Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions
aa r X i v : . [ m a t h . A P ] F e b Global existence and infinite time blow-up of classical solutions tochemotaxis systems of local sensing in higher dimensions
Kentaro Fujie ∗ Tohoku University,Sendai, 980-8578, Japan
Takasi Senba † Fukuoka University,Fukuoka, 814-0180, JapanFebruary 25, 2021
Abstract
This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimen-sions. Despite the striking similarity between this system and the Keller–Segel system, we prove theabsence of finite-time blow-up phenomenon in this system even in the supercritical case. It meansthat for any regular initial data, independently of the magnitude of mass, the classical solutionexists globally in time in the higher dimensional setting. Moreover, for the exponential decayingmotility case, it is established that solutions may blow up at infinite time for any magnitude ofmass. In order to prove our theorem, we deal with some auxiliary identity as an evolution equationwith a time dependent operator. In view of this new perspective, the direct consequence of theabstract theory is rich enough to establish global existence of the system.
Key words: chemotaxis; Keller–Segel system; global existence
AMS Classification:
Consider the following initial-boundary value problem for the fully parabolic system: u t = ∆( γ ( v ) u ) ( x, t ) ∈ Ω × (0 , ∞ ) ,v t = ∆ v − v + u ( x, t ) ∈ Ω × (0 , ∞ ) ,∂ ν u = ∂ ν v = 0 ( x, t ) ∈ ∂ Ω × (0 , ∞ ) ,u ( x,
0) = u ( x ) , v ( x,
0) = v ( x ) , x ∈ Ω , (1.1)where Ω ⊂ R n ( n ≥
3) is a smooth bounded domain. The initial data ( u , v ) satisfies( u , v ) ∈ ( W ,p (Ω)) with some p > n, u ≥ , v ≥ , u , (1.2)and for γ we assume γ ∈ C [0 , ∞ ) , γ > , γ ′ ≤ , ∞ ) , (1.3) ∗ [email protected] † [email protected] s →∞ γ ( s ) = 0 . (1.4)The system (1.1) is motivated from biological background (see [10, 13, 4]). From a mathematicalview point, this system resembles the fully parabolic Keller–Segel system. Especially, in the case γ ( v ) = e − v , the system (1.1) has same mathematical structures as one of the Keller–Segel system([9, 5, 2]). Indeed, smooth nonegative solutions of (1.1) satisfy the mass conservation law k u ( t ) k L (Ω) = k u k L (Ω) t > , and the Lyapunov functional is constructed: ddt F ( u ( t ) , v ( t )) + Z Ω ue − v |∇ (log u − v ) | dx = 0 t > , where F ( u, v ) := Z Ω u log u dx − Z Ω uv dx + 12 Z Ω v dx + Z Ω |∇ v | dx. We mention the Keller–Segel system has the same Lyapunov functional ([14]). Moreover, these systemsshares the same stationary problem ([9, 5]) and the scaling structure. For the simplified case ( u t = ∆( e − v u ) ( x, t ) ∈ R n × (0 , ∞ ) ,v t = ∆ v + u ( x, t ) ∈ R n × (0 , ∞ ) , the scaled function ( u λ ( x, t ) , v λ ( x, t )) := ( λ u ( λx, λ t ) , v ( λx, λ t )) ( λ >
0) satisfies the simplifiedsystem and k u λ k L ( R n ) = λ − n k u k L ( R n ) . Thus the mass critical dimension is n = 2. Actually, in [9, 5] the critical mass phenomenon is observedin the two dimensional setting: • for small mass, solutions exists globally and remains bounded uniformly in time; • for large mass, solutions exists globally and may blow up at infinite time.In the present paper, we focus on the supercritical case ( n ≥ γ ( v ) = e − v global existence of very weak solutions is established in arbitrary dimensional setting. Global existenceand boundedness of solutions to (1.1) with some polynomial decaying function γ are obtained undersmallness conditions on some parameters in [18, 3, 6]. For the simplified case (the second equationof (1.1) is replaced by the elliptic equation), in [8] global existence is established for any magnitudeof mass in the higher dimensional setting. On the other hand, the following theorem claims globalexistence of classical solutions for the fully parabolic system independently of the magnitude of mass. Theorem 1.1.
Let n ≥ . Assume γ satisfies (1.3) and (1.4) . For any given initial data ( u , v ) satisfying (1.2) , The system (1.1) permits a unique global classical solution ( u, v ) ∈ ( C ([0 , ∞ ); W ,p (Ω)) ∩ C , (Ω × (0 , ∞ ))) . γ ( v ) = e − v , the higher dimensional case ( n ≥
3) is supercritical as mentioned above.The next theorem guarantees an infinite time blow-up for any magnitude of mass.
Theorem 1.2.
Let n ≥ , Ω = B R (0) and γ ( s ) = e − s . For any m > there exists some radiallysymmetric initial datum ( u , v ) satisfying (1.2) such that • R Ω u = m ; • the global classical solution ( u, v ) of (1.1) blows up at infinite time, that is, lim sup t ր∞ (cid:0) k u ( · , t ) k L ∞ (Ω) + k v ( · , t ) k L ∞ (Ω) (cid:1) = ∞ . Remark 1.3.
For the Keller–Segel system, a finite time blowup phenomenon occurs in the supercriticalcase ([17]). Although the system (1.1) shares several features of the Keller–Segel system, solutions of (1.1) exist globally even in the supercritical case (Theorem 1.1) and a blow-up phenomenon takes placeat infinite time (Theorem 1.2).
We recall some useful lemmas. Local existence and uniqueness of classical solutions to system (1.1)are established by applying the abstract theory (cf. [7, Theorem 1]).
Lemma 2.1.
Assume that γ satisfies (1.3) and ( u , v ) satisfies (1.2) . There exists T max ∈ (0 , ∞ ] such that problem (1.1) permits a unique classical solution ( u, v ) ∈ ( C ([0 , T max ); W ,p (Ω)) ∩ C , (Ω × (0 , T max ))) . If T max < ∞ , then lim sup t ր T max (cid:0) k u ( · , t ) k L ∞ (Ω) + k v ( · , t ) k L ∞ (Ω) (cid:1) = ∞ . Moreover the solution is positive on Ω × (0 , T max ) and the mass conservation law holds: Z Ω u ( · , t ) dx = Z Ω u dx for all t ∈ (0 , T max ) . Let ( u, v ) be a solution of (1.1) in Ω × (0 , T max ). We introduce the auxiliary function w ( x, t ) suchthat ( − ∆ w + w = u ( x, t ) ∈ Ω × (0 , T max ) ,∂ ν w = 0 ( x, t ) ∈ ∂ Ω × (0 , T max ) . Here we remark w = (1 − ∆) − u ∈ C , (Ω × (0 , T max )) by the elliptic regularity theory. The followinglemmas are established in [5, Lemma 5, Lemma 7]. Lemma 2.2.
For any < t < T max , there holds w t + γ ( v ) u = ( I − ∆) − [ γ ( v ) u ] . Moreover, for any x ∈ Ω and t ∈ [0 , T max ) , it follows ≤ w ( x, t ) ≤ w ( x ) e γ (0) t . emma 2.3. There exists
K > depending on the initial data and γ such that for all ( x, t ) ∈ Ω × [0 , T max ) , v ( x, t ) ≤ K ( w ( x, t ) + 1) . By Lemma 2.2 and Lemma 2.3, it follows for any
T > ≤ v ( x, t ) ≤ v ∗ ( T ) := K ( w ( x ) e γ (0) T + 1) for all x ∈ Ω , t ∈ (0 , T ) . (2.1) Lemma 3.1.
Let ( u, v ) be a solution of (1.1) in Ω × (0 , T max ) . If T max < ∞ , then there exists some α ∈ (0 , such that v ∈ C α, α (Ω × [ T max , T max ]) . Proof.
As noted in Lemma 2.2, ( u, v, w ) ∈ ( C , (Ω × (0 , T max ))) satisfies w t − γ ( v )(∆ w − w ) = ( I − ∆) − [ γ ( v ) u ] (3.1)classically on (0 , T max ). Here functions γ ( v ) and ( I − ∆) − [ γ ( v ) u ] are bounded on [ T max , T max ].Indeed, it follows by (2.1) that for all t ∈ [ T max , T max ), γ ( v ∗ ( T max )) ≤ γ ( v ( t )) ≤ γ (0) . By the maximum principle we have0 ≤ ( I − ∆) − [ γ ( v ) u ] ≤ ( I − ∆) − [ γ (0) u ]= γ (0)( I − ∆) − [ u ]= γ (0) w, and then by Lemma 2.2 there exists some C ( T max ) > t ∈ (0 , T max ),0 ≤ ( I − ∆) − [ γ ( v ) u ] ≤ C ( T max ) . (3.2)Since w is a bounded weak solution of (3.1) on [ T max , T max ] with w (0) = w ( T max ), the H¨olderregularity estimate ([12, Corollary 7.51]) guarantees some α ∈ (0 ,
1) such that w ∈ C α, α (Ω × [ T max , T max ]) . Let z ( x, t ) := (1 − ∆) − v ( x, t ) . By the definition of w and z , it follows z t = ∆ z − z + w, (3.3)where z is bounded and w ∈ C α, α (Ω × [ T max , T max ]) as proved above. By applying the parabolicregularity estimate ([11, Chapter IV, Theorem 5.3]) to (3.3), we have z ∈ C α, α (Ω × [ T max , T max ]) , where we used the fact z ( T max ) satisfies the compatibility condition. Thus by the elliptic regularityestimate it follows v ∈ C α, α (Ω × [ T max , T max ]) . We conclude the proof. 4 roof of Theorem 1.1.
Assume to the contrary that T max < ∞ . Define the operator A ( t ) ϕ := − γ ( v ( t + T max ))(∆ ϕ − ϕ )for ϕ ∈ { ψ ∈ W ,p (Ω) | ∂ ν ψ = 0 on ∂ Ω } with p ∈ (1 , ∞ ). Since γ ( v ) is H¨older continuous (Lemma3.1), the operator − A ( t ) generates analytic semigroup in L p (Ω) for any p ∈ (1 , ∞ ). Denoting W ( s ) := w ( s + T max ) ,F ( s ) := (1 − ∆) − ( γ ( v ( s + T max )) u ( s + T max ))for s ∈ [0 , T max ), we can check that W ∈ C ([0 , T max ); L p (Ω)) ∩ C ((0 , T max ); L p (Ω)) is a solutionof the evolution equation ( ddt W + A ( t ) W = F t ∈ [0 , T max ) ,W (0) = w ( T max ) . Since the elliptic regularity theorem implies F ∈ C ([0 , T max ); L p (Ω)) for any p ∈ (1 , ∞ ), in view ofthe abstract theory ([15, Theorem 5.2.2]), the solution W can be represented by the integral equation: W ( t ) = U ( t, w ( T max ) + Z t U ( t, s ) F ( s ) ds t ∈ [0 , T max ) , where U ( t, s ) is the fundamental solution. Due to the fact that w ( T max ) and F ( s ) satisfy the Neumannboundary condition, we can apply the estimate of fundamental solutions ([15, Theorem 5.2.1]) to have k A ( t ) W ( t ) k L p (Ω) ≤ k A ( t ) U ( t, A (0)) − A (0) w ( T max ) k L p (Ω) + Z t k A ( t ) U ( t, s )( A ( s )) − A ( s ) F ( s ) k L p (Ω) ds ≤ C ( p ) k A (0) w ( T max ) k L p (Ω) + C ( p ) Z t k A ( s ) F ( s ) k L p (Ω) ds with some C ( p ) >
0. By the definition of W , it follows k A ( t ) W ( t ) k L p (Ω) = k γ ( v ( t + T max ))(∆ w ( t + T max ) − w ( t + T max )) k L p (Ω) = k γ ( v ( t + T max )) u ( t + T max ) k L p (Ω) . We also obtain for s ∈ [0 , T max ), A ( s ) F ( s ) = − γ ( v ( s + T max ))(∆ − (cid:2) (1 − ∆) − ( γ ( v ( s + T max )) u ( s + T max )) (cid:3) = ( γ ( v ( s + T max ))) u ( s + T max ) . Since γ ( v ) is bounded, for any p ∈ (1 , ∞ ) it follows for t ∈ [0 , T max ) k u ( t + T max )) k L p (Ω) ≤ C ′ ( p ) k (∆ − w ( T max ) k L p (Ω) + C ′ ( p ) Z t k u ( s + T max ) k L p (Ω) ds C ′ ( p ) >
0. By the Gronwall inequality, for any p ∈ (1 , ∞ ) there exists some C ′′ ( p ) > t → T max k u ( t ) k L p (Ω) ≤ C ′′ ( p ) . We can pick up some p > n in the above and by Moser’s iteration argument (see [1]) it followslim sup t → T max k u ( t ) k L ∞ (Ω) < ∞ , which contradicts the assumption T max < ∞ .As mentioned in Introduction, the Lyapunov functional and the stationary problem of (1.1) withthe case γ ( v ) = e − v are same as one of the Keller–Segel system. We can directly employ calculationsin [16, Section 3]. Proof of Theorem 1.2.
On the contrary, the solution ( u, v ) is uniformly bounded in time. By theparabolic Schauder theory and the compactness argument, there exist a sequence of time t k → ∞ anda stationary solution ( u ∞ , v ∞ ) such that the solution ( u ( · , t k ) , v ( · , t k )) converges to ( u ∞ , v ∞ ) in C (Ω)as t k → ∞ , where the following inequality holds F ( u ∞ , v ∞ ) ≤ F ( u , v ) (3.4)(see [16, Lemma 3.1]). Through all radially symmetric stationary solutions ( u ∞ , v ∞ ), the infimum ofthe functional F ( u ∞ , v ∞ ) is bounded from below ([16, Lemma 3.4]). On the other hand, for any m > u , v ) with R Ω u = m having sufficiently large negative energy F ( u , v ) ([16, Lemma 3.2]), which contradicts (3.4). Therefore the corresponding global solution( u, v ) must be unbounded. Acknowledgments
K. Fujie is supported by Japan Society for the Promotion of Science (Grant-in-Aid for Early-CareerScientists; No. 19K14576). T. Senba is supported by Japan Society for the Promotion of Science(Grant-in-Aid for Scientific Research(C); No. 18K03386)
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