Kaplan’s eigenfunction method.This yields in particular the upper part of the bounds in (1.21).(iii) As for the more delicate lower bounds in (1.21), they are proved in three steps.The first step (Proposition 4.1) is to establish a nondegeneracy property which guaranteesthat ρ ∈ (0 , R ) is not a blowup point whenever( T ∗ − t ) α u ( t, ρ ) ≤ η and ( T ∗ − t ) β v ( t, ρ ) ≤ η (1.23)at some time t and some ρ ∈ (0 , ρ ) with η > u, v ), hence of global bound on the rescaled solution. This is overcome, after truncatingthe domain, by carefully comparing with a modified solution. The latter is obtained by asuitable reflection and supersolution procedure, taking advantage of the local upper boundin (1.22) (see step 1 of the proof of Proposition 4.1). After passing to similarity variables,the modified solution is now uniformly bounded, but at the expense of additional terms,generated by the reflection procedure, which appear in the PDE’s. However, these termscan be localized exponentially far away in space for large time, and thus taken care of inthe smoothing effect arguments.(iv) As a second step in the proof of the lower bounds in (1.21), we prove (see Section 5)that solutions rescaled around a blow-up point behave, in a suitable sense, like a continuousdistribution solution of the following system of ordinary differential inequalities (ODI): ( φ ′ + αφ ≥ c ψ p ,ψ ′ + βψ ≥ c φ q , (1.24) N. MAHMOUDI, PH. SOUPLET, AND S. TAYACHI on ( −∞ , ∞ ). This is proved by a further use of similarity variables, along with the spacemonotonicity. Moreover, we single out a simple but crucial property of local interpendenceof components for such solutions of (1.24); namely, φ (0) = 0 if and only if ψ (0) = 0.(v) Then, as a last step (Section 6), we show that, if one of the lower bounds in (1.21) isviolated, then, owing to point (iv), we have convergence of rescaled solutions to a solutionof (1.24) such that φ (0) = 0 and ψ (0) = 0. Restated in terms in ( u, v ), this leads to thedegeneracy condition (1.23) at some time t . But, in view of point (iii), this contradicts ρ being a blowup point.We note that, in [17], the study of the particular system (1.1) led to the system ofequalities ( φ ′ + αφ = c ψ p ,ψ ′ + βψ = c φ q , (1.25)instead of (1.24), and a complete classification of entire solutions of (1.25) was obtained,which enabled one to deduce the more precise behavior (1.20) at the left of an allegednonzero blowup point. We stress that, thanks to the new possibility of arguing throughthe weaker estimates (1.21), we can now avoid such a classification (which is not availablefor the general system (1.24)).The organization of the rest of this paper is as follows. In Section 2, we prove Theo-rem 1.2 (hence Theorem 1.1) assuming the local upper and lower type I estimates (1.21)near blow-up points. Sections 3-6 are next devoted to proving these estimates. In Sec-tion 3, we establish upper blowup estimates away from the origin (Proposition 3.1). InSection 4 we prove the key nondegeneracy property Proposition 4.1. In Section 5 weshow the ODI behavior for rescaled solutions and the local interpendence of componentsfor the ODI system. In Section 6 we then prove the lower bounds in (1.21) by using acontradiction argument and the results of Sections 3-5. Finally, in Section 7, we establishthe pointwise lower bounds on the blow-up profiles, i.e., Theorem 1.3, and we verify theassertions in Examples 1.1.2. Proof of Theorem 1.2 assuming local upper and lower type I estimates
The local upper and lower type I estimates, in case of existence of nonzero blow-uppoints, are formulated in the following proposition.
Proposition 2.1.
Let
Ω = B (0 , R ) , p, q > , δ > . Assume (1.3)–(1.5) and (1.10)–(1.12) and let the solution ( u, v ) of (1.2) satisfy T ∗ < ∞ . Assume that there exists INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 9 ρ ∈ (0 , R ) such that lim sup t → T ∗ (cid:0) u ( t, ρ ) + v ( t, ρ ) (cid:1) = ∞ and let [ ρ , ρ ] be a compact subinterval of (0 , ρ ) . Then, there exist constants C , C > (possibly depending on the solution ( u, v ) and on ρ , ρ , ρ ), such that C ≤ ( T ∗ − t ) α u ( t, ρ ) ≤ C on [ T ∗ / , T ∗ ) × [ ρ , ρ ] (2.1) and C ≤ ( T ∗ − t ) β v ( t, ρ ) ≤ C on [ T ∗ / , T ∗ ) × [ ρ , ρ ] . (2.2) In particular, there exist C ′ , C ′ > such that C ′ ≤ u q +1 ( t, ρ ) v p +1 ( t, ρ ) ≤ C ′ on [ T ∗ / , T ∗ ) × [ ρ , ρ ] . (2.3)As already explained in Section 1.2, the proof of Proposition 2.1 will be developed inSections 3-6, and we shall now prove Theorem 1.2 assuming Proposition 2.1.We introduce the auxiliary J, J functions defined by J ( t, ρ ) = u ρ + εc ( ρ ) u γ , J ( t, ρ ) = v ρ + εc ( ρ ) v γ , (2.4)with c ( ρ ) = sin (cid:18) π ( ρ − ρ ) ρ − ρ (cid:19) , c ( ρ ) = κ c ( ρ ) , ρ ≤ ρ ≤ ρ , (2.5)where γ , γ > ε , κ , ρ > ρ > J , J ∈ C ((0 , T ∗ ) × [0 , R ]) ∩ W , kloc ((0 , T ∗ ) × [0 , R )), for all 1 < k < ∞ , by parabolic L p -regularity. Lemma 2.1.
Under the hypotheses of Theorem 1.2, assume that there exists ρ ∈ (0 , R ) such that lim sup t → T ∗ (cid:0) u ( t, ρ ) + v ( t, ρ ) (cid:1) = ∞ and let ρ = ρ / and ρ = ρ / . Then there exist γ , γ > , κ > and T ∈ (0 , T ∗ ) , suchthat, for any ε ∈ (0 , , the functions J and J defined in (2.4)–(2.5) satisfy J t − δJ ρρ − δ n − ρ J ρ + δ n − ρ J ≤ F v ( u, v ) J + h F u ( u, v ) − εδγc ′ u γ − i J,J t − J ρρ − n − ρ J ρ + n − ρ J ≤ G u ( u, v ) J + h G v ( u, v ) − εγ c ′ v γ − i J , (2.6) for a.e. ( t, x ) ∈ [ T , T ∗ ) × ( ρ , ρ ) . Proof.
Step 1.
Computation of a parabolic operator on J and ¯ J . Let H = u γ . By differentiation of (2.4), we have J t − δJ ρρ = ( u ρ ) t + εcH t − δ ( u ρρ ) ρ − δεc ′′ H − δεc ′ H ρ − δεcH ρρ = ( u t − δu ρρ ) ρ + ε (cid:16) c (cid:0) H t − δH ρρ (cid:1) − δc ′ H ρ − δc ′′ H (cid:17) . By the first equation in (1.2), we get( u t − δu ρρ ) ρ = (cid:18) δ n − ρ u ρ + F ( u, v ) (cid:19) ρ = δ n − ρ u ρρ − δ n − ρ u ρ + F u u ρ + F v v ρ and H t − δH ρρ = γu γ − u t − δγ ( γ − u γ − u ρ − δγu γ − u ρρ ≤ γu γ − (cid:0) u t − δu ρρ (cid:1) = γu γ − (cid:18) δ n − ρ u ρ + F (cid:19) . Here and in the sequel, we omit the arguments u, v when no confusion may arise. Usingthis, along with u ρ = J − εcu γ and v ρ = J − εcv γ , we obtain J t − δJ ρρ ≤ δ n − ρ ( J − εcu γ ) ρ − δ n − ρ ( J − εcu γ )+ F u ( J − εcu γ ) + F v (cid:0) J − εcv γ (cid:1) + εu γ − (cid:20) γc (cid:18) δ n − ρ u ρ + F (cid:19) − γδc ′ u ρ − δc ′′ u (cid:21) = δ n − ρ J ρ − δε n − ρ c ′ u γ − δεc n − ρ γu γ − u ρ − δ n − ρ J + δε n − ρ cu γ + F u ( J − εcu γ ) + F v (cid:0) J − εcv γ (cid:1) + εu γ − (cid:20) γc (cid:18) δ n − ρ u ρ + F (cid:19) − δγc ′ ( J − εcu γ ) − δc ′′ u (cid:21) . Consequently, J t − δJ ρρ − δ n − ρ J ρ + δ n − ρ J ≤ F v J + h F u − εδγc ′ u γ − i J + εH , (2.7)with H := − cu γ F u − cv γ F v + u γ − (cid:20) γcF + 2 δεγc ′ cu γ + δu (cid:18) n − ρ (cid:16) cρ − c ′ (cid:17) − c ′′ (cid:19)(cid:21) . For convenience, we set ξ ( ρ ) = n − ρ (cid:16) ρ − c ′ c (cid:17) − c ′′ c , ρ < ρ < ρ and, on (0 , T ∗ ) × ( ρ , ρ ), e H := H c u γ − = − uF u − κ v γ − u γ − vF v + γF + 2 δ εγc ′ u γ + δξ ( ρ ) u. (2.8)Note that, up to now, our calculations made use of (1.2) through the first PDE only. Thus,by replacing δ with 1 and exchanging the roles of u, F, γ, c and v , G, γ, c , we get J t − J ρρ − n − ρ J ρ + n − ρ J ≤ G u J + h G v − εγ c ′ v γ − i J + εH , (2.9)with e H := H c v γ − := − vG v − κ u γ − v γ − uG u + γG + 2 εγ c ′ v γ + ξ ( ρ ) v. (2.10)Next setting ℓ = ρ − ρ = ρ /
4, we have − c ′ c = − πℓ cot (cid:16) π ( ρ − ρ ) ℓ (cid:17) and − c ′′ c = − π ℓ cot (cid:16) π ( ρ − ρ ) ℓ (cid:17) + 2 π ℓ hence, ξ ( ρ ) = n − ρ + 2 π ℓ − πℓ (cid:20) n − ρ + πℓ cot (cid:16) π ( ρ − ρ ) ℓ (cid:17)(cid:21) cot (cid:16) π ( ρ − ρ ) ℓ (cid:17) . It follows that ξ ( ρ ) −→ ρ → ρ +1 −∞ and ξ ( ρ ) −→ ρ → ρ − −∞ . Since ξ is continuous on ( ρ , ρ ), then there exists C = C ( n, ρ ) > ξ ( ρ ) ≤ C , for all ρ ∈ ( ρ , ρ ) . (2.11)By (2.8), (2.10) and (2.11), we obtain, for some C = C ( δ, ρ ) > e H ≤ − uF u − κ v γ − u γ − vF v + γF + C δγu γ + δC u (2.12)and e H ≤ − vG v − κ u γ − v γ − uG u + γG + C γv γ + C v. (2.13) Step 2.
Estimation of the remainder terms e H , e H with help of the local lower andupper type I estimates. Assume that γ satisfies 1 < γ < p q + 1 p + 1 (2.14)and set γ = 1 + p + 1 q + 1 ( γ −
1) (2.15) which, in turn, guarantees 1 < γ < q p + 1 q + 1 . (2.16)Let the constants C , C > (cid:16) C C − p +1 q +1 (cid:17) γ − = C γ − C ¯ γ − ≤ u γ − v γ − ≤ C γ − C ¯ γ − = (cid:16) C C − p +1 q +1 (cid:17) γ − on [ T ∗ / , T ∗ ) × ( ρ , ρ ) . (2.17)Next, by (2.1)-(2.3) and assumption (1.13) (with C ′ , C ′ in place of C , C ), there exist κ , κ , µ > κ κ < T ∈ ( T ∗ / , T ∗ ), such that uF u + κ vF v ≥ (1 + 2 µ ) F on [ T , T ∗ ) × ( ρ , ρ ) (2.18)and vG v + κ uG u ≥ (1 + 2 µ ) G on [ T , T ∗ ) × ( ρ , ρ ) . (2.19)Choose κ in (2.5) such that κ < κ < /κ . Then taking γ > κ v γ − u γ − ≥ κ and 1 κ u γ − v γ − ≥ κ on [ T , T ∗ ) × ( ρ , ρ ) , (2.20)and we may also assume that γ ≤ µ, ¯ γ ≤ µ (2.21)and that (2.14), (2.16) are satisfied. On the other hand, since F ≥ c v p and G ≥ c u q , itfollows from (2.3), (2.14) and (2.16) that there exists T ∈ ( T , T ∗ ) such that C δγu γ + C δu ≤ Cv p +1 q +1 γ ≤ µF on [ T , T ∗ ) × ( ρ , ρ ) (2.22)and C ¯ γu ¯ γ + C v ≤ Cu q +1 p +1 ¯ γ ≤ µG on [ T , T ∗ ) × ( ρ , ρ ) . (2.23)Combining (2.12), (2.13) with (2.15)-(2.23) and using F v , G u ≥
0, we deduce that e H ≤ − uF u − κ vF v + (1 + 2 µ ) F ≤ T , T ∗ ) × ( ρ , ρ )and e H ≤ − vG v − κ uG u + (1 + 2 µ ) G ≤ T , T ∗ ) × ( ρ , ρ )and the Lemma follows from (2.7)–(2.10). (cid:3) INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 13
With Proposition 2.1 and Lemma 2.1 at hand, we can now conclude the proof of The-orem 1.2.
Proof of Theorem 1.2.
Let ( u, v ) be a solution of system (1.2) satisfying the hypothesesof Theorem 1.2 and assume for contradiction that there exists ρ ∈ (0 , R ) such thatlim sup t → T ∗ ( u ( t, ρ ) + v ( t, ρ )) = ∞ . (2.24)Also, since ( u, v ) (0 , u, v > , T ∗ ) × [0 , R ), hence u ρ ( t, · ) v ρ ( t, · ) t ∈ (0 , T ∗ ). Next, we have u t − δu ρρ − δ n − ρ u ρ = f ( t, ρ )on (0 , T ∗ ) × (0 , R ) , with f ( t, ρ ) = F ( u, v ). Since, u ρ , v ρ ≤ F v ≥
0, a strongmaximum principle (which can be seen from straightforward modifications of the proof of[16, Lemma 52.18, p. 519]) then guarantees u ρ < , T ∗ ) × (0 , R ), (2.25)and similarly v ρ < , T ∗ ) × (0 , R ). (2.26)Set ρ = ρ / , ρ = ρ / J, J , T be given by Lemma 2.1. Since c ( ρ ) = c ( ρ ) =0, we have J, J ≤ (cid:0) ( T , T ∗ ) × { ρ } (cid:1) ∪ (cid:0) ( T , T ∗ ) × { ρ } (cid:1) . Taking ε > J, J ≤ { T } × [ ρ , ρ ]. Then, owing toassumption (1.5), we may use the maximum principle (as in, e.g., [17]), to obtain J, J ≤ T , T ∗ ) × [ ρ , ρ ] . Consequently, − u ρ ≥ εc ( ρ ) u γ on ( T , T ∗ ) × [ ρ , ρ ] . By integration, we obtain u − γ ( t, ρ ) ≥ ( γ − ε Z ρ ρ sin (cid:18) π ( ρ − ρ ) ρ − ρ (cid:19) dρ > T ≤ t < T ∗ . It follows that u ( t, ρ ) is bounded for T ≤ t < T ∗ , and similarly v ( t, ρ ) is bounded for T ≤ t < T ∗ . Since u ρ , v ρ ≤
0, this leads to a contradiction with (2.24) and proves thetheorem. (cid:3) Upper type I estimates away from the origin
Proposition 3.1.
Let
Ω = B (0 , R ) , p, q > , δ > . Assume that (1.3)–(1.5) are satisfiedand that, for some c > , F ( u, v ) ≥ c v p , G ( u, v ) ≥ c u q , for all u, v ≥ . (3.1) Let the solution ( u, v ) of (1.2) satisfy T ∗ < ∞ . Then, there exists a constant M > (depending only on n, p, q, δ, c , R, T ∗ ) such that u ( t, ρ ) ≤ M ρ − n ( T ∗ − t ) − α and v ( t, ρ ) ≤ M ρ − n ( T ∗ − t ) − β , (3.2) for all t ∈ [0 , T ∗ ) and < ρ ≤ R. The argument, which is based on Kaplan’s eigenfunction method, is well known forscalar equations (see e.g. [12] and [14, Propositions 4.4, 4.6 and Corollary 4.5, pp. 895-896]) and can be easily adapted to systems.
Proof.
We denote by λ the first eigenvalue of − ∆ in H ( B (0 , R )) and ϕ the correspond-ing eigenfunction such that ϕ > R B (0 , R ) ϕ ( x ) dx = 1. Multiplying (1.2) by ϕ ,using (3.1) and integrating by parts, we obtain, on (0 , T ∗ ), ddt Z B (0 , R ) u ( t, x ) ϕ ( x ) dx ≥ c Z B (0 , R ) v p ( t, x ) ϕ ( x ) dx − δλ Z B (0 , R ) u ( t, x ) ϕ ( x ) dx,ddt Z B (0 , R ) v ( t, x ) ϕ ( x ) dx ≥ c Z B (0 , R ) u q ( t, x ) ϕ ( x ) dx − λ Z B (0 , R ) v ( t, x ) ϕ ( x ) dx. Let y ( t ) = R B (0 , R ) u ( t, x ) ϕ ( x ) dx and z ( t ) = R B (0 , R ) v ( t, x ) ϕ ( x ) dx . By Jensen’s inequal-ity, we deduce that y ′ ( t ) ≥ c z p ( t ) − δλ y ( t ) , z ′ ( t ) ≥ c y q ( t ) − λ z ( t ) . We put Y ( t ) = e δλ t y ( t ) and Z ( t ) = e λ t z ( t ) . Then, there exists
C > Y ′ ( t ) ≥ CZ p ( t ) , Z ′ ( t ) ≥ CY q ( t ) on (0 , T ∗ ).Here and in the rest of the proof, C denotes a positive constant depending only on T ∗ , δ, p, q, n, R and which may vary from line to line. By [17, Lemma 32.10, p. 284],there exists C such that Y ( t ) ≤ C ( T ∗ − t ) − α , Z ( t ) ≤ C ( T ∗ − t ) − β on [0 , T ∗ ),where α, β are given by (1.8). Therefore, y ( t ) ≤ C ( T ∗ − t ) − α , z ( t ) ≤ C ( T ∗ − t ) − β on [0 , T ∗ ).For 0 < ρ ≤ R/
2, since u, v are radially symmetric and radially nonincreasing, we deducethat ρ n u ( t, ρ ) ≤ C Z B (0 , R/ u ( t, | x | ) dx ≤ C Z B (0 , R/ u ( t, | x | ) ϕ ( x ) dx ≤ C ( T ∗ − t ) − α ,ρ n v ( t, ρ ) ≤ C Z B (0 , R/ v ( t, | x | ) dx ≤ C Z B (0 , R/ v ( t, | x | ) ϕ ( x ) dx ≤ C ( T ∗ − t ) − β . INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 15
The case when R/ < ρ < R then follows from the radial nonincreasing property. Thiscompletes the proof. (cid:3) A non-degeneracy criterion for blow-up points
The main objective of this subsection is the following result, which gives a sufficient,local smallness condition, at any given time sufficiently close to T ∗ , for excluding blow-upat a given point different from the origin. Proposition 4.1.
Let
Ω = B (0 , R ) , p, q > , δ > . Assume (1.3)–(1.5), (1.10)–(1.12)and let the solution ( u, v ) of (1.2) satisfy T ∗ < ∞ . Let d , d satisfy < d < d < R .There exist η, τ > such that if, for some t ∈ [ T ∗ − τ , T ∗ ) , we have ( T ∗ − t ) α u ( t , d ) ≤ η and ( T ∗ − t ) β v ( t , d ) ≤ η, (4.1) then d is not a blow-up point of ( u, v ) , i.e. ( u, v ) is uniformly bounded in the neighborhoodof ( T ∗ , d ) . Here, the numbers η, τ depend only on p, q, r, s, δ, c , c , d , d , n, R, T ∗ . As in [17], the proof uses similarity variables and delayed smoothing effects. However,as explained in Section 1.2, a new difficulty arises, caused by the absence of global type Iinformation on the blow-up rate. For this reason, we consider only radial and radiallydecreasing solutions (whereas the analogous criterion in [17] was established for any so-lution). In this more delicate situation, the current formulation, slightly different fromthat in [17], turns out to be more convenient. Namely, instead of expressing the localnon-blow-up criterion itself with the weighted L norm of rescaled solution, it is expressedin terms of pointwise smallness on (( T ∗ − t ) α u, ( T ∗ − t ) β v ) at a point d < d and at sometime close to T ∗ .4.1. Similarity variables and delayed smoothing effects.
In view of the proof ofProposition 4.1 we introduce the well-known similarity variables (cf. [8]). More precisely,for any given d ∈ R , we define the (one-dimensional) similarity variables around ( T ∗ , d ),associated with ( t, ρ ) ∈ (0 , T ∗ ) × R , by: σ = − log( T ∗ − t ) ∈ [ˆ σ, ∞ ) , θ = ρ − d √ T ∗ − t = e σ/ ( ρ − d ) ∈ R , (4.2)where ˆ σ = − log T ∗ . For given δ >
0, let U be a (classical) solution of U t − δU ρρ = H ( t, ρ ) , < t < T ∗ , ρ ∈ R (where the smooth functions H will be specified later). Then V = V d ( σ, θ ) = ( T ∗ − t ) α U ( t, y ) = e − ασ U (cid:0) T ∗ − e − σ , d + θe − σ/ (cid:1) is a solution of V σ − L δ V + αV = e − ( α +1) σ H (cid:0) T ∗ − e − σ , d + θe − σ/ (cid:1) , σ > ˆ σ, θ ∈ R , (4.3)where L δ = δ∂ θ − θ ∂ θ = δK − δ ∂ θ ( K δ ∂ θ ) , K δ ( θ ) = (4 πδ ) − / e − θ δ . We denote by ( T δ ( σ )) σ ≥ the semigroup associated with L δ . More precisely, for each φ ∈ L ∞ ( R ) , we set T δ ( σ ) φ := w ( σ, . ), where w is the unique solution of ( w σ = L δ w, θ ∈ R , σ > ,w (0 , θ ) = φ ( θ ) , θ ∈ R . (4.4)For any φ ∈ L ∞ ( R ) , we put k φ k L mKδ = (cid:18)Z R | φ ( θ ) | m K δ ( θ ) dθ (cid:19) /m , ≤ m < ∞ . Let 1 ≤ k < m < ∞ and δ >
0, then, by Jensen’s inequality, k φ k L kKδ ≤ k φ k L mKδ , ≤ k < m < ∞ . (4.5)The semigroups ( T δ ( σ )) σ ≥ have the following properties, which will be useful when dealingwith system (1.2) with unequal diffusivities: Lemma 4.1. (1) (Contraction) For any ≤ m < ∞ , we have k T δ ( σ ) φ k L mKδ ≤ k φ k L mKδ , for all δ > , σ ≥ , φ ∈ L ∞ ( R ) . (4.6) Moreover, for all < δ ≤ λ < ∞ , we have k T δ ( σ ) φ k L mKλ ≤ (cid:16) λδ (cid:17) / k φ k L mKλ , for all σ ≥ , φ ∈ L ∞ ( R ) . (4.7)(2) (Delayed regularizing effect) For any ≤ k < m < ∞ , there exist ˆ C, σ ∗ > suchthat k T δ ( σ ) φ k L mKδ ≤ ˆ C k φ k L kKδ , for all δ > , σ ≥ σ ∗ , φ ∈ L ∞ ( R ) . (4.8) Moreover, for all < δ ≤ λ < ∞ , we have k T δ ( σ ) φ k L mKλ ≤ ˆ C (cid:16) λδ (cid:17) / k φ k L kKλ , for all σ ≥ σ ∗ , φ ∈ L ∞ ( R ) . (4.9) Proof.
We put w ( σ, θ ) = ( T δ ( σ ) φ )( √ δ θ ) . Then, by (4.4), it follows that w is the solutionof ( w σ = L w, θ ∈ R , σ > ,w (0 , θ ) = φ ( √ δ θ ) , θ ∈ R . INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 17
Then w ( σ, θ ) = (cid:2) T ( σ ) φ ( √ δ . ) (cid:3) ( θ ) . (4.10)By (4.10) and [17, Lemma 3.1(i), p.176], we obtain k T δ ( σ ) φ k L mKδ = k ( T δ ( σ ) φ )( √ δ . ) k L mK = k T ( σ ) φ ( √ δ . ) k L mK ≤ k φ ( √ δ . ) k L mK = k φ k L mKδ , for all σ ≥ . Let next 0 < δ ≤ λ < ∞ . Denote by ( S δ ( t )) t ≥ the semigroup associated with δ∂ y in R and let the functions u ( t, y ) and w ( σ, θ ) be related by the following backward self-similartransformation (with T ∗ = 1, d = 0): σ = − log(1 − t ) ∈ [0 , ∞ ) , θ = y √ − t = e σ/ y ∈ R , w ( σ, θ ) = u ( t, y ) . We have, for all σ ≥ (cid:12)(cid:12)(cid:2) T δ ( σ ) φ (cid:3) ( θ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:2) S δ ( t ) u (cid:3) ( y ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (4 πδt ) − / Z R e −| y − z | δt u ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) λδ (cid:17) / (4 πλt ) − / Z R e −| y − z | λt | u ( z ) | dz = (cid:16) λδ (cid:17) / (cid:2) S λ ( t ) (cid:12)(cid:12) u (cid:12)(cid:12)(cid:3) ( y ) = (cid:16) λδ (cid:17) / (cid:2) T λ ( σ ) (cid:12)(cid:12) φ (cid:12)(cid:12)(cid:3) ( θ ) . Inequality (4.7) then follows from (4.6).To prove assertion (2), we recall that, by e.g. [17, Lemma 3.1(ii), p.176], for any 1 ≤ k < m < ∞ , there exist ˆ C, σ ∗ > k T ( σ ) φ k L mK ≤ ˆ C k φ k L kK , for all σ ≥ σ ∗ , φ ∈ L ∞ ( R ) . We may then argue similarly as for assertion (1). (cid:3)
Proof of Proposition 4.1.
The proof is long and technical. We split it in severalsteps. Assume p ≥ q without loss of generality, hence α ≥ β . Step 1.
Definition of suitably modifed solutions.
As mentioned before we lack a globaltype I blow-up estimate. However, we have a local type I blow-up estimate, away fromthe origin. Indeed, by (3.2) in Proposition 3.1, we know that( T ∗ − t ) α u ( t, y ) ≤ N , ( T ∗ − t ) β v ( t, y ) ≤ N , ≤ t < T ∗ , d ≤ y < R, (4.11)with N = M d − n . We shall thus truncate the radial domain and consider suitablycontrolled extensions of the solution to the real line. We first define the following extensions e u, e v ≥ u, v by setting: e u ( t, y ) := u ( t, y ) , y ∈ [ d , R ] , , y ∈ R \ [ d , R ] , for any t ∈ [0 , T ∗ ), (4.12)and e v ( t, y ) similarly.Next, let M ≥ N to be chosen below. For given t ∈ [0 , T ∗ ), let ( u, v ) = ( u ( t ; · , · ) ,v ( t ; · , · )) be the solution of the following auxiliary problem: u t − δu yy = F ( e u, e v ) , t < t < T ∗ , y ≥ d ,v t − v yy = G ( e u, e v ) , t < t < T ∗ , y ≥ d ,u ( t, d ) = M ( T ∗ − t ) − α , t < t < T ∗ ,v ( t, d ) = M ( T ∗ − t ) − β , t < t < T ∗ ,u ( t , y ) = e u ( t , y ) , y ≥ d ,v ( t , y ) = e v ( t , y ) , y ≥ d . (4.13)It is clear that u, v ≥ t , T ∗ ) × [ d , ∞ ). Also, using (4.11) and M ≥ N , wededuce from the maximum principle that e u ≤ u, e v ≤ v on [ t , T ∗ ) × [ d , ∞ ). (4.14)Now choosing M = max (cid:16) N , c α − ( N p + N r + T ∗ α +1 ) , c β − ( N q + N s + T ∗ β +1 ) (cid:17) , (4.15)where c is from (1.10)–(1.11), and using (1.10)–(1.12), (4.11), (4.12), (4.15), we have F ( e u, e v ) ≤ c ( e v p + e u r + 1) ≤ c (cid:0) ( N p + N r )( T ∗ − t ) − α − + 1 (cid:1) ≤ αM ( T ∗ − t ) − α − and G ( e u, e v ) ≤ c ( e u q + e v s + 1) ≤ c (cid:0) ( N q + N s )( T ∗ − t ) − β − + 1 (cid:1) ≤ βM ( T ∗ − t ) − β − . We may thus use M ( T ∗ − t ) − α (resp., M ( T ∗ − t ) − β ) as a supersolution of the inhomo-geneous, linear heat equation in (4.13), verified by u (resp. v ) on [ t , T ∗ ) × [ d , ∞ ), andinfer from the maximum principle that0 ≤ u ≤ M ( T ∗ − t ) − α , ≤ v ≤ M ( T ∗ − t ) − β on [ t , T ∗ ) × [ d , ∞ ) . (4.16)We next extend ( u, v ) by odd reflection for y < d , i.e., we set: u ( t, d − y ) = 2 M ( T ∗ − t ) − α − u ( t, d + y ) , t ≤ t < T ∗ , y > ,v ( t, d − y ) = 2 M ( T ∗ − t ) − β − v ( t, d + y ) , t ≤ t < T ∗ , y > . INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 19
From (4.16), along with (4.14) and (4.12), we have0 ≤ u ≤ M ( T ∗ − t ) − α , ≤ v ≤ M ( T ∗ − t ) − β on [ t , T ∗ ) × R (4.17)and e u ≤ u, e v ≤ v on [ t , T ∗ ) × R . (4.18)It is easy to see that the functions u, v ∈ C , (( t , T ∗ ) × R ) and that we have ( u t − δu yy = F ( t, y ) , t < t < T ∗ , y ∈ R ,v t − v yy = G ( t, y ) , t < t < T ∗ , y ∈ R , where F ( t, y ) := αM ( T ∗ − t ) − α − − F ( e u, e v )( t, d − y ) , y < d ,F ( e u, e v )( t, y ) , y ≥ d , (4.19) G ( t, y ) := βM ( T ∗ − t ) − β − − G ( e u, e v )( t, d − y ) , y < d ,G ( e u, e v )( t, y ) , y ≥ d . (4.20) Step 2.
Self-similar rescaling of modifed solutions.
We now fix d ∈ ( d , d ) (say, d = ( d + d ) /
2) and pass to self-similar variables ( σ, θ ) around ( T ∗ , d ), cf. (4.2). Inthese variables, we first define the rescaled solution ( e w, e z ) = ( e w d , e z d ), associated with theextended solution ( e u, e v ), namely, ( e w ( σ, θ ) = ( T ∗ − t ) α e u ( t, y ) , ˆ σ ≤ σ < ∞ , θ ∈ R , e z ( σ, θ ) = ( T ∗ − t ) β e v ( t, y ) , ˆ σ ≤ σ < ∞ , θ ∈ R , (4.21)where ˆ σ = − log T ∗ . For given t ∈ [0 , T ∗ ) (cf. Step 1), we also define ( w, z ) =( w d ( t ; · , · ) , z d ( t ; · , · )), associated with the modifed solution ( u ( t ; · , · ) , v ( t ; · , · )), givenby ( w ( σ, θ ) = ( T ∗ − t ) α u ( t, y ) , σ ≤ σ < ∞ , θ ∈ R ,z ( σ, θ ) = ( T ∗ − t ) β v ( t, y ) , σ ≤ σ < ∞ , θ ∈ R , (4.22)where σ = − log( T ∗ − t ) ≥ ˆ σ . At this point, we stress that ( w, z ) depends on the choiceof σ (or t ), whereas ( e w, e z ) does not. Actually, in Step 3, the ( w, z ) will be used asauxiliary functions in order to establish suitable estimates on ( e w, e z ) itself.Set ℓ = d − d >
0. Owing to (4.17), (4.18), we have e w ≤ w ≤ M, e z ≤ z ≤ M on [ σ , ∞ ) × R (4.23)and, for all σ ≥ ˆ σ , θ e w ( σ, θ ) and θ e z ( σ, θ ) are nonincreasing for θ ∈ [ − ℓe σ/ , ∞ ), (4.24) due to (1.6). Then, using (4.3), (4.19), (4.20), α + 1 = pβ, β + 1 = qα , α ( r − − β ( s − − w, z ) is a solution of ( w σ − L δ w + αw = F ( σ, θ ) , σ < σ < ∞ , θ ∈ R ,z σ − L z + βz = G ( σ, θ ) , σ < σ < ∞ , θ ∈ R , (4.25)where F ( σ, θ ) = e − ( α +1) σ F (cid:0) T ∗ − e − σ , d + θe − σ/ (cid:1) ≤ c (cid:16)e z p ( σ ) + e w r ( σ ) + e − ( α +1) σ (cid:17) + 2 αM χ { θ< − ℓe σ/ } (4.26)and G ( σ, θ ) = e − ( β +1) σ G (cid:0) T ∗ − e − σ , d + θe − σ/ (cid:1) ≤ c (cid:16) e w q ( σ ) + e z s ( σ ) + e − ( β +1) σ (cid:17) + 2 βM χ { θ< − ℓe σ/ } (4.27)Also, using the last two conditions in (4.13), along with (4.21), (4.22) and (4.23), we seethat w ( σ ) ≤ e w ( σ ) + 2 M χ { θ< − ℓe σ / } and z ( σ ) ≤ e z ( σ ) + 2 M χ { θ< − ℓe σ / } . (4.28)In the next steps, we shall estimate ( e w, e z ) by using semigroup and delayed smoothingarguments. As compared with the situation in [17], we have here additional terms whichcome from the reflection procedure. However, thanks to the self-similar change of variables,whose center d is shifted to the right of the reflection point d , the contribution of theseterms, as σ → ∞ , will be localized exponentially far away at −∞ in space and thuscan be made arbitrarily small for τ small. Also, the need to handle two semigroups,due to the different diffusivities, as well as added nonlinear terms, cause some technicalcomplications, which require for instance an additional interpolation argument. Step 3.
First semigroup estimates for ( e w, e z ). We claim that, for all σ ≥ ˆ σ and σ > e w ( σ + σ ) ≤ e − ασ T δ ( σ ) h e w ( σ ) + 2 M χ { θ< − ℓe σ / } i + c Z σ e − α ( σ − τ ) T δ ( σ − τ ) (cid:16)e z p ( σ + τ ) + e w r ( σ + τ ) + e − ( α +1)( σ + τ ) (cid:17) dτ + 2 αM Z σ e − α ( σ − τ ) T δ ( σ − τ ) χ { θ< − ℓe ( σ τ ) / } dτ (4.29) INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 21 and e z ( σ + σ ) ≤ e − βσ T ( σ ) he z ( σ ) + 2 M χ { θ< − ℓe σ / } i + c Z σ e − β ( σ − τ ) T ( σ − τ ) (cid:16) e w q ( σ + τ ) + e z s ( σ + τ ) + e − ( β +1)( σ + τ ) (cid:17) dτ + 2 βM Z σ e − β ( σ − τ ) T ( σ − τ ) χ { θ< − ℓe ( σ τ ) / } dτ, (4.30)and that, moreover, e w ( σ + σ ) + e z ( σ + σ ) ≤ e M σ S ( σ ) h e w ( σ ) + e z ( σ ) + 4 M χ { θ< − ℓe σ / } i + c Z σ e M ( σ − τ ) S ( σ − τ ) h e − ( α +1)( σ + τ ) + e − ( β +1)( σ + τ ) i dτ + 2 αM Z σ e M ( σ − τ ) S ( σ − τ ) χ { θ< − ℓe ( σ τ ) / } dτ, (4.31)where ( S ( σ )) σ ≥ = ( T δ ( σ ) + T ( σ )) σ ≥ and M = c max (cid:0) (2 M ) p − , (2 M ) q − , (2 M ) r − , (2 M ) s − (cid:1) . (Note that, as announced, estimates (4.29)-(4.31) do not involve ( w ( t ; · , · ) , z ( t ; · , · )) any-more.)Let us first verify (4.29)-(4.30). We fix σ ≥ ˆ σ and consider ( w, z ) = (cid:0) w d ( t ; · , · ) ,z d ( t ; · , · ) (cid:1) , defined in (4.22) with σ = − log( T ∗ − t ). we use (4.25) and the variation ofconstants formula to write w ( σ + σ ) = e − ασ T δ ( σ ) w ( σ ) + Z σ e − α ( σ − τ ) T δ ( σ − τ ) F ( σ + τ, · ) dτ for all σ >
0, hence, by (4.26), w ( σ + σ ) ≤ e − ασ T δ ( σ ) w ( σ ) + 2 αM Z σ e − α ( σ − τ ) T δ ( σ − τ ) χ { θ< − ℓe ( σ τ ) / } dτ + c Z σ e − α ( σ − τ ) T δ ( σ − τ ) (cid:16)e z p ( σ + τ ) + e w r ( σ + τ ) + e − ( α +1)( σ + τ ) (cid:17) dτ. (4.32)Similarly, by exchanging the roles of w , e w , p, r, α, and z, e z, q , s, β , we obtain z ( σ + σ ) ≤ e − βσ T ( σ ) z ( σ ) + 2 βM Z σ e − β ( σ − τ ) T ( σ − τ ) χ { θ< − ℓe ( σ τ ) / } dτ + c Z σ e − β ( σ − τ ) T ( σ − τ ) (cid:16) e w q ( σ + τ ) + e z s ( σ + τ ) + e − ( β +1)( σ + τ ) (cid:17) dτ. (4.33)Inequalities (4.29)-(4.30) then follow from (4.32), (4.33), (4.23) and (4.28). To verify (4.31), we set H := w + z . Adding up (4.32) and (4.33), and recalling α ≥ β ,we easily get H ( σ + σ ) ≤ S ( σ ) H ( σ ) + Z σ S ( σ − τ ) (cid:2) M H ( σ + τ ) + D ( τ ) (cid:3) dτ, σ ≥ , (4.34)where D ( τ, · ) = c (cid:2) e − ( α +1)( σ + τ ) + e − ( β +1)( σ + τ ) (cid:3) + 2 αM χ { θ< − ℓe ( σ τ ) / } , τ ≥ . Set b H ( σ + σ ) := e M σ S ( σ ) H ( σ ) + Z σ e M ( σ − τ ) S ( σ − τ ) D ( τ ) dτ, σ ≥ . (4.35)By direct computation, using the semigroup properties of ( S ( σ )) σ ≥ and Fubini’s theorem,we see that b H ( σ + σ ) = S ( σ ) H ( σ ) + Z σ S ( σ − τ ) (cid:2) M b H ( σ + τ ) + D ( τ ) (cid:3) dτ, σ > . (4.36)Combining (4.34), (4.36) and using the positivity-preserving property of ( S ( σ )) σ ≥ , weobtain [ H − b H ] + ( σ + σ ) ≤ M Z σ S ( σ − τ )[ H − b H ] + ( σ + τ ) dτ, σ > . (4.37)Letting now ¯ δ = max( δ,
1) and K = K ¯ δ , we deduce from (4.7) in Lemma 4.1 that k S ( σ ) φ k L kK ≤ e C k φ k L kK , σ ≥ , φ ∈ L ∞ ( R ) , ≤ k < ∞ , (4.38)with e C = e C ( δ ) ≥
1. Therefore, it follows from (4.37) that (cid:13)(cid:13) [ H − b H ] + ( σ + σ ) (cid:13)(cid:13) L K ≤ e CM Z σ (cid:13)(cid:13) [ H − b H ] + ( σ + τ ) (cid:13)(cid:13) L K dτ, σ > , and we infer from Gronwall’s Lemma that H ( σ + σ ) ≤ b H ( σ + σ ) for all σ ≥
0. Inequality(4.31) then follows from (4.23) and (4.28).
Step 4.
Small time estimate of rescaled solutions.
At this point, we set, as before,¯ δ = max( δ,
1) and K = K ¯ δ , and we fix m > max h p, q, s, r, r ( r − α − β ) i (4.39)and let σ ∗ be given by Lemma 4.1(2), with k = 1. We note that, by Lemma 4.1, we have k S ( σ ) φ k L mK ≤ e C k φ k L K , σ ≥ σ ∗ , φ ∈ L ∞ ( R ) , (4.40) INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 23 with e C = e C ( p, q, s, r, δ ) ≥
1. Also, by (4.38), we have k S ( σ ) χ { θ< − A } k L kK ≤ e C k χ { θ< − A } k L kK = e C (cid:18) (4 πδ ) − / Z − A −∞ exp (cid:16) − θ δ (cid:17) dθ (cid:19) /k ≤ C exp( − (8 kδ ) − A ) , for all A > ≤ k ≤ m , (4.41)with C = C ( p, q, s, r, δ ) ≥ η >
0. We claim that there exists τ ∈ (0 , T ∗ ), depending only on η and and onthe parameters p, q, r, s, δ, c , c , d , d , n, R, T ∗ , (4.42)such that:For any t ∈ [ T ∗ − τ , T ∗ ) satisfying (4.1) and σ = − log( T ∗ − t ), we have k e w ( σ + σ ) k L K + k e z ( σ + σ ) k L K ≤ e C η, < σ ≤ σ ∗ , (4.43)with e C = 3 e Ce M σ ∗ > . To prove the claim, we choose σ = σ in (4.31). Observe that, by assumption (4.1)and owing to (1.6), we have e w ( σ , · ) , e z ( σ , · ) ≤ η on R , hence k e w ( σ ) k L K + k e z ( σ ) k L K ≤ η. (4.44)Using (4.31), (4.38), (4.41), (4.44), e σ = ( T ∗ − t ) − ≥ τ − and assuming τ <
1, wededuce that, for 0 ≤ σ ≤ σ ∗ , k e w ( σ + σ ) k L K + k e z ( σ + σ ) k L K ≤ e Ce M σ ∗ η + 4 e CC M e M σ ∗ exp( − (8¯ δτ ) − ℓ )+2 C e Cσ ∗ e M σ ∗ τ β +11 + 2 αC e Cσ ∗ M e M σ ∗ exp (cid:0) − (8¯ δτ ) − ℓ (cid:1) ≤ e Ce M σ ∗ (cid:2) η + c σ ∗ τ β +11 + C M ( ασ ∗ + 2) exp( − (8¯ δτ ) − ℓ ) (cid:3) . For τ ∈ (0 , T ∗ ) sufficiently small, depending only on η and on the parameters in (4.42),we finally get (4.43) with e C = 3 e Ce M σ ∗ . Step 5.
Large time estimate of rescaled solutions.
We claim that there exist η > τ ∈ (0 , τ ( η )], depending only on the parameters in (4.42), such that:for any t ∈ [ T ∗ − τ , T ∗ ) satisfying (4.1), we have A η, t = (0 , ∞ ), (4.45)where σ = − log( T ∗ − t ) and A η,t = n σ > e ατ k e w ( σ + σ ∗ + τ ) k L K + e βτ k e z ( σ + σ ∗ + τ ) k L K ≤ e C e C η, τ ∈ [0 , σ ] o . First observe that A η, t = ∅ , due to (4.43) and the continuity of the function σ e ασ k e w ( σ + σ ∗ + σ ) k L K + e βσ k e z ( σ + σ ∗ + σ ) k L K . We denote T = sup A η, t ∈ (0 , ∞ ] . Assume for contradiction that
T < ∞ . Then by (4.43), recalling that α ≥ β , we have k e w ( σ + σ ∗ + σ ) k L K + k e z ( σ + σ ∗ + σ ) k L K ≤ e C e C ηe − βσ , − σ ∗ ≤ σ ≤ T . (4.46)For 0 ≤ τ ≤ T , we apply (4.31) with σ = σ + τ and σ = σ ∗ . Using (4.38), (4.40), (4.41),(4.44), (4.46), e σ = ( T ∗ − t ) − ≥ τ − and assuming τ <
1, we get k e w ( σ + σ ∗ + τ ) k L mK + k e z ( σ + σ ∗ + τ ) k L mK ≤ e C e M σ ∗ (cid:16) k e w ( σ + τ ) k L K + k e z ( σ + τ ) k L K (cid:17) + 8 C M e M σ ∗ exp (cid:0) − (8¯ δτ m ) − ℓ e τ (cid:1) +4 c e Cσ ∗ e M σ ∗ τ β +10 e − ( β +1) τ + 4 αM e CC σ ∗ e M σ ∗ exp (cid:0) − (8¯ δτ m ) − ℓ e τ (cid:1) ≤ e C e C e Ce M σ ∗ ηe − β ( τ − σ ∗ ) + 4 c e Cσ ∗ e M σ ∗ τ β +10 e − ( β +1) τ +4 C (2 + α e Cσ ∗ ) M e M σ ∗ exp (cid:0) − (8¯ δτ m ) − ℓ e τ (cid:1) . Put e C = 5 e C e C e Ce ( M + β ) σ ∗ . For τ ∈ (0 , τ ( η )] sufficiently small, depending only on η and on the parameters in (4.42), it follows that k e w ( σ + σ ∗ + τ ) k L mK + k e z ( σ + σ ∗ + τ ) k L mK ≤ e C ηe − βτ , ≤ τ ≤ T . (4.47)Next let 0 < σ ≤ T . Now using (4.29) with σ = σ + σ ∗ , (4.38), (4.41), T δ ( σ ) ≤ S ( σ )and e σ ≥ τ − , we obtain e ασ k e w ( σ + σ ∗ + σ ) k L K ≤ k T δ ( σ ) e w ( σ + σ ∗ ) k L K + 2 M k T δ ( σ ) χ { θ< − ℓe ( σ σ ∗ ) / } k L K + c Z σ e ατ k T δ ( σ − τ ) e z p ( σ + σ ∗ + τ ) k L K dτ + c Z σ e ατ k T δ ( σ − τ ) e w r ( σ + σ ∗ + τ ) k L K dτ + c Z σ e ατ k T δ ( σ − τ ) e − ( α +1)( σ + σ ∗ + τ ) k L K dτ +2 αM Z σ e ατ k T δ ( σ − τ ) χ { θ< − ℓe ( σ σ ∗ + τ ) / } k L K dτ INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 25 hence, e ασ k e w ( σ + σ ∗ + σ ) k L K ≤ e C k e w ( σ + σ ∗ ) k L K + 2 C M exp( − (8 δτ ) − ℓ )+ c e C Z σ e ατ k e z p ( σ + σ ∗ + τ ) k L K dτ + c e C Z σ e ατ k e w r ( σ + σ ∗ + τ ) k L K dτ + c e Cτ α +10 + 2 αC M Z σ e ατ exp (cid:0) − (8 δτ ) − ℓ e τ (cid:1) dτ. By taking τ possibly smaller (dependence as above), we may ensure that c e Cτ α +10 + 2 C M exp( − (8 δτ ) − ℓ ) + 2 αC M Z ∞ e ατ exp (cid:0) − (8 δτ ) − ℓ e τ (cid:1) dτ ≤ η , hence, e ασ k e w ( σ + σ ∗ + σ ) k L K ≤ e C k e w ( σ + σ ∗ ) k L K + η + c e C Z σ e ατ k e z ( σ + σ ∗ + τ ) k pL pK dτ + c e C Z σ e ατ k e w ( σ + σ ∗ + τ ) k rL rK dτ. (4.48)To estimate the last integral, setting ν = ( m − r ) / ( m − ∈ (0 ,
1) and interpolatingbetween (4.47) and the fact that τ ∈ A η, t , we write k e w ( σ + σ ∗ + τ ) k L rK ≤ k e w ( σ + σ ∗ + τ ) k νL K k e w ( σ + σ ∗ + τ ) k − νL mK ≤ (2 e C e C ηe − ατ ) ν ( e C ηe − βτ ) − ν = e C ηe − ( αν + β (1 − ν )) τ , with e C = (2 e C e C ) ν e C − ν . Using this, along with (4.5) and (4.47), we obtain e ασ k e w ( σ + σ ∗ + σ ) k L K ≤ e C k e w ( σ + σ ∗ ) k L K + η + c e C ( e C η ) p Z σ e ατ e − βpτ dτ + c e C ( e C η ) r Z σ e ατ e − ( αν + β (1 − ν )) rτ dτ. Since α − βp = − α = αr − ν := 1 − ( α − β )(1 − ν ) r >
0, owing to (4.39), wededuce that e ασ k e w ( σ + σ ∗ + σ ) k L K ≤ e C k e w ( σ + σ ∗ ) k L K + η + c e C e C p η p + c e C e C r η r ν . (4.49)Similarly as (4.48), by using (4.30) instead of (4.29), we get e βσ k e z ( σ + σ ∗ + σ ) k L K ≤ e C k e z ( σ + σ ∗ ) k L K + η + c e C Z σ e βτ k e w ( σ + σ ∗ + τ ) k qL qK dτ + c e C Z σ e βτ k e z ( σ + σ ∗ + τ ) k sL sK dτ. Therefore, by (4.47), e βσ k e z ( σ + σ ∗ + σ ) k L K ≤ e C k e z ( σ + σ ∗ ) k L K + η + c e C ( e C η ) q Z σ e βτ e − βqτ dτ + c e C ( e C η ) s Z σ e βτ e − βsτ dτ. This time, the above interpolation is not necessary. Indeed, using β = βs −
1, we directlyget e βσ k e z ( σ + σ ∗ + σ ) k L K ≤ e C k e z ( σ + σ ∗ ) k L K + η + c e C e C q η q β ( q −
1) + c e C e C s η s . (4.50)Finally, for σ = T in (4.49) and (4.50), by definition of T and by using (4.43) with σ = σ ∗ , we obtain 2 e C e C η = e αT k e w ( σ + σ ∗ + T ) k L K + e βT k e z ( σ + σ ∗ + T ) k L K ≤ e C k e w ( σ + σ ∗ ) k L K + e C k e z ( σ + σ ∗ ) k L K +2 η + c e C e C p η p + c e C e C r η r ν + c e C e C q η q β ( q −
1) + c e C e C s η s ≤ e C e C η + C [ η + η p + η r + η q + η s ] , hence e C e C ≤ C ( η + η p − + η r − + η q − + η s − ), where C > p, q, r, s >
1, choosing η > τ ), we reach a contradiction. Consequently, T = ∞ and the claim is proved. Step 6.
Conclusion.
Let η, τ be as in Step 5 and let t ∈ [ T ∗ − τ , T ∗ ) satisfy (4.1).It follows from the definition of A η, t thatΛ = sup σ ≥ σ + σ ∗ (cid:16) e ασ k e w ( σ ) k L K + e βσ k e z ( σ ) k L K (cid:17) < ∞ . (4.51)Set L := R − K ( θ ) dθ >
0. For all t ∈ [ ˆ T ∗ − ℓ − , T ∗ ), recalling (4.2), we have ℓe σ/ ≥ e w ( σ, ≤ L − Z − e w ( σ, θ ) K ( θ ) dθ, e z ( σ, ≤ L − Z − e z ( σ, θ ) K ( θ ) dθ, (4.52)owing to (4.24). Let then ˆ t = T ∗ − min (cid:0) ℓ − , e − ( σ + σ ∗ ) (cid:1) . It follows from (4.12), (4.21),(4.51), (4.52) that, for all t ∈ [ˆ t , T ∗ ), u ( t, d ) + v ( t, d ) = e ασ e w ( σ,
0) + e βσ e z ( σ, ≤ L − (cid:16) e ασ k e w ( σ ) k L K + e βσ k e z ( σ ) k L K (cid:17) ≤ L − Λ . Using (1.6), we conclude that d > d is not a blow-up point. (cid:3) INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 27 Convergence of rescaled solutions to solutions of a system ofordinary differential inequalities
For given ρ ∈ (0 , R ), we again switch to similarity variables around ( T ∗ , ρ ), alreadyused in the previous section. Namely, we set: σ = − log( T ∗ − t ) , θ = ρ − ρ √ T ∗ − t = e σ/ ( ρ − ρ ) , (5.1)and consider the rescaled solution ( W, Z ) = ( W ρ , Z ρ ) associated with ( u, v ): W ( σ, θ ) = ( T ∗ − t ) α u ( t, ρ ) , Z ( σ, θ ) = ( T ∗ − t ) β v ( t, ρ ) , (5.2)defined for σ ∈ [ˆ σ, ∞ ) with ˆ σ = − log T ∗ and θ ∈ ( − ρ e σ/ , ( R − ρ ) e σ/ ).The goal of this section is to show that any such rescaled solution ( W, Z ) behaves, in asuitable sense as σ → ∞ and θ → ∞ , like a (distribution) solution of the following systemof ordinary differential inequalities: ( φ ′ + αφ ≥ c ψ p ,ψ ′ + βψ ≥ c φ q (5.3)on the whole real line ( −∞ , ∞ ) (however, we shall eventually only use the fact that ( φ, ψ )solves (5.3) on some bounded open interval). Moreover, we single out a simple but crucialproperty of local interpendence of components for solutions of (5.3). Proposition 5.1.
Let
Ω = B (0 , R ) , p, q > , δ > . Assume (1.4)–(1.5), (1.10)–(1.12)and let the solution ( u, v ) of (1.2) satisfy T ∗ < ∞ . Let ρ ∈ (0 , R ) and let ( W, Z ) bedefined by (5.1)-(5.2).(i) Then, for all sequence σ j → ∞ , there exists a subsequence (not relabeled) such that,for each σ ∈ R , φ ( σ ) = lim θ →∞ (cid:16) lim j →∞ W ( σ + σ j , θ ) (cid:17) , ψ ( σ ) = lim θ →∞ (cid:16) lim j →∞ Z ( σ + σ j , θ ) (cid:17) (5.4) exist and are finite, where the limits in j are uniform for ( σ, θ ) in bounded subsets of R × R ,and the limits in θ are monotone nonincreasing.(ii) The functions φ, ψ defined in (5.4) belong to BC ( R ) and ( φ, ψ ) is a nonnegativesolution in D ′ ( R ) of system (5.3).(iii) Let I ⊂ R be an open interval containing . For any nonnegative functions φ, ψ ∈ C ( I ) satisfying (5.3) in D ′ ( I ) , we have φ (0) = 0 if and only if ψ (0) = 0 .Proof. (i) Let A = min( ρ / , R − ρ ) >
0. By (3.2), we have that(
W, Z ) is bounded on the set D = { ( σ, θ ) ∈ R × R , σ > ˆ σ, | θ | ≤ Ae σ/ } (5.5) and ( W, Z ) solves the system W σ − δW θθ + h θ − δ ( n − e − σ/ ρ + θe − σ/ i W θ + αW = e − ( α +1) σ F ( e ασ W, e βσ Z ) Z σ − Z θθ + h θ − ( n − e − σ/ ρ + θe − σ/ i Z θ + βZ = e − ( β +1) σ G ( e ασ W, e βσ Z ) in D . (5.6)Moreover, by (1.8), (1.10)–(1.12), it follows that c Z p ≤ e − ( α +1) σ F (cid:0) e ασ W, e βσ Z (cid:1) ≤ c (cid:0) Z p + W r + e − ( α +1) σ (cid:1) . (5.7) c W q ≤ e − ( β +1) σ G (cid:0) e ασ W, e βσ Z (cid:1) ≤ c (cid:0) W q + Z s + e − ( β +1) σ (cid:1) . (5.8)Denoting the time-translates W j ( σ, θ ) := W ( σ + σ j , θ ) and Z j ( σ, θ ) := Z ( σ + σ j , θ ) andsetting µ j ( σ, θ ) = ( n − e − ( σ + σ j ) / ρ + θe − ( σ + σ j ) / , ε j ( σ ) = e − ( α +1)( σ + σ j ) , e ε j ( σ ) = e − ( β +1)( σ + σ j ) , we have, by (5.6)-(5.8), ( c Z pj ≤ ∂ σ W j − δ∂ θ W j + (cid:2) θ − δµ j (cid:3) ∂ θ W j + αW j ≤ c (cid:0) Z pj + W rj + ε j (cid:1) c W qj ≤ ∂ σ Z j − ∂ θ Z j + (cid:2) θ − µ j (cid:3) ∂ θ Z j + βZ j ≤ c (cid:0) W qj + Z sj + e ε j (cid:1) in D .(5.9)For each compact Q of R × R , the sequences ( Z pj + W rj + ε j ) and ( W qj + Z sj + e ε j ) aredefined on Q for j large enough and, owing to (5.5), they are bounded in L m ( Q ) foreach m ∈ (1 , ∞ ). Therefore, by (5.9) and parabolic estimates (see, e.g. [16, p.438]), thesequences ( W j ) and ( Z j ) are bounded in W , m ( Q ) for each compact Q of R × R andeach m ∈ (1 , ∞ ). Fixing α ∈ (0 ,
1) and using the compact embeddings W , m ( Q ) ⊂⊂ C α, α/ ( Q ) for m large, we deduce that, for some subsequence (not relabeled), ( W j , Z j )converges, in C α, α/ for each compact Q of R × R , to some pair of nonnegative, boundedfunctions ( w, z ), with w, z ∈ W , mloc ( R × R ) for each m ∈ (1 , ∞ ).Moreover, since u ρ , v ρ ≤ ∂ θ W j , ∂ θ Z j ≤ D and therefore, foreach σ ∈ R , R ∋ θ w ( σ, θ ) and R ∋ θ z ( σ, θ ) are nonincreasing. (5.10)Since w and z are bounded and nonincreasing, we may define φ ( σ ) = lim θ → + ∞ w ( σ, θ ) , ψ ( σ ) = lim θ → + ∞ z ( σ, θ ) , which proves assertion (i).(ii) We first observe that the properties of the sequence obtained in the previousparagraph allow us to pass to the limit in the distribution sense in (5.9) and, recall-ing ∂ θ W j , ∂ θ Z j ≤ D , it follows in particular that ( w, z ) is a (continuous bounded) INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 29 solution of ( w σ − δw θθ + αw ≥ c z p ,z σ − z θθ + βz ≥ c w q , in D ′ ( R ). (5.11)We can then obtain (5.3) by the following simple argument. We check for instance thefirst inequality in (5.3), the other being completely similar. Fix χ, ξ ∈ D ( R ), with χ, ξ ≥ R R χ = 1. For j ∈ N , replacing θ by θ + j in (5.11) and testing with ξ ( σ ) χ ( θ ), weobtain Z R Z R (cid:2) c z p − αw (cid:3) ( σ, θ + j ) ξ ( σ ) χ ( θ ) dθdσ = D(cid:2) c z p − αw (cid:3) ( · , · + j ) , ξ ⊗ χ E ≤ D(cid:0) w σ − δw θθ (cid:1) ( · , · + j ) , ξ ⊗ χ E = Z R Z R ( − ξ σ ( σ ) χ ( θ ) − δξ ( σ ) χ θθ ( θ )) w ( σ, θ + j ) dθdσ. (5.12)Due to the boundedness of w, z , we may therefore apply the dominated convergence the-orem on the first and last terms of (5.12). Taking R R χ = 1 and R R χ θθ = 0 into account,we thus obtain Z R (cid:2) c ψ p − αφ (cid:3) ( σ ) ξ ( σ ) dσ = Z R Z R (cid:2) c ψ p − αφ (cid:3) ( σ ) χ ( θ ) ξ ( σ ) dθdσ ≤ Z R Z R (cid:0) − ξ σ ( σ ) χ ( θ ) − δξ ( σ ) χ θθ ( θ ) (cid:1) φ ( σ ) dθdσ = Z R − ξ σ ( σ ) φ ( σ ) dσ and the conclusion follows.(iii) Assume for contradiction that, for instance, φ (0) = 0 and ψ (0) >
0. Then, bycontinuity, there exists η > c ψ p − αφ ]( σ ) ≥ η on ( − η, η ) ⊂ I . Consequently φ ′ ≥ η in D ′ ( − η, η ). It is well known that this guarantees φ ( y ) − φ ( x ) ≥ Z yx η dσ = η ( y − x ) for − η < x < y < η .In particular φ ( x ) ≤ φ (0) + ηx = ηx < x ∈ ( − η, (cid:3) Completion of proof of Proposition 2.1
In this section, by using a contradiction argument and the results of Sections 3-5, wecomplete the proof of Proposition 2.1.
Proof of Proposition 2.1.
The upper estimates in (2.1)-(2.2) follow from (3.2) in Proposi-tion 3.1. To prove the lower estimates, since u ρ , v ρ ≤ u, v > T ∗ / , T ∗ ) × [0 , R ) by the strong maximum principle, it suffices to show that, for each ρ ∈ (0 , ρ ),lim inf t → T ∗ ( T ∗ − t ) α u ( t, ρ ) > t → T ∗ ( T ∗ − t ) β v ( t, ρ ) > . We argue by contradiction and assume for instance that there exist ρ ∈ (0 , ρ ) and asequence t j → T ∗ such that lim j →∞ ( T ∗ − t j ) α u ( t j , ρ ) = 0 . Set σ j := − log( T ∗ − t j ) → ∞ , let ( W, Z ) be defined by (5.1)-(5.2) and let ( φ, ψ ) be givenby Proposition 5.1(i). Since W ( σ, θ ) ≤ W ( σ,
0) for all θ ∈ [0 , ( R − ρ ) e σ/ ] due to (1.6), itfollows from (5.4) that φ (0) = lim θ →∞ (cid:16) lim j →∞ W ( σ j , θ ) (cid:17) ≤ lim j →∞ W ( σ j ,
0) = lim j →∞ ( T ∗ − t j ) α u ( t j , ρ ) = 0 . By Proposition 5.1(ii) and (iii), it follows that ψ (0) = φ (0) = 0. Therefore, with η givenby Proposition 4.1, we deduce from (5.4) that there exists θ > j →∞ W ( σ j , θ ) ≤ η/ , lim j →∞ Z ( σ j , θ ) ≤ η/ . Then, for all j sufficiently large, we have W ( σ j , θ ) ≤ η, Z ( σ j , θ ) ≤ η hence, in view of (5.1)-(5.2),( T ∗ − t j ) α u ( t j , ρ + θ p T ∗ − t j ) ≤ η, ( T ∗ − t j ) β v ( t j , ρ + θ p T ∗ − t j ) ≤ η. Taking j large enough so that ρ + θ p T ∗ − t j < ( ρ + ρ ) / T ∗ − t j ≤ τ , we concludefrom Proposition 4.1 that ρ is not a blow-up point: a contradiction. (cid:3) Proof of Theorem 1.3 and verification of Examples 1.1.
As a preliminary to the proof of Theorem 1.3, we prove the following proposition.
Proposition 7.1.
Under the assumptions of Theorem 1.3, there exists a constant
C > such that sup Q t u q +1 p +1 ≤ C sup Q t v, T ∗ / < t < T ∗ (7.1) and sup Q t v p +1 q +1 ≤ C sup Q t u, T ∗ / < t < T ∗ , (7.2) INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 31 where Q t = (0 , t ) × B (0 , R ) . Proof.
As in [20], we define the functions U , V by: U ( t ) = sup Q t u and V ( t ) = sup Q t v. (7.3)Then U and V are positive continuous and nondecreasing on (0 , T ∗ ). Also, since ( u, v ) is ablowing-up solution, it follows that U or V diverges as t ր T ∗ . We argue by contradiction.Assume that (7.1) fails. Then there exists a sequence t j ր T ∗ as j → ∞ such that V ( t j ) U − q +1 p +1 ( t j ) → j → ∞ . It follows that U must diverge as t ր T ∗ . In the rest of the proof, we use the notation λ j := U − α ( t j ) → j →∞ , where α is given by (1.8).Let ( t ′ j , x ′ j ) ∈ (0 , t j ] × B (0 , R ) be such that u ( t ′ j , x ′ j ) ≥ (1 / U ( t j ) . We have t ′ j → T ∗ as j → ∞ . Now, we rescale the functions U and V by setting: φ j ( σ, y ) := λ α u ( λ j σ + t ′ j , λ j y + x ′ j ) ,ψ j ( σ, y ) := λ β v ( λ j σ + t ′ j , λ j y + x ′ j ) , where ( σ, y ) ∈ ( − λ − j t ′ j , λ − j ( T ∗ − t ′ j )) × ( − λ − j | x ′ j | , λ − j ( R − | x ′ j | )) =: D j and α, β aregiven by (1.8). Then, If we restrict σ to ( − λ − j t ′ j , , we obtain0 ≤ φ j ≤ , φ j (0 , ≥ / ≤ ψ j ≤ V ( t j ) U − q +1 p +1 ( t j ) → j → ∞ . (7.4)On the other hand, ( φ j , ψ j ) solves the system: c ψ p ≤ φ σ − δ ∆ φ ≤ c (cid:16) ψ p + φ r + λ α +1) j (cid:17) ,c φ q ≤ ψ σ − ∆ ψ ≤ c (cid:16) φ q + ψ s + λ β +1) j (cid:17) , on D j . By using interior parabolic estimates, there exists a subsequence, still denoted by( φ j , ψ j ), converging uniformly on compact subsets of ( −∞ , × R n to ( φ, ψ ) a nonnegative(strong) solution of ( φ σ − δ ∆ φ ≤ c ( ψ p + φ r ) ,ψ σ − ∆ ψ ≥ c φ q . By (7.4), it follows that φ (0 , ≥ / ψ ≡ . But the second equation implies φ ≡ u , p , r, α and v , q , s, β . (cid:3) Proof of Theorem 1.3.
Recall that, under the assumptions of Theorem 1.3, we know that k u ( t ) k ∞ = u ( t, k v ( t ) k ∞ = v ( t,
0) and u ( T ∗ ,
0) = v ( T ∗ ,
0) = ∞ . By Proposition 7.1,it follows that there exists C > v p ( t, ≤ Cu r ( t, . (7.5)and u q ( t, ≤ Cv s ( t, . (7.6)Here and in the rest of the proof, C denotes a positive constant which may vary from lineto line.On the other hand, since v t ≥ , u ρ ≤ v ρ ≤ ∂∂ρ (cid:18) v ρ + c v ( u q + v s + 1)) (cid:19) = ( v ρρ + c ( u q + v s + 1)) v ρ + c qvu q − u ρ + c sv s v ρ ≤ ( v ρρ + F ( u, v )) v ρ + c qvu q − u ρ + c sv s v ρ = (cid:18) v t − n − ρ v ρ (cid:19) v ρ + c qvu q − u ρ + c sv s v ρ ≤ . Consequently, (cid:18) v ρ + vF ( u, v ) (cid:19) ( t, ρ ) ≤ (cid:18) v ρ + c v ( u q + v s + 1) (cid:19) ( t, ρ ) ≤ (cid:18) v ρ + c v ( u q + v s + 1) (cid:19) ( t, ≤ c v ( u q + v s + 1)( t, . Moreover, by (7.5), there exists
C > v ( u q + v s + 1)( t, ≤ Cv s +1 ( t, v ρ ( t, ρ ) ≤ Cv s +1 ( t, , for all t ∈ ( T ∗ / , T ∗ ) and ρ ∈ [0 , R ] . Therefore, k v ρ ( t ) k ∞ ≤ Cv ( s +1) / ( t,
0) = Cv β +1 ( t, , for all t ∈ ( T ∗ / , T ∗ ) . Arguing as in [17, p. 187], we deduce that there exist ε , ε > v ( T ∗ , | x | ) ≥ ε | x | − β , for all | x | ∈ (0 , ε ) . The inequality on G is obtained similarly. (cid:3) Finally, we verify the assertions made in Examples 1.1.(i) Let
F, G be given by (1.14)-(1.15). Properties (1.4)–(1.5) are clear (for u, v > , INGLE-POINT BLOW-UP FOR REACTION-DIFFUSION SYSTEMS 33 of the products u r i v s i with r i > r i = 0 being immediate). This follows fromYoung’s inequality applied with the exponent p ( q +1) r i ( p +1) >
1, writing u r i v s i ≤ u p ( q +1) p +1 + v sip ( q +1) p ( q +1) − ri ( p +1) ≤ u p ( q +1) p +1 + C ( v p + 1) , where we used s i p ( q +1) p ( q +1) − r i ( p +1) ≤ p due to (1.15). Property (1.11) is obtained similarly.It thus remains to verify (1.13). Fixing C > C >
0, this amounts to finding µ, A, κ , κ > κ κ <
1, such that R := λ ( κ p − − µ ) v p + m X i =1 λ i (cid:0) r i + κ s i − − µ (cid:1) u r i v s i ≥ R := λ ( κ q − − µ ) v p + m X i =1 λ i (cid:0) κ r i + s i − − µ (cid:1) u r i v s i ≥ { u, v ≥ A | C ≤ u q +1 v p +1 ≤ C } . Fix 1 /p < κ <
1, 1 /q < κ < I = (cid:8) i ∈ { , . . . , m } ; r i p +1 q +1 + s i = p (cid:9) , I = (cid:8) i ∈ { , . . . , m } ; r i + s i q +1 p +1 = q (cid:9) . Observe that if i ∈ I , then r i + κ s i − ≥ r i + 1 p (cid:16) p − r i p + 1 q + 1 (cid:17) − r i pq − p ( q + 1)and we may also assume r i > r i = 0, s i = p and λ i u r i v s i can be includedinto the main term λv p ). Similarly, if i ∈ I , then κ r i + s i − ≥ q (cid:16) q − s i q + 1 p + 1 (cid:17) + s i − s i pq − q ( p + 1)and we may also assume s i >
0. Choosing0 < µ < min (cid:18) κ p − , κ q − , pq − p ( q + 1) min i ∈ I r i , pq − q ( p + 1) min i ∈ I s i (cid:19) , it follows that R ≥ µv p + X i ∈{ ,...,m }\ I λ i (cid:0) r i + κ s i − − µ (cid:1) u r i v s i , (7.7) R ≥ µu q + X i ∈{ ,...,m }\ I λ i (cid:0) κ r i + s i − − µ (cid:1) u r i v s i . (7.8)Now consider i ∈ { , . . . , m } \ I . We have r i p +1 q +1 + s i < p by (1.15). Therefore, on the set D A := { u, v ≥ A | C ≤ u q +1 v p +1 ≤ C } , we have u r i v s i − p ≤ C r i / ( q +1)2 v − p + s i + r i ( p +1) / ( q +1) ≤ C r i / ( q +1)2 A − p + s i + r i ( p +1) / ( q +1) → as A → ∞ . We get the similar property for i ∈ { , . . . , m } \ I . By (7.7)-(7.8), we concludethat R , R ≥ D A by taking A large enough.(ii) Let F , G be given by (1.16)-(1.17). Properties (1.4) and (1.10)–(1.12) are clear. Inorder to verify (1.5) and (1.13), since F u = G v = 0, it clearly suffices to find η > vF v ( u, v ) ≥ (1 + η ) F ( u, v ) , v ≥ uG u ( u, v ) ≥ (1 + η ) G ( u, v ) , u ≥ . Setting X = k log(1 + v ), we compute vF v − (1 + η ) F = v p h ( p − − η )(1 + λ sin X ) + 2 λk v v cos X sin X i . Using | X sin X | ≤ cos X √ λ + √ λ sin X = 1 + λ sin X √ λ , we get vF v − (1 + η ) F ≥ v p h p − − η − λk √ λ i (1 + λ sin X ) ≥ , v ≥ , under assumption (1.17) if we choose η > G is similar. References [1] Andreucci D., Herrero M. A., Velázquez J. J. A.,
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