A generalized solution concept for the Keller Segel system with logarithmic sensitivity: global solvability for large nonradial data


 Tove Thoresen
 4 dager siden
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Transkript
1 Nonlinear Differ. Eu. Al. 17) 4:49 c 17 Sringer International Publishing AG 1197/17/4133 ublished online July, 17 DOI 1.17/s Nonlinear Differential Euations and Alications NoDEA A generalized solution concet for the Keller Segel system with logarithmic sensitivity: global solvability for large nonradial data Johannes Lankeit and Michael Winkler Abstract. The chemotaxis system { u ut =Δu χ v), v v t =Δv v + u, is considered in a bounded domain R n with smooth boundary, where χ>. An aarently novel tye of generalized solution framework is introduced within which an extension of reviously known ranges for the key arameter χ with regard to global solvability is achieved. In articular, it is shown that under the hyothesis that if n =, χ< 8 if n =3, n if n 4, n for all initial data satisfying suitable assumtions on regularity and ositivity, an associated noflux initialboundary value roblem admits a globally defined generalized solution. This solution inter alia has the roerty that u L 1 loc [, )). Mathematics Subject Classification. 35K55, 35D99, 9C17. Keywords. Chemotaxis, Logarithmic sensitivity, Global existence, Generalized solution.
2 49 Page of 33 J. Lankeit and M. Winkler NoDEA 1. Introduction We consider the Keller Segel system with logarithmic sensitivity, as given by the initialboundary value roblem u u t =Δu χ v ), v x, t>, u ν =, x, t>, 1.1) ux, ) = u x), x, couled to the arabolic roblem v t =Δv v + u, x, t>, v ν =, x, t>, vx, ) = v x), x, 1.) where is a bounded domain in R n, n, with smooth boundary, χ is a ositive arameter and the given initial data u and v satisfy suitable regularity and ositivity assumtions. This system can be viewed as a rototyical arabolic model for selfenhanced chemotactic migration rocesses in which crossdiffusion occurs in accordance with the Weber Fechner law of stimulus ercetion [9, 15], and accordingly a considerable literature is concerned with its mathematical analysis. However, u to now it seems yet unclear to which extent the articular mechanism of taxis inhibition at large signal densities in 1.1) is sufficient to revent henomena of blowu, known as the robably most striking ualitative feature of the classical Keller Segel system: Indeed, in its fully arabolic version, as determined by the choice τ := 1 in { ut =Δu χ u v), 1.3) τv t =Δv v + u, the latter admits solutions blowing u in finite time for any choice of χ> whenever n [8, ], and in the simlified arabolic ellitic case obtained on choosing τ := it is even known that some radial solutions to an associated Cauchy roblem in the whole lane collase into a ersistent Diractye singularity in the sense that a globally defined measurevalued solution exists which has a singular art beyond some finite time and asymtotically aroaches a Dirac measure cf. e.g. [19] or also [1]). As oosed to this, the literature has identified various circumstances under which henomena of this tye are ruled out in 1.1) 1.): For instance, when χ<χ n) with some χ ) > 1.15 and χ n) := n for n 3, global bounded classical solutions exist for all reasonably regular ositive initial data [, 4, 11, 13, 1, 4]; in the corresonding arabolic ellitic analogue, the same conclusion holds with χ ) = [5] and with χ n) := n when n 3and the satial setting is radially symmetric [14], cf. also [6] for a related result addressing a variant with its second euation being τv t =Δv v + u for small τ>), whereas it is known that some exloding solutions exist if n 3and
3 NoDEA A generalized solution concet for the Keller Segel system Page 3 of [14]. As for larger values of χ in the fully arabolic roblem 1.1) 1.), in some cases at least certain global generalized solutions can be found which satisfy χ> n n u L 1 loc [, )) 1.4) and thereby indicate the absence of strong singularity formation of the flavor described above. Such constructions are ossible in the context of natural weak solution concets if n + χ< 1.5) 3n 4 [1] and within a slightly more generalized framework if merely n χ< 1.6) n but in addition the solutions are suosed to be radially symmetric [17]. To the best of our knowledge, however, the uestion how far 1.5) is otimal with resect to the existence of not necessarily radial solutions fulfilling 1.4) isyet unsolved; in articular, it aears to be unknown whether in nonradial lanar settings such solutions do exist also beyond the range χ< determined by 1.5). Main results. The urose of this work is to design a novel concet of generalized solvability which is yet suitably strong so as to reuire 1.4), but which on the other hand is mild enough so that it enables us to construct corresonding global solutions without any symmetry hyotheses and under conditions somewhat weaker than 1.5) and actually also than 1.6). More recisely, considering 1.1) 1.) under the assumtions that { u C ) is such that u inandu, and that v W 1, 1.7) ) satisfies v > in, we can state our main results as follows. Theorem 1.1. Let n and R n be a bounded domain with smooth boundary, and let χ> be such that if n =, χ< 8 if n =3, 1.8) n n if n 4. Then for any u and v fulfilling 1.7), the roblem 1.1) 1.) has at least one global generalized solution u, v) in the sense of Definition.4 below. In articular, this solution satisfies 1.4), and moreover we have u,t)= u for a.e. t>. 1.9) Plan of the aer. A first substantial task will be related to the design of a suitable family of aroximate versions of 1.1) 1.) in which, on the one hand, the crucial nonlinear interaction is regularized in such an effective
4 49 Page 4 of 33 J. Lankeit and M. Winkler NoDEA manner that even global classical solutions exist, but which on the other hand retains, as far as ossible, a fundamental dissiative roerty of 1.1) 1.). As is essentially wellknown, namely, the functional u v enjoys certain uasientroy features along trajectories of 1.1) 1.), rovided that the crucial ositive arameter therein satisfies < 1 χ and > is chosen adeuately. After introducing the regularization 3.1) of1.1) 1.) aroriate for our uroses, in Sect. 4 we will derive a rigorous counterart of this entroylike roerty for the corresonding aroximate solutions. The main challenge now consists in taking aroriate advantage of accordingly imlied a riori estimates obtained in Sects. 5, 6 and 7, which inter alia seem far from sufficient to warrant L 1 bounds for the crossdiffusive flux χ u v v esecially in cases when χ is large and hence needs to be chosen small. In the rearatory Sect. 3, we will therefore resort to a solution framework involving certain sublinear owers of u rather than u itself, thus reminiscent of the celebrated concet of renormalized solutions [3]. This idea has artially been adated to the resent context in [17] already, but in the resent work we shall further weaken the reuirements on solutions to a considerable extent: Namely, for the crucial first subroblem 1.1) to be solved we shall only reuire that the couled uantity u v, with certain ositive and, satisfies a arabolic ineuality associated with 1.1) 1.) in a weak form, and that moreover u,t) u for a.e. t>; a key observation, to be made in Lemma.5, will reveal that if we furthermore assume the comonent v to fulfill 1.) ina natural weak sense, then we indeed obtain a concet consistent with that of classical solvability in 1.1) 1.) for all suitably smooth functions. As seen in Sect. 8 by means of aroriate comactness arguments, the reviously gained estimates in fact enable us to construct a global solution within this framework.. A concet of generalized solvability In secifying the subseuently ursued concet of weak solvability, we first reuire certain roducts u v to satisfy an ineuality which can be viewed as generalizing a classical suersolution roerty of this uantity with regard to 1.1) 1.). Definition.1. Let, 1) and, 1), and suose that u and v are measurable functions on, ) such that u>and v>a.e. in, ), that u v L 1 loc [, )) and u +1 v 1 L 1 loc [, )),.1) and that u and v belong to L 1 loc, )) and are such that v u L loc [, )) and u v L loc [, ))..) Then u, v) will be called a global weak, )suersolution of 1.1) if u v ϕ t u v ϕ, )
5 NoDEA A generalized solution concet for the Keller Segel system Page 5 of ) 4 1 ) χ v u ϕ χ +1 ) 4χ +1 ) + u 1 )χ + v χ +1 ) v u ϕ χ u v u ϕ + 1 χ ) u v Δϕ u v ϕ + u +1 v 1 ϕ.3) for all nonnegative ϕ C [, )) such that ϕ ν =on, ) and if moreover u v > a.e. on, )..4) Remark.. i) Observing that.1) in articular ensures that u v L loc [, )), and that hence.1) and.) warrant that ) ) u v u = u v v u L 1 loc [, )) and similarly u v v L 1 loc [, )), it follows that under the above reuirements all integrals in.3) are indeed welldefined. ii) According to.1) and.), for a.e. t>, u,t)v,t) W 1, ) so that u v,t) L ) exists in the sense of traces, giving meaning to the ositivity reuirement in.4). Aart from that, we will reuire the second roblem 1.) to be satisfied in the following rather natural weak sense. Definition.3. Aairu, v) of functions { u L 1 loc [, )), v L 1 loc [, ); W 1,1 )) will be named a global weak solution of 1.) if vϕ t v ϕ, ) = v ϕ for all ϕ C [, )). vϕ +.5) uϕ.6) Following an aroach already ursued in [3] in a considerably less involved related context, in order to comlete our solution concet we will comlement the above two reuirements by merely ostulating an uer bound for the mass functional u in terms of u : Definition.4. A coule of nonnegative measurable functions u and v defined on, ) will be said to be a global generalized solution of 1.1) 1.) if u, v) is a global weak solution of 1.) according to Definition.3, if there exist
6 49 Page 6 of 33 J. Lankeit and M. Winkler NoDEA, 1) and, 1) such that u, v) is a global weak, )suersolution of 1.1) in the sense of Definition.1, and if moreover u,t) u for a.e. t>..7) This is indeed consistent with the concet of classical solvability in the following sense. Lemma.5. Let χ>, and suose that u, v) C [, )) C,1, ))) is such that u, v) is a global generalized solution of 1.1) 1.) in the sense of Definition.4. Thenu, v) satisfies 1.1) 1.) classically in, ). Proof. By means of standard arguments relying on the assumed regularity roerties of v, it can easily be verified that v solves 1.) classically. According to the maximum rincile, v hence is strictly ositive in [, ) andv 1 is uniformly bounded in every set [,T)forT, ). Positivity of v ensures that by.4) u> on a dense subset of, ) which moreover is oen in, ) by continuity of u. For arbitrary ψ C ) with ψ and ψ ν =, testing.3) by ϕx, t) :=ψx)1 1 t) +,, 1), which is ermissible by Weierstrass theorem, and invoking Lebesgue s dominated convergence theorem and continuity of t u,t)v,t)att = in taking we readily achieve u, )v, )ψ u v ψ for all ψ C ),ψ, ψ ν =, showing that u, )v, ) u v throughout. Because of v, ) = v > and the monotonicity of ) 1 we obtain u, ) u in and from continuity of u and.7) we can conclude that u, ) = u in. In the first two integrals on the right of.3) straightforward comutations yield 41 ) v u 41 )χ 4χ +1 ) since = + { u v { 41 ) + 4χ +1 ) ) +8 u v u v u v = 4χ +1 ) 1 )χ + χ +1 u χ χ +) v u v + 4χ +1 ) v u 1 )χ +) χ +1 ) } v u u 1 )χ + v χ +1 ) v u + 41 ) 4 1 ) χ v u,.8) χ +1 ) }
7 NoDEA A generalized solution concet for the Keller Segel system Page 7 of ) 1 )χ +) χ +1 ) = 41 )χ +1 ) 1 )χ +) χ +1 ) = 41 ) 4 1 ) χ. χ +1 ) In rearation of the following calculations we also note that for each ositive function w C ) and any r>, we have the ointwise identities w r r r r ) Δw = w w r w = w r rr ) w r 4 w + r ) w r Δw and = rr ) 4 4 w r w + r wr 1 Δw = r w r + r r wr 1 Δw.9) Δw r = rw r 1 w) =rr 1)w r w + rw r 1 Δw 4r 1) = w r + rw r 1 Δw.1) r The ositivity reuirement on w in.9) and.1) romts us to erform the following calculations only for test functions ϕ comactly suorted in {u >} := {x, t) [, ) : ux, t) > }, ensuring strict ositivity of u and boundedness of u 1 on su ϕ. Accordingly, for all nonnegative ϕ C, )) with su ϕ {u> } and ϕ ν, ) =,by.9) alied to u and, an integration by arts in the integral in.3) containing ϕ yields χ u v u χ ϕ = v u ϕ + 4χ + χ = χ )χ + + χ u v v u ϕ u v Δu χ ϕ v u ϕ + 4χ v u ϕ u 1 v Δuϕ χ u v u ν ϕ u v v u ϕ u v u ν ϕ,.11) whereas integrating by arts twice in the integral containing Δϕ in.3), by.1) alied to u, and v,, resectively, leads to 1 χ ) u v Δϕ = 1 χ ) v Δu )ϕ + 1 χ ) u Δv )ϕ
8 49 Page 8 of 33 J. Lankeit and M. Winkler NoDEA + 1 χ ) u u v v ϕ 1 χ ) u u ν v ϕ 1 χ ) u v v ν ϕ 4 1) = 1 χ ) v u ϕ 4 1) + 1 χ ) u v ϕ + 1 χ ) v u 1 Δuϕ + 1 χ ) u v 1 Δvϕ +8 1 χ ) u v u v ϕ 1 χ ) u u ν v ϕ.1) for any such ϕ, for we already know that v ν =on, ). If we combine.3) with.8),.11) and.1), we obtain { 41 ) u v ) t ϕ + χ 4 1) + 1 χ ) } )χ + v u ϕ { 4χ +1 ) 4 1) χ ) } u v ϕ { 41 )χ χ +8 8χ } u v u v ϕ { χ χ ) } v u 1 Δuϕ + 1 χ ) u v 1 Δvϕ u v ϕ + u +1 v 1 ϕ χ u v u ν ϕ 1 χ ) u u ν v ϕ { 4χ = u v 4χ u v u v + v u 1 Δu χu v 1 Δv } ϕ + u v 1 {Δv v + u} ϕ u u ν v ϕ.13) for every ϕ C, )) satisfying ϕ throughout, ) and ϕ, ) = as well as su ϕ {u>}. ν
9 NoDEA A generalized solution concet for the Keller Segel system Page 9 of The observations that u 1 v Δu χ u ) v v) = u 1 v Δu χ u 1 v 1 u v + χ u v v χu v 1 Δv = u 1 v Δu 4χ u v u v + 4χ u v χ u v 1 Δv, and that v solves 1.), now turn.13) into { u )} u 1 v u t ϕ u 1 v Δu χ v v ϕ u 1 v u ν ϕ.14) for all nonnegative ϕ C, )) with su ϕ {u > } and ϕ ν, ) =. Secializing this to nonnegative ϕ C, ) {u > }) byadu BoisReymond lemma tye argument we conclude u ) u t Δu χ v v in {u >}..15) Density of {u >} in, ), obtained from the assumtion that u> a.e., and continuity show that.15) actually holds on all of, ). We ick t > and some nonnegative ψ C 1 ) with ψ ν =such that su ψ {u,t ) > } := {x : ux, t ) > }. Then by continuity of u we can find some τ>such that su ψ t t τ,t +τ){u,t) > }. Alying.14) to functions of the form ϕx, t) =ζt)ψx), ζ C t τ,t + τ)) by once more invoking a Weierstrass tye density argument and the Du BoisReymond lemma, we see that { u )} u 1 v u t,t)ψ u 1 v Δu χ v v,t)ψ u 1 v u ν,t)ψ for every nonnegative ψ C 1 ) such that ψ ν =, su ψ {u,t ) > } and for almost every t t τ,t + τ) and due to continuity esecially for t = t. In articular inserting ψ x) :=1 1 dist x, )) + ψ and Lebesgue s theorem show that for every t>, ψ C 1 ), ψ with ψ ν =and su ψ {u,t) > }, u 1 v u ψ..16) ν Since the integral only deends on ψ and not on the values of ψ inside, it can be seen that.16) actually holds for any t> and any nonnegative ψ C ) such that su ψ {u,t) > }.
10 49 Page 1 of 33 J. Lankeit and M. Winkler NoDEA If u ν x,t ) < for some x,t ), ) with ux,t ) >, we ick ψ 1 C ) such that ψ 1 x ) >, ψ 1, and su ψ 1 {x : u ν x, t ) <,ux, t ) > } =: M. Moreover, we let d := dist su ψ 1, \M) or d = 1 if \ M = ) and let ψ be the solution to Δψ = in, ψ = u 1,t )v u,t )ψ 1 ν on. Then, thanks to the choice of su ψ 1, ψ is nonnegative on the boundary and hence by the maximum rincile in. Defining ψ 3 x) :=ψ x)1 d dist x, M)) + we obtain a nonnegative continuous function on whose suort intersects the boundary only in {x : ux, t) > } and which hence is a ermissible test function in.16). We conclude that u 1 v u ν ψ 3 = u ν ψ 1 and hence in articular u ν x,t ) =, which is a contradiction. We conclude that u on, ) {u>} and hence, by continuity of u ν set that u on, ). ν ν and density of this Finally integrating.15) over,t) and taking.7) into consideration, we see that t t u ) u u,t) u + Δu χ v v t u = u + ν by Gauss theorem and v u ν =, which firstly shows that ν =on, ) and secondly that.15) actually is an euality. 3. Global smooth solutions to aroximate roblems Now in order to aroximate solutions by means of a convenient regularization of 1.1) 1.), for, 1) we consider u u t =Δu χ 1+u )v v ), x, t >, v t =Δv v + u, x, t >, u ν = 3.1) v ν =, x, t >, u x, ) = u x), v x, ) = v x), x, and then first obtain the following. Lemma 3.1. For all, 1), the roblem 3.1) admits a global classical solution u,v ) C [, )) C,1, ))) for which u > in, ) and v > in [, ). Proof. The local existence of a solution can be obtained in a standard manner cf. [1, Lemma 3.1] for a related general setting). Boundedness of the sensitivity u term χ 1+u )v, due to a strict ositivity roerty of v on,t) to be made more recise in Lemma 3.3 below allows for an iterative rocedure converting boundedness information of v,t) L ) for some >1into n bounds for u,t) L ) for 1, n ) + ) that in turn yield better estimates for v through alication of semigrou estimates in the first and second
11 NoDEA A generalized solution concet for the Keller Segel system Page 11 of euation, resectively. Finally, this serves to rovide a uniform bound on u on,t), in light of the extensibility criterion [1, 3.3)] thus ensuring global existence of the solution. Positivity of u follows from a classical strong maximum rincile. These aroximate solutions clearly reserve mass: Lemma 3.. Let, 1). Then u,t)= u for all t>. 3.) Proof. This directly results on integrating the first euation in 3.1). Moreover, the assumed ositivity of v enables us to control v from below at least locally in time: Lemma 3.3. For each, 1), we have ) v x, t) inf v x) e t for all x and t>. x Proof. As u is nonnegative, this is a straightforward conseuence of a comarison argument alied to the second euation in 3.1). By means of wellknown smoothing estimates of the heat semigrou, the mass conservation roerty 3.) readily imlies some basic regularity features of the second comonent. Lemma 3.4. Let r 1 and s 1 be such that r< n n n and s< n 1.Then there exists C> such that for each, 1), v,t) r C for all t> 3.3) and Proof. The reresentation of v as v,t) s C for all t>. 3.4) v,t)=e tδ 1) v + t e t s)δ 1) u,s)ds makes it ossible to aly wellknown estimates for the Neumann heatsemigrou cf. [, Lemma 1.3]), which rovide ositive constants c 1,c,c 3 and c 4 such that v,t) Lr ) t c 1 v L r ) + c 1 + t s) n 1 1 r ) )e t s) u,s) ds L1 ) and
12 49 Page 1 of 33 J. Lankeit and M. Winkler NoDEA v,t) Ls ) c 3 v W 1, ) + c 4 t 1 + t s) 1 n 1 1 s ) )e t s) u,s) L 1 )ds for all t> and all, 1), so that Lemma 3. and finiteness of 1 + τ n 1 1 r ) )e τ dτ and 1 + τ 1 n 1 1 s ) )e τ dτ due to the conditions on r and s rove the lemma. 4. A fundamental identity and first conseuences thereof Let us next formulate an identity which aarently reflects a fundamental structural roety of 1.1) 1.), as already used in a slightly modified form and for more restricted choices of χ in [1]. In Lemma 4.3 alied to ϕ 1, this will serve as a source for some essential a riori estimates for 3.1), whereas in Lemma 8.8 we will make use of the freedom to choose widely arbitrary test functions here in order to verify.3) for the limit coule u, v) tobe constructed in Lemma 8.1. Lemma 4.1. Let, 1) and, 1), and assume that T > and that ϕ C [,T]) is such that ϕ ν =on,t). Then T u vϕ t + u,t)v,t)ϕ,t) u v ϕ, ) T 41 ) 4 1 ) χ 1+u = ) χ v u ϕ T T T T 1+u +1 ) 4 χ +1 1+u ) u v [1 )u ]χ 1 + u ) u v u ϕ χ ) 1 u vδϕ 1 + u ) T u vϕ + Proof. Using 3.1), we comute t u v) ϕ = = 1 ) 1 )χ u 1 u v 1 1 )χ 1+u + χ 1+u +1 ) v u ϕ u +1 v 1 ϕ for all, 1). 4.1) ) vϕ u ) u χ v 1 + u )v ) ϕ v u vϕ + u u vϕ u 1 1+u v 1 u v )ϕ u +1 v 1 ϕ
13 NoDEA A generalized solution concet for the Keller Segel system Page 13 of χ u 1 v 1 u v )ϕ u v v ϕ + 1 ) 1+u u u v v ϕ u 1 v u ϕ + χ v 1 v ϕ 1+u u v 1 v ϕ u vϕ + u +1 v 1 ϕ 41 ) = v u ϕ 41 )χ u v u v )ϕ 1+u 8 u v u v )ϕ + 4χ u v 41 ) ϕ + u 1+u v ϕ u v u ϕ + χ u v ϕ u 1+u v ϕ u vϕ + u +1 v 1 ϕ for all t>. 4.) Here a straightforward rearrangement in the first five integrands on the right along the lines of.8) shows that 41 ) v u 41 )χ 8u v u v ) u + 4χ v + 1+u = 4 ) χ +1 u 1+u v )χ ) 1+u + v 4 +1 u χ 1+u u ) v u v 1+u 41 ) u v 1 )χ 1+u + χ 1+u +1 u v u v + 41 ) 4 1 ) χ 1+u ) χ v 1+u +1 ) u in, ). 4.3)
14 49 Page 14 of 33 J. Lankeit and M. Winkler NoDEA Moreover, in two of the three summands in 4.) which contain ϕ we once more integrate by arts to see that u v u ϕ + χ u v ϕ u v ϕ 1+u = u v u ϕ χ u 1 v u ϕ 1+u + χ u 1 + u ) v u ϕ χ u vδϕ 1+u + u 1 v u ϕ + u vδϕ = u v u ϕ χ u v u ϕ 1+u + χ u u ) v u ϕ χ u vδϕ 1+u + u v u ϕ + u vδϕ [1 )u ]χ = 1 + u ) u v u ϕ χ ) + 1 u vδϕ in,t) 1 + u ) thanks to the assumtion that ϕ =on,t). Combining this with 4.) and ν 4.3) establishes 4.1). An elementary but crucial observation now identifies a condition on the relationshi between the exonents and which ensure ositivity of the coefficient aearing in the first summand on the righthand side in 4.1). Lemma 4.. Given χ> and, 1) such that < 1 χ,let + ), 1) and ), + )) be defined by ± ) := 1 1 ± ) 1 χ. 4.4) Then for any choice of ), + )) and all, 1), 41 ) 4 1 ) χ χ 1+u ) 1+u +1 ) 41 ) 4 1 ) χ > χ +1 ) in, ). 4.5) Proof. We use that 1 + u 1 to firstly obtain 41 ) 4 1 ) χ 1 + u ) 41 ) 4 1 ) χ in, ). Since here our hyothesis ), + )) guarantees that ) ) 41 ) 4 1 ) χ = 4 + ) ) >,
15 NoDEA A generalized solution concet for the Keller Segel system Page 15 of we may once again estimate 1 + u 1 to infer that indeed both ineualities in 4.5) hold. As a conseuence, for and as in Lemma 4. we can readily derive the following from Lemma 4.1 when combined with the ointwise lower estimate for v in Lemma 3.3. Lemma 4.3. Let, 1) be such that < 1 χ,andlet ), + )) with ± ) taken from 4.4). Then for each T>there exists C,, T ) > fulfilling T v u C,, T ) 4.6) and as well as T and for all, 1). T ) 4 χ +1 u v 1+u u C,, T ) 4.7) 1 )χ 1+u + χ 1+u +1 ) v u C,, T ) 4.8) T u +1 v 1 C,, T ) 4.9) Proof. According to Lemma 4., our assumtion ), + )) ensures that with some c 1 > wehave 41 ) 4 1 ) χ χ 1+u +1 ) 1+u ) c 1 for all, 1), whence alying Lemma 4.1 to ϕ 1 shows that T c 1 v u T ) 4 χ 1 )χ + +1 u v 1+u + χ 1+u 1+u +1 ) v u ϕ T + u +1 v 1 T u,t)v,t) u v + u v for all, 1). 4.1)
16 49 Page 16 of 33 J. Lankeit and M. Winkler NoDEA Now by the Hölder ineuality, { } { u v u so that since } 1 v 1 for all t>, 1 < +) 1 = 1+ 1 χ < 1 < n n, we may combine 3.) with Lemma 3.4 to find c 3 > fulfilling u v c 3 for all t> whenever, 1). The estimates in 4.6), 4.8) and 4.9) therefore result from 4.1), whereuon 4.7) is a conseuence of 4.6) and the fact that Lemma 3.3 along with 1.7) says that given T>wecan find c > such that v x, t) c for all x, t,t)and, 1). 5. A further conseuence: a bound for u in L r for some r>1 Now in view of the desired integrability feature in 1.4), a crucial ste in our analysis will consist in deriving a satiotemoral euiintegrability roerty of u. This will result from bounds therefor in some reflexive L r saces, to be obtained by an interolation between 4.9) and 3.3). The following statement identifies the minimal ossible choice of an integrability exonent arising in the course of this argument cf. 5.6) below), and will thereby form the core of our reuirement 1.8) on χ. Lemma 5.1. Let χ>, andfor, min{1, 1 χ }) let ± ) be as in 4.4). Then 1 1 if χ 1, inf = χ if χ 1, ), 5.1),1), < 1 χ 1+ χ ), +)) 4 if χ. Proof. By an evident monotonicity roerty, 1 inf,1), < 1 χ ), +)) = inf,1), < 1 χ = inf,1), < 1 χ = inf,1), < ) χ ) 1+ 1 ) 1 χ ) =: Iχ) χ 5.)
17 NoDEA A generalized solution concet for the Keller Segel system Page 17 of for any χ>. Since 1 ) 1 χ < 1 and thus 1+ 1 ) 1 χ > 1 for all, min{1, 1 χ }), and since on the other hand for χ 1wehave 1+ 1 ) 1 χ Iχ) lim inf =1, 1 this firstly imlies that Iχ) = 1 for any such χ. In the case χ>1, having in mind the substitution ξ = 1 χ in 5.), we note that satisfies ρξ) := 1 ξ 1+ χ 1 1 ξ χ ) ξ 1 ξ χ = 1 { χ } 1 ξ 1 ξ)+1+ξ = 1 { χ } 1+ξ +1+ξ, ξ [, 1), χ ρ ξ) = 1 + ξ) + 1 for all ξ, 1), so that ρ attains a zero at ξ = χ 1, 1) if and only if χ 1, ), while ρ throughout, 1) if χ. Therefore, inf ξ [,1) ρξ) =ρχ 1) = χ if χ 1, ), whereas inf ξ [,1) ρξ) = lim ξ 1 ρξ) = 1 + χ 4 if χ. In conjunction with 5.), these observations verify 5.1). Now under the assumtions on χ from Theorem 1.1, the announced interolation argument indeed bears fruit of the desired flavour. Lemma 5.. Suose that χ> is such that 1.8) holds. Then there exists r>1 such that for any T>one can find CT ) > with the roerty that T u r CT ) for all, 1). 5.3) Proof. As a conseuence of Lemma 5.1, our assumtion on χ warrants that we can ick, min{1, 1 χ })and ), + )) such that 1 < n n. 5.4) Indeed, if n = this is obvious, while if n 4 this is immediate from 5.1), n because then due to the fact that n, the hyothesis 1.8) in articular reuires that χ<, so that in both cases χ 1andχ>1, 5.1) shows that the assumtion χ< imlies that n n 1 inf = max{1,χ} < n,1), < n. 1 χ ), +))
18 49 Page 18 of 33 J. Lankeit and M. Winkler NoDEA If n = 3, in the case χ< we similarly obtain that 1 inf,1), < 1 χ ), +)) = max{1,χ} < < 3= n n, whereas when χ we use our restriction χ< 8 to infer from 5.1) that 1 inf =1+ χ,1), < 4 < 3 1 χ ), +)) and that thus 5.4) can be achieved also in this case. Henceforth keeing and fixed such that 5.4) holds, e.g. by means of a continuity argument we can ick r>1sufficiently close to 1 such that still +1 r>and 1 )r +1 r < n n. 5.5) Then using Young s ineuality, for T>wecan estimate T T ) r u r = u +1 v 1 1 )r v T T u +1 v 1 1 )r +1 r + v for all, 1), 5.6) so that 5.3) results on using 4.9) and alying 3.3) together with 5.5). 6. A weighted L bound for v In order to comlement 4.7) by an analogous L estimate for v merely involving v but not u as a weight function, indeendently from the above we aly a standard testing techniue to the second euation in 3.1) with the following outcome. Lemma 6.1. For all, 1) and any T>one can find CT ) > such that T v CT ) for all, 1). 6.1) Proof. Thanks to the ositivity of v, we may use v 1 the second euation in 3.1) to see that 1 d dt v =1 ) 1 ) v v v v v + u v 1 where according to Lemma 3.4 and the fact that <1 < c 1 > such that v c 1 for all t>. as a test function in v for all t>, 6.) n n, we can find
19 NoDEA A generalized solution concet for the Keller Segel system Page 19 of On integration, we thus obtain from 6.) that 41 ) T T v =1 ) v v 1 T v,t)+ v for all, 1). c 1 + Tc 1 7. Time regularity As a final rearation for our limit rocedure, we establish some regularity features of the time derivatives in 3.1), beginning with a conveniently transformed version of the first solution comonent. Lemma 7.1. Assume 1.8), and let, 1) be such that < 1 χ. Then for all T>there exists CT ) > such that T ) t u,t)+1 W 1, )) dt CT ) for all, 1). 7.1) Proof. We fix ψ C ) such that ψ W 1, ) 1 and use 3.1) and Young s ineuality as well as the trivial estimate u +1 1 to see that t u +1) ψ = ) u +1) 4 u ψ u +1) u ψ 4 )χ 4 + χ ) χ u u +1) 4 u v )ψ 1 + u )v u u +1) v ψ 1 + u )v )χ u +1) 4 u + u +1) u +1) u v v v v ) +1) 4 u u + 4 )χ + u +1) u 8 )χ v + 8 v + χ +1) 4 u + χ v 4 v u +1) u u +1) u + 4 for all t>and, 1).
20 49 Page of 33 J. Lankeit and M. Winkler NoDEA Since Lemma 3.3 rovides c 1 > such that v c 1 in,t) and hence v,t) 9c 3 1 v 1 3,t) for all t,t)and, 1), v and since u,t)+1) u + for all t>and, 1) by 3.), we thus infer that there exists c > such that for all, 1), ) t u,t)+1 W = su ) 1, )) t u,t)+1 ψ ψ C ) ψ W 1, ) 1 { c u,t) + v 1 3,t) +1 } for all t,t). Thanks to the outcomes of Lemmas 4.3 and 6.1, an integration over t,t) therefore yields 7.1). As for the second comonent, we can directly address the uantity v t. Lemma 7.. Let χ>. Then there exists C>such that whenever, 1), v t,t) W 1, )) dt C for all t>. 7.) Proof. We again fix ψ C ) fulfilling ψ W 1, ) 1, and using 3.1) we find that v t ψ = v ψ v ψ + u ψ v + v + u for all t>and, 1). Therefore, { v t,t) W 1, )) su v,τ) + τ> for all t>and, 1), so that 7.) results from Lemmas 3.4 and 3.). } v,τ)+ u,τ) 8. Construction of limit functions. Proof of Theorem 1.1 Collecting the above estimates, by means of a straightforward extraction rocedure we can ass to the limit in the following sense. Lemma 8.1. Suose that 1.8) holds, and let, 1) and, 1) be such that < 1 χ and ), + )). Then there exist j ) j N, 1) and functions u and v defined on, ) such that j as j, that u and v a.e. in, ), and that u u in L 1 loc [, )) and a.e. in, ), 8.1)
21 NoDEA A generalized solution concet for the Keller Segel system Page 1 of u u in L loc [, )), 8.) v v in L 1 loc [, )) and a.e. in, ), 8.3) v v in L 1 loc [, )) and 8.4) v v in L loc [, )) 8.5) as = j, and u,t)= u for a.e. t>. 8.6) Proof. We fix, 1) such that < 1 χ and combine Lemma 4.3 with 3.) and Lemma 7.1 to see that ) u +1) is bounded in L loc[, ); W 1, )),1) and that ) t u +1),1) is bounded in L 1 loc[, ); W 1, )) ). Aart from that, Lemmas 3.4 and 7. show that there exists r>1 such that v ),1) is bounded in L r loc[, ); W 1,r )) and v t ),1) is bounded in L, ); W 1, )) ). Therefore, by means of two alications of an AubinLions lemma [16, Cor. 8.4] we can find j ) j N, 1) such that j asj, that u u in L loc [, )) and a.e. in, ) as = j, and that 8.), 8.3) and 8.4) hold with some nonnegative functions u and v defined on, ). Since thus also u u a.e. in, ) as = j, making use of the euiintegrability roerty imlied by Lemma 5. we may invoke the Vitali convergence theorem to infer that in fact also 8.1) holds, whereuon 8.6) becomes a conseuence of 3.). The additional convergence statement in 8.5) finally results from Lemma 6.1 and 8.3) in a straightforward manner. Our next aim is to make sure that the functions u and v we have just constructed form a generalized solution of 1.1) 1.). We begin with the second euation. Lemma 8.. If 1.8) holds, then the air u, v) obtained in Lemma 8.1 is a global weak solution of 1.) in the sense of Definition.3. Proof. From 8.1), 8.3) and 8.4) we immediately see that the regularity roerties in.5) hold, and that moreover for arbitrary ϕ C [, )), in the identity v ϕ t v ϕ, )= v ϕ v ϕ+ u ϕ, valid for all, 1) due to 3.1), we may let = j ineachintegral searately to readily verify.6).
22 49 Page of 33 J. Lankeit and M. Winkler NoDEA Another imortant art of Definition.4 are ositivity reuirements, which will be established in Lemma 8.6. The following technical lemmas reare the main argument therein, where we will derive a differential ineuality for ln u, and where further exloting the latter will in articular reuire some indeendent lower bound for this functional at some suitable initial value, desite the fact that 1.7) does not guarantee finiteness of ln u.an aroriate relacement, to be rovided by Lemma 8.5, is entailed by the comarisontye Lemma 8.3 in combination with a differential ineuality, the derivation of which rests on Lemma 8.4. Lemma 8.3. Let a>, b> and T>, andlety :,T) R be a continuously differentiable function satisfying y t) ay t)+b for all t,t) at which yt) >. Then b yt) a coth abt) for all t,t). { } b Proof. Let η>. Then M η := t,t):yt) > a + η is either emty or) can be written as union of its connected comonents, i.e. M η = k N I k with disjoint oen intervals I k. If we consider any nonemty I k with inf I k, b by continuity yinf I k )= a + η and hence y inf I k ) abη aη <, b contradicting the definition of inf I k as infimum of a set where y> a + η. b Hence there is t η [,T) such that y a +η in t b η,t) and that y> a +η in,t η ) so that b ay is negative in,t η ) and for t,t η )andt t,t η ) we find that t y s) t t t b ay s) ds = 1 a b yt) 1 ab a b yt) 1 z dz = 1 ab {arcoth a ) b yt) a ) } arcoth b yt ), leading to a ) abt t ) + arcoth b yt a ) ) arcoth b yt) and hence to b abt yt) a coth a ) ) b ) t )+arcoth b yt ) a coth abt t ). Using { that t,t)andη } > were arbitrary, we conclude that yt) b max a coth b b abt), a = a coth abt). The following statement essentially goes back to an observation made in [18].
23 NoDEA A generalized solution concet for the Keller Segel system Page 3 of Lemma 8.4. Let η >. Then there exists C > such that every ositive function ϕ C 1 ) fulfilling {x ; ϕx) >δ} >η for some δ> satisfies ϕ { ϕ C ln δ or ln ϕ} δ ϕ <. Proof. This directly follows from the ineuality rovided by [18, Lemma 4.3] uon suaring. A reuirement on convexity of the domain, as additionally made there in order to allow for an alication of the Poincaré ineuality from [1, Cor 9.1.4] in the roof, can actually be removed by relacing the latter with Lemma 9.1. We can now ass to our derivation of lower bounds for ln u in the following form. Lemma 8.5. There exists T>such that for every t,t), ln u,t) >. inf,1) Proof. We let M t) :=su τ [,t] u,τ) L ) for t, ) and, 1), and ick >n.fromknownl L estimates for the Neumann heat semigrou [, Lemma 1.3 iii) and ii)] we obtain c 1 > andc > such that v,t) L ) c 1 v W 1, ) + c t 1 + t s) 1 )e t s) u L ) ds c M t)) for all t, ), { } where c 3 = max c 1 v W 1, ),c 1 + τ 1 )e τ dτ. Invoking further semigrou estimates in the form of [, Lemma 1.3 iv)]) we find c 4 > such that t u,t) L ) u L ) + χc t s) 1 n ) u v,s)1 + u,s)) v,s) ds u L ) L ) + χc 4 e t c M t)) u 1 inf v L 1 ) M t)) 1 t 1 + τ 1 n )dτ for all t, ) and all, 1), because u,t) 1+u,t) u,t) L L ) ) u 1 L 1 ) u,t) 1 L ) for all t, ), so that in conclusion we can find c 5 > fulfilling M t) u L ) +c 5t + t 1 n )e t 1+M t))m t)) 1 for all t, ),
24 49 Page 4 of 33 J. Lankeit and M. Winkler NoDEA and hence by the fact that for all a, b [, ), γ, 1) we can achieve that su{x [, ); x a + bx γ } a 1 γ + b 1 1 γ M t) u L ) c + 5 t + t 1 n )e t 1 + M t)) } If we let T := su {t, ) :M t) u L ) +1, then certainly { T > min 1, 1 ) 1 } c 5 e + u ) 1 n L ) =: T. In conclusion, this means that for all, 1), u,t) L ) u L ) +1=:M and v,t) W 1, ) c M) for all t,t). In articular with δ := 1 u and η := 1 M u we have {u,t) δ} η for every t,t) and each, 1), because u = u,t)= u,t) {u,t) δ} + u,t) M {u,t) δ} + δ = M {u,t) δ} + 1 u {u,t)<δ} and therefore η = 1 u {u,t) δ}. M From Lemma 8.4 we hence obtain c 6 > such that d dt δ ln u,t) ) = for every t,t)atwhich ln claim. u,t) u,t) + χ ) u )u,t)v,t) u,t) v,t) 1 u,t) u + χ v,t),t) v,t) c ) 6 δ ln + χ e T c 31 + M) u,t) inf v ) δ u,t) >. Lemma 8.3 hence roves the This enables us to verify the ositivity reuirements from Definition.4 without any assumtions on the initial data beyond 1.7). Lemma 8.6. The functions u, v obtained in Lemma 8.1 satisfy v>, u> a.e. in, ) and u v > a.e. on, ).
25 NoDEA A generalized solution concet for the Keller Segel system Page 5 of Proof. According to Lemma 3.3 and 8.3), essinf x vx, t) > inf v e t for any t>, and 8.) and 8.5) together with 8.1) or8.3), resectively, show that u and v can be evaluated on, ) in the sense of traces. For the roof of the remaining ositivity roerties u> a.e. in, ) andu> a.e. on, ), we intend to rove ln u L loc, ); W 1, )), 8.7) which entails ln u L loc, )) and, due to the embedding W 1, ) L ), also ln u L loc, )), roving ositivity of u a.e. in the resective sets. By Lemmas 3.3 and 3., [ ] d ln u χ u 1 ln v = dt u + χ u v 1 + u ) u v χ v v + χ χ 1 u u + χ e t u inf v u v χ v + χ for any t >. Now icking τ >, according to Lemma 8.5 we can find τ,τ)andc 1 > such that ln u,τ ) >. 8.8) inf,1) For any fixed T>τwethen obtain ln u,t)+ 1 t τ + χ ln v,t) v,τ ) + χ T + ln u χ inf v e T v ln u,τ ) u 8.9) for t τ,t), where according to 8.8) and by Lemmas 3.3 and 3.3) the righthand side is bounded indeendently of. We invoke the Poincaré ineuality to find c 1 > such that ϕ W 1, ) c 1 ϕ L ) + ϕ L 1 )) for all ϕ W 1, ), and since the elementary estimate ln s s ln s valid for all s>, Lemmas 3. and 8.9) rovidec > such that ln u,t) c for all t τ,t), we conclude that for every τ>andt>τthere exists c 3 > such that for all, 1) we have ln u L τ,t);w 1, )) c 3, which by a weak comactness argument immediately results in 8.7). In order to demonstrate that u, v) satisfies.3), we reare the following.
26 49 Page 6 of 33 J. Lankeit and M. Winkler NoDEA Lemma 8.7. Let φ ),1) C [, )) L, )) be such that su,1) φ L, )) < 8.1) and that there exists φ C [, )) such that φ φ in L loc[, )) as. 8.11) Then given χ> such that 1.8) holds and taking u, v and j ) j N, 1) from Lemma 8.1, for all, min{1, 1 χ }), any ), + )) and each T>we have φ u )v u φu)v u in L,T)) as = j. 8.1) Proof. Let T >. Then since from 8.1) we know that there exists c 1 > such that φ s) c 1 for all s>and, 1) 8.13) and hence T φ u )v u T c1 v u for all, 1), writing w := φ u )v u,, 1), we infer from 4.6) thatw ),1) is bounded in L,T)) and hence relatively comact therein with resect to the weak toology. According to a standard argument, in order to verify 8.1) it is thus sufficient to make sure that whenever jk ) k N is a subseuence of j ) j N with the roerty that w w in L,T)) as = jk 8.14) with some w L,T)), we have w = φu)v u a.e. in,t). To achieve this, we note that due to <1and 8.3), v v in L,T)). By 8.13) and the ointwise convergence asserted in 8.1) together with Lebesgue s dominated convergence theorem, we therefore even have φ u )v φu)v in L,T)). Furthermore taking into account 8.), we see that φ u )v u φu)v u in L 1,T)), which ensures w = φu)v u a.e. in,t). We can now make sure that indeed the obtained air u, v) has all the roerties reuired in Definition.1. Lemma 8.8. Suose that 1.8) holds, and let, 1) and, 1) be such that < 1 χ and ), + )). Then the functions u and v constructed in Lemma 8.1 form a global weak, )suersolution of 1.1) in the framework of Definition.1. Proof. Writing φ 1) s) := 1 )χ 1+s + χ 1+s + 1 )
27 NoDEA A generalized solution concet for the Keller Segel system Page 7 of and as well as φ ) s) := χ ) 1+s φ 3) s) := 41 ) 4 1 ) χ 1+s) +1 ) χ 1+s and φ 4) [1 )s ]χ s) := 1 + s) for, 1) and s, we first observe that φ k) {1,, 3, 4} with φ 1) 1 )χ + c 1 :=, χ 1 ) φ ) c := χ +1, φ 3) c 3 := φ 4) c 4 := χ 41 ) 4 1 ) χ [χ 1 )] is welldefined for k and 8.16) in L loc [, )) as. Using these auxiliary functions, given T>wenow invoke Lemma 4.3 to fix c 5 > andc 6 > such that T φ 1) u )v u φ ) u )u v c5 8.17) and T u +1 v 1 c ) for all, 1). Next, since evidently φ 1) ),1),φ 3) ),1) and φ 4) ),1) are bounded in L, )), three alications of Lemma 8.7 on the basis of 8.16) show that φ 1) u )v u c 1 v u in L,T)) 8.19) and φ 3) u )v u c 3 v u in L,T)) 8.) as well as φ 4) u )v u c 4 v u in L,T)) 8.1)
28 49 Page 8 of 33 J. Lankeit and M. Winkler NoDEA as = j. We moreover observe that 8.1) imlies that c φ ) u ) c a.e. in,t)as = j, so that since T φ ) u )u c T u = c T u for all, 1) by 3.), the Vitali convergence theorem ensures that φ ) u )u c u in L,T)) as = j due to the fact that >. Since v v in L,T)) as = j according to Lemma 8.1, we thus infer that φ ) u )u v c u v in L 1,T)) as = j. But since φ 1) u )v u φ ) u )u v ),1) is relatively comact with resect to the weak toology in L,T)) by 8.17), together with 8.19) the latter guarantees that in fact φ ) u )u v c u v in L,T) 8.) as = j. Along with e.g. 8.), this articularly asserts the regularity reuirements in.), whereas those in.1) result from 8.18) and the fact that by Young s ineuality, 3.) and Lemma 3.4 there exists c 7 > such that T 1 T T u v + u + v c 7 for all, 1) and hence, as = j, u v u v in L 1,T)) 8.3) thanks to Lemma 8.1 and again the Vitali convergence theorem, because + <+ + ) < 1. Positivity roerties of u, v and u v, and the validity of.6) are ensured by Lemma 8.6 and 8., resectively. Now for the verification of.3) we let ϕ C [, )) be such that ϕ ν =on, ) and fix T>such that ϕ in T, ). Then an alication of Lemma 4.1 shows that T + φ 3) T φ 1) T = + T u )v u ϕ φ 4) u )u v u ϕ u )v u φ ) u )u v T ϕ + u vϕ t u v ϕ, ) T 1 u +1 χ ) u 1 + u ) vδϕ v 1 u v ϕ for all, 1), 8.4) ϕ
29 NoDEA A generalized solution concet for the Keller Segel system Page 9 of where again emloying 8.3) we see that T u vϕ t T u v ϕ t 8.5) and T T u vϕ u v ϕ 8.6) as well as T χ ) 1 u 1 + u ) vδϕ 1 χ ) T u v Δϕ 8.7) as = j, the derivation of the latter additionally relying on an alication of the dominated convergence theorem. Since 8.3) together with Lemma 8.1 clearly warrants that u v u v in L,T)) and hence T φ 4) u )u v u ϕ = T T T = c 4 u v ) ) u v u v u ϕ φ 4) u )v u ) ϕ ) c 4 v u ϕ as = j, in view of Fatou s lemma and a standard argument based on lower semicontinuity of the norm in L,T)) with resect to weak convergence it follows from 8.4), 8.), 8.19), 8.) and 8.5) 8.7) that T c 3 v u ϕ+ T u v ϕ t T c 4 + T T c 1 v u u v ϕ, ) u v u ϕ 1 χ u v ϕ, c u v ϕ+ ) T u v Δϕ which is euivalent to.3) and thus comletes the roof. Our main result thereby becomes evident. T u +1 v 1 ϕ Proof of Theorem 1.1. We only need to combine Lemma 8. with Lemma 8.8. Acknowledgements The authors acknowledge suort of the DFG within the roject Analysis of chemotactic crossdiffusion in comlex frameworks.
30 49 Page 3 of 33 J. Lankeit and M. Winkler NoDEA 9. Aendix: A Poincaré ineuality in nonconvex domains Our roof of Lemma 8.5 relies on Lemma 8.4, which in its original formulation in [18, Lemma 4.] reuires convexity of the domain due to the version of Poincaré s ineuality [1, Corollary 9.1.4] used. In this aendix we state this Poincaré ineuality without any such convexity condition, and since we could not find any reference to this in the literature, we briefly outline an argument. 1 Here and in the following, by u X we denote the average X ux)dx for X u L 1 ) and any measurable set X with ositive measure. Lemma 9.1. Let R n be a bounded with smooth boundary, and let δ> and [1, ). Then there exists C = C,δ,) with the roerty that for all u W 1, ), C,δ,) ) 1 u u B holds for any measurable set B with B = δ. A derivation of this can be based on the following. Du ) 1 Lemma 9.. Let R n be a bounded domain with smooth boundary, and let δ>. Then there exists C>such that for every measurable set B with B = δ, andforeachu W 1,1 ) we have for almost every x. ux) u B C Duz) dz 9.1) x z n 1 Proof. We follow the roof of [7, Theorem 1], where 9.1) is shown for B =, and indicate necessary changes. With B being a certain ball in, defined as in the roof of [7, Theorem 1], in [7, 14)] it is shown that there is c 1 > such that for every u W 1,1 ) and almost every x the estimate uz) ux) u B c 1 dz 9.) x z n 1 holds, whose roof relies on a Poincaré ineuality on balls that takes into account the deendence of the constant on the radius and on the existence of a chain of balls connecting B with x that allows for certain estimates indeendently of x cf. [7,. 119]). Whereas the first summand in the righthand side of ux) u B ux) u B + u B u B 9.3) is immediately covered by 9.), as to the second we observe that, again by 9.), u B u B 1 B c 1 B B B u B uy) dy uz) dzdy y z n 1
31 NoDEA A generalized solution concet for the Keller Segel system Page 31 of c 1 1 uz) dydz. 9.4) B B y z n 1 In estimating 1 B y z dy we emloy the fact that with some c n 1 = cn), for all z R n and any measurable E R n, dy E y z c n 1 E 1 n [7, 13)]), because if Ẽ is a ball centered in z with E = Ẽ and radius R = c n) E 1 n, dy y z n 1 dy R y z n 1 = r n 1 1 r n 1 dr = c Ẽ 1 n. E Ẽ diam )n 1 x z n 1 We moreover use that 1 for any x, z. With these observations, 9.4) turns into u B u B c 1c B B 1 n uz) dz c 1 c B 1 n 1 diam ) n 1 uz) dz. x z n 1 9.5) Noting that B 1 n 1 = δ 1 n 1 and combining 9.) and 9.5) with 9.3) roves 9.1). Proof of Lemma 9.1. For convex domains, this is exactly Corollary of [1], which follows from [1, Lemma 9.1.3] and [1, Lemma 9.1.], the latter of which a continuity roerty of the Riesz otential oerator) oses no convexity condition on. As relacement of the former, in the case of general we now rather rely on Lemma 9.. Remark 9.3. In Lemma 9. and hence in Lemma 9.1), for the domain it is actually sufficient to be bounded and) a John domain, instead of having smooth boundary. In articular, any bounded domain satisfying the interior cone condition is admissible in these lemmata. For details concerning this, we once more refer the reader to [7]. Remark 9.4. With Lemma 9.1, it is also ossible to remove the convexity condition on the domain in [18]. References [1] Bellomo, N., Bellouuid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller Segel models of attern formation in biological tissues. Math. Models Methods Al. Sci. 5, ) [] Biler, P.: Global solutions to some arabolic ellitic systems of chemotaxis. Adv. Math. Sci. Al. 91), ) [3] Di Perna, R.J., Lions, P.L.: On the Cauchy roblem for Boltzmann euations: global existence and weak stability. Ann. Math. 13, ) [4] Fujie, K.: Boundedness in a fully arabolic chemotaxis system with singular sensitivity. J. Math. Anal. Al. 44, )
32 49 Page 3 of 33 J. Lankeit and M. Winkler NoDEA [5] Fujie, K., Senba, T.: Global existence and boundedness in a arabolic ellitic Keller Segel system with general sensitivity. Discret. Contin. Dyn. Syst. B 1, ) [6] Fujie, K., Senba, T.: Global existence and boundedness of radial solutions to a two dimensional fully arabolic chemotaxis system with general sensitivity. Nonlinearity 9, ) [7] Haj lasz, P.: Sobolev ineualities, truncation method, and John domains. In: Paers on analysis, Re. Univ. Jyväskylä De. Math. Stat. vol. 83, ) [8] Herrero, M.A., Velázuez, J.J.L.: A blowu mechanism for a chemotaxis model. Ann. Sc. Norm. Su. Pisa Cl. Sci. 4, ) [9] Hillen, T., Painter, K.J.: A user s guide to PDE models for chemotaxis. J. Math. Biol. 581), ) [1] Jost, J.: Partial Differential Euations. Graduate Texts in Mathematics, nd edn. Sringer, New York 7) [11] Lankeit, J.: A new aroach toward boundedness in a twodimensional arabolic chemotaxis system with singular sensitivity. Math. Methods Al. Sci. 39, ) [1] Luckhaus, S., Sugiyama, Y., Velázuez, J.J.L.: Measure valued solutions of the D Keller Segel system. Arch. Ration. Mech. Anal. 6, ) [13] Mizukami, M., Yokota, T.: A unified method for boundedness in fully arabolic chemotaxis systems with signaldeendent sensitivity. Math. Nachr. 17). doi:1.1/mana [14] Nagai, T., Senba, T.: Global existence and blowu of radial solutions to a arabolic ellitic system of chemotaxis. Adv. Math. Sci. Al. 8, ) [15] Rosen, G.: Steadystate distribution of bacteria chemotactic toward oxygen. Bull. Math. Biol. 4, ) [16] Simon, J.: Comact sets in the sace L,T; B). Ann. Mat. Pura Al. 146, ) [17] Stinner, C., Winkler, M.: Global weak solutions in a chemotaxis system with large singular sensitivity. Nonlinear Anal. Real World Al 1, ) [18] Tao, Y., Winkler, M.: Persistence of mass in a chemotaxis system with logistic source. J. Differ. E. 59, ) [19] Tello, J.I., Winkler, M.: Reduction of critical mass in a chemotaxis system by external alication of a chemoattractant. Ann. Sc. Norm. Su. Pisa Cl. Sci. 1, ) [] Winkler, M.: Aggregation vs. global diffusive behavior in the higherdimensional Keller Segel model. J. Differ. E. 48, )
33 NoDEA A generalized solution concet for the Keller Segel system Page 33 of [1] Winkler, M.: Global solutions in a fully arabolic chemotaxis system with singular sensitivity. Math. Methods Al. Sci. 34, ) [] Winkler, M.: Finitetime blowu in the higherdimensional arabolic arabolic Keller Segel system. J. Math. Pures Al. 1, ). arxiv: v1 [3] Winkler, M.: Largedata global generalized solutions in a chemotaxis system with tensorvalued sensitivities. SIAM J. Math. Anal. 47, ) [4] Zhao, X., Zheng, S.: Global boundedness of solutions in a arabolic arabolic chemotaxis system with singular sensitivity. J. Math. Anal. Al. 443, ) Johannes Lankeit and Michael Winkler Institut für Mathematik Universität Paderborn 3398 Paderborn Germany Johannes Lankeit Received: 5 January 17. Acceted: 6 July 17.
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