A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system
AA double critical mass phenomenon in ano-flux-Dirichlet Keller-Segel system
Jan Fuhrmann ∗ Jülich Supercomputing Centre, Forschungszentrum Jülich52428 Jülich, Germanyand Frankfurt Institute for Advanced Studies60438 Frankfurt/Main, Germany
Johannes Lankeit † Leibniz Universität Hannover, Institut für Angewandte Mathematik,Welfengarten 1, 30167 Hannover, Germany
Michael Winkler ‡ Institut für Mathematik, Universität Paderborn,33098 Paderborn, Germany
Abstract
Derived from a biophysical model for the motion of a crawling cell, the evolution system (cid:40) u t = ∆ u − ∇ · ( u ∇ v ) , v − kv + u, ( (cid:63) ) is investigated in a finite domain Ω ⊂ R n , n ≥ , with k ≥ . Whereas a comprehensive literatureis available for cases in which ( (cid:63) ) describes chemotaxis-driven population dynamics and henceis accompanied by homogeneous Neumann-type boundary conditions for both components, thepresently considered modeling context, besides yet requiring the flux ∂ ν u − u∂ ν n to vanish on ∂ Ω ,inherently involves homogeneous Dirichlet boundary conditions for the attractant v , which in thecurrent setting corresponds to the cell’s cytoskeleton being free of pressure at the boundary.This modification in the boundary setting is shown to go along with a substantial change withrespect to the potential to support the emergence of singular structures: It is, inter alia, revealedthat in contexts of radial solutions in balls there exist two critical mass levels, distinct from eachother whenever k > or n ≥ , that separate ranges within which ( i ) all solutions are global in timeand remain bounded, ( ii ) both global bounded and exploding solutions exist, or ( iii ) all nontrivialsolutions blow up in finite time. While critical mass phenomena distinguishing between regimes oftype ( i ) and ( ii ) belong to the well-understood characteristics of ( (cid:63) ) when posed under classicalno-flux boundary conditions in planar domains, the discovery of a distinct secondary critical masslevel related to the occurrence of ( iii ) seems to have no nearby precedent.In the planar case with the domain being a disk, the analytical results are supplemented with somenumerical illustrations, and it is discussed how the findings can be interpreted biophysically for thesituation of a cell on a flat substrate. Key words:
Keller-Segel; blow-up; critical mass
MSC: ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ m a t h . A P ] J a n Introduction
A Keller-Segel type model for crawling keratocytes.
This study is concerned with the cross-diffusion problem u t = ∆ u − ∇ · ( u ∇ v ) , x ∈ Ω , t > , v − kv + u, x ∈ Ω , t > ,u ( x,
0) = u ( x ) , x ∈ Ω , (1.1)in a bounded domain Ω ⊂ R n , n ≥ . During the past decades, this system has received noticeableinterest when used as a parabolic-elliptic simplification of the celebrated Keller-Segel model to describecollective behavior in microbial populations with movement chemotactically biased by a chemical signal,and hence typically found accompanied by no-flux boundary conditions in the literature ([18], [15], [21]).In contrast to this, the context to be considered in the present paper necessitates to supplement (1.1)by the requirements ∂u∂ν − u ∂v∂ν = v = 0 , x ∈ ∂ Ω , (1.2)on the boundary of the domain Ω ⊂ R n , as intrinsically linked to the role which, quite independentlyof the above, (1.1) plays when derived from a biomechanical model for a single crawling keratocyte,or rather a keratocyte fragment, that has been introduced in [2] for space dimension n = 2 . Thesefragments are similar to lamellipodia, i.e., very flat structures, and can in good approxomation bedescribed as two-dimensional entities. The computational model presented in [2] was reduced and an-alyzed in [4], and similar models in one space dimension have been investigated in, e.g., [33]. From thephysical model in [2], a reduced free boundary problem has been derived in [4] by combining bulk andshear components of the stress in the actin gel in a phenomenological way, allowing for the stress tensorto be represented as a scalar multiple of the identity matrix. This step used the fact that cytoskeletongels are rather unusual viscoelastic fluids with the stress not being shear dominated. This led to a freeboundary problem for two variables, in our context named v for the stress in the cytoskeleton and u for the density of myosin motor proteins. The latter actively generate stress by binding to and pullingon the actin filaments constituting the cytoskeleton meshwork.The first equation in (1.1) is thus interpreted as a diffusion-advection equation for the concentrationof myosin molecules which are either freely diffusing inside the cytoplasm or are bound to the actingel and hence convected with the velocity ∇ v which is the divergence of the stress tensor, v I . Thesecond equation describes the force balance in the actin gel with the term u representing the activelygenerated stress due to the myosin motors, which is assumed to be proportional to the density ofthese motors. The term − kv models the dissipation of stress via traction with the substrate to whichthe actin gel is linked by adhesion molecules. The distribution of these adhesions is supposed to beuniform and constant in time for a resting cell. Moreover, the second equation being elliptic assumesthat stresses equilibrate on a much faster time scale than the motion of the actin gel, indicated by verylow Deborah numbers reported for moving, let alone resting cells [34]. This simply means that the gelbehaves more like a viscous fluid than an elastic solid on the relevant time scale. The parameter k is the typical stress stored in the actin gel relative to the typical stress generated by myosin motors.The second parameter present in the model is the size of the domain Ω which is measured in multiplesof √ k L , where L is the viscous length of the actin gel which describes how far the locally generatedstress acts through the network before being dissipated away. It is defined as square root of the ratio2f the viscosity and the traction coefficient.Whereas both [2] and [4] were interested in traveling wave solutions to their respective free boundaryproblems to describe steady cell motion, we will focus here on the behavior of steady states and thepossibility of finite time blow up. Steady state solutions clearly correspond to a resting cell althoughwe should mention that stationarity in (1.1) does not imply that there is no motion inside the cell;recall that the velocity of the actin gel is ∇ v . More strikingly, solutions blowing up in finite time areinterpreted as the cell being physically disrupted by too much contractile activity of myosin motorsas represented by a large total myosin mass m = (cid:82) Ω u which is obviously a conserved quantity for(1.1)-(1.2). While the bifurcation from rest to motion at subcritical values of m described in [4] refersto a dynamic instability of the free boundary problem modeling a potentially motile cell switching fromrest to directed motion, blow up of solutions for large m in system (1.1) with fixed boundary relatesto the observed disruption of immobile cells upon variations of myosin activity or adhesion strength ashas been seen experimentally ([1], cf. e.g. [35] for mechanism of fragmentation of actin filaments bymyosin generated forces). Mechanical breakdown due to enhanced myosin activity and concomitantconcentration of myosin is also associated with physiological processes such as programmed cell death,or apoptosis, as described in [11].To rule out possible issues of self intersection of the moving boundary as mechanism for the break downof solutions we fixed the shape of the domain Ω occupied by the cell. Physically, this may be achievedby letting the cell sit on a particularly sticky substrate or by providing it with an adhesive patch ofsubstrate of a given shape Ω and making the surrounding region, viz. R \ Ω , particularly hostile bycoating with adverse substances or no coating at all. Keeping the stress-free boundary condition v = 0 and the no-flux condition for the myosin molecules from the original model ([2]), we finally arrive at(1.1)-(1.2) which differs from the classical parabolic-elliptic Keller-Segel system most significantly inthe boundary conditions. The peculiar condition v = 0 on ∂ Ω arises from the fact that myosin motorsat the boundary are not supposed to generate stress since there is nothing outside the cell to be pulledagainst. There is no contradiction in the cytoskeleton gel’s velocity being different from zero at theboundary. In fact, in a resting cell, actin is polymerized at the boundary, leading on averaege to aradial expansion of the cytoskeleton, which is counteracted by the actin gel constantly moving towardthe center where the actin filaments are depolymerized. This retrograde flow means that the gel movesaway from the boundary at non-zero velocity. Detecting explosion-related dichotomies in Keller-Segel systems.
Over the past decades,significant effort in the analysis of chemotaxis problems has been directed towards excluding (e.g. [30])or detecting blow-up ([16, 14, 29]) and the study of additional qualitative properties (e.g. [36, 26, 38,8, 9]) in (1.1) and related variants, e.g. further simplified like in [16], or rather fully parabolic andhence more complex. Among the apparently most striking characteristics of such Keller-Segel systems,the literature has identified situations in which the occurrence of blow-up depends on the size of theconserved total mass (cid:82) Ω u in a crucial manner. Specifically, when posed along with homogeneous Neu-mann boundary conditions for both components in planar bounded domains Ω , (1.1) with arbitrary k > is known to exhibit a sharp and well-understood critical mass phenomenon in the sense thatwhenever ≤ u is sufficiently regular with (cid:82) Ω u < π , an associated initial-boundary value problemwith u | t =0 = u admits a globally defined bounded solution, whereas for any m > π one can findsmooth initial data with (cid:82) Ω u = m such that the corresponding solution blows up in finite time ([29]);a restriction to radially symmetric solutions in balls increases this separating mass level to the value3 π ([29]). Similar dichotomies have been detected in Neumann problems for further parabolic-ellipticand for fully parabolic relatives of (1.1) [27, 7, 14, 30]; cf. also [9, 39] for some related findings forCauchy problems on the whole plane Ω = R ). A secondary critical mass phenomenon enforced by Dirichlet conditions for v . Main re-sults. The present study will now reveal that when considered along with the boundary conditionsin (1.2), the system (1.1) may gain a further dynamical facet that is linked to the presence of a sec-ondary, and apparently yet undiscovered, critical mass phenomenon.To appropriately formulate and embed our findings in this regard, let us first summarize some fun-damental properties thereof, as can readily be verified upon straightforward adaptation of argumentsknown from the literature (cf. e.g. [37] for Part i), [37], [29] for Part ii), and [28] for Part iii)):
Theorem A
Let n ≥ and Ω ⊂ R n be a bounded domain with smooth boundary, and let k ≥ .i) If n = 2 and u ∈ C (Ω) is nonnegative with (cid:90) Ω u < π, then (1.1) - (1.2) possesses a global classical solution ( u, v ) which is bounded in the sense that there exists C > such that (cid:107) u ( · , t ) (cid:107) L ∞ (Ω) ≤ C for all t > . (1.3) ii) If n = 2 , then for all m > π there exists some nonnegative u ∈ C (Ω) with (cid:82) Ω u = m such thatthe corresponding solution of (1.1)-(1.2) blows up in finite time in the sense specified in Proposition2.1 below. Here, if Ω = B R (0) with some R > , then u can be chosen to be radially symmetric withrespect to x = 0 .iii) In the case n ≥ and if Ω is star-shaped, for all m > one can find nonnegative u ∈ C (Ω) with (cid:82) Ω u = m , radially symmetric if Ω is a ball, such that the solution of (1.1)-(1.2) blows up. As a direct consequence for the general, not necessarily radial case, this implies the following essentiallywell-known statement identifying the number π as a k -independent critical mass in (1.1)-(1.2) when n = 2 , whereas if n ≥ then a corresponding critical mass phenomenon seems absent: Corollary B
Let n ≥ , Ω ⊂ R n be a bounded domain with smooth boundary, and k ≥ . Then M (cid:63) (Ω , k ) := inf (cid:26) m > (cid:12)(cid:12)(cid:12)(cid:12) There exists some nonnegative u ∈ C (Ω) with (cid:90) Ω u = m such that the solution of (1.1) - (1.2) blows up (cid:27) (1.4) is well-defined and satisfies M (cid:63) (Ω , k ) = (cid:40) π if n = 2 , if n ≥ . (1.5)Now the first of our main results identifies a secondary mass threshold which, as can already be statedat this stage, at least in the case n ≥ indeed differs from the value M (cid:63) (Ω , k ) = 0 .4 heorem 1.1 Let n ≥ and Ω ⊂ R n be a bounded domain with smooth boundary which is strictlystar-shaped with respect to ∈ Ω in the sense that γ := inf x ∈ ∂ Ω x · ν ( x ) > . (1.6) Then for all k ≥ , M (cid:63) (Ω , k ) := inf (cid:26) m > (cid:12)(cid:12)(cid:12)(cid:12) For all nonnegative u ∈ C (Ω) with (cid:90) Ω u = m, the solution of (1.1) - (1.2) blows up (cid:27) (1.7) is well-defined and finite with π ≤ M (cid:63) (Ω , k ) ≤ | ∂ Ω | γ + 2 k | Ω | if n = 2 (1.8) and < M (cid:63) (Ω , k ) ≤ n | ∂ Ω | γ + 2 k | Ω | if n ≥ . (1.9)In two-dimensional domains, however, the situation will turn out to be more subtle, involving a crucialqualitative dependence on whether or not the parameter k is positive. As a first step toward revealingthis, let us concentrate on the special situation when Ω is a ball, in which the above enables us torather explicitly estimate this secondary critical mass, and to thereby detect, in particular, coincidenceof both mass thresholds in the planar case when k = 0 in such geometries. Corollary 1.2
Let n ≥ , R > and Ω = B R (0) ⊂ R n . Then for all k ≥ , π = M (cid:63) ( B R (0) , k ) ≤ M (cid:63) ( B R (0) , k ) ≤ π + 2 kπR if n = 2 and M (cid:63) ( B R (0) , k ) < M (cid:63) ( B R (0) , k ) ≤ ω n R n n + 2 nkω n R n − if n ≥ , where ω n denotes the ( n − -dimensional measure of the unit sphere ∂B (0) . In particular, for k = 0 , M (cid:63) ( B R (0) ,
0) = M (cid:63) ( B R (0) ,
0) = 8 π for all R > if n = 2 . On further specializing the setup by resorting henceforth to radially symmetric solutions in balls
Ω = B R (0) ⊂ R n , n ≥ , R > , emanating from initial data in the space C rad (Ω) := { ϕ ∈ C (Ω) | ϕ is radially symmetric with respect to x = 0 } , we can rephrase part of Theorem A as follows. Corollary C
Let n ≥ , R > , and Ω = B R (0) ⊂ R n , and let k ≥ . Then m (cid:63) (Ω , k ) := inf (cid:26) m > (cid:12)(cid:12)(cid:12)(cid:12) There exists some nonnegative u ∈ C rad (Ω) with (cid:90) Ω u = m such that the solution of (1.1)-(1.2) blows up (cid:27) (1.10)5 s well-defined with m (cid:63) ( n, R, k ) = M (cid:63) ( B R (0) , k ) = (cid:40) π if n = 2 , if n ≥ . Now the second of our main results makes sure that a corresponding secondary mass threshold, definedin the spirit of Theorem 1.1, plays the role of a genuinely new critical mass for radial solutions not onlywhen n ≥ and k ≥ , but also when n = 2 and k > is arbitrary, thus complementing the outcomeof Corollary 1.2 in quite a sharp manner: Theorem 1.3
Let n ≥ , R > , and Ω = B R (0) ⊂ R n . Then for all k ≥ , m (cid:63) (Ω , k ) := inf (cid:26) m > (cid:12)(cid:12)(cid:12)(cid:12) For all nonnegative u ∈ C rad (Ω) with (cid:90) Ω u = m, the solution of (1.1)-(1.2) blows up (cid:27) (1.11) satisfies M (cid:63) ( B R (0) , k ) = m (cid:63) ( n, R, k ) ≤ m (cid:63) ( n, R, k ) ≤ M (cid:63) ( B R (0) , k ) . (1.12) Moreover, m (cid:63) (2 , R,
0) = m (cid:63) (2 , R,
0) = 8 π, (1.13) but π = m (cid:63) (2 , R, k ) < m (cid:63) (2 , R, k ) for all k > , (1.14) and apart from that, m (cid:63) ( n, R, k ) < m (cid:63) ( n, R, k ) for all k ≥ if n ≥ . (1.15)For the special case k = 0 , the finiteness of M (cid:63) (in n -dimensional balls, n ≥ , but for possiblynonradial u ) was already observed in [5] and that of m (cid:63) in [6]. It is remarkable that the values of m (cid:63) and m (cid:63) , which coincide for k = 0 and n = 2 , differ for positive k . In this sense linear signal degradationaffects the blow-up affinity of (1.1) and makes it possible to find two separate critical masses in thesame system. Let us first adapt an essentially well-established contraction-based reasoning to see that similar toits no-flux type relative, the problem (1.1)-(1.2) admits local smooth solutions which can cease toexist within finite time only when becoming unbounded with respect to the L ∞ norm in their firstcomponent. Proposition 2.1
Let n ≥ and Ω ⊂ R n be a bounded domain with smooth boundary, let k ≥ , andsuppose that u ∈ C (Ω) is nonnegative. Then there exist T max ∈ (0 , ∞ ] and a uniquely determinedpair ( u, v ) of nonnegative functions (cid:40) u ∈ C (Ω × [0 , T max )) ∩ C , (Ω × (0 , T max )) and v ∈ C , (Ω × (0 , T max )) (2.1)6 hich solve (1.1)-(1.2) classically in Ω × (0 , T max ) , and which are such thatif T max < ∞ , then ( u, v ) blows up at t = T max , (2.2) where we say that ( u, v ) blows up at t = T max if and only if lim sup t (cid:37) T max (cid:107) u ( · , t ) (cid:107) L ∞ (Ω) = ∞ .Furthermore, (cid:107) u ( · , t ) (cid:107) L (Ω) = (cid:90) Ω u for all t ∈ (0 , T max ) . (2.3) Proof.
We fix some p > n and let M := (cid:107) u (cid:107) L p (Ω) + 1 . With T > to be determined later, we set X M,T := (cid:8) u ∈ C ([0 , T ]; L p (Ω)) | (cid:107) u (cid:107) L ∞ ((0 ,T ); L p (Ω)) ≤ M, u ( · ,
0) = u (cid:9) . Given any u ∈ X T := C ([0 , T ]; L p (Ω)) , for t ∈ (0 , T ) letting v ( · , t ) ∈ W , (Ω) denote the weaksolution of the Dirichlet problem for v ( · , t ) − kv ( · , t ) + u ( · , t ) we obtain a function v = v ( u ) ∈ C ([0 , T ]; W ,p (Ω) ∩ W ,p (Ω)) and note that due to our choice of p , elliptic regularity theory (see e.g.[25, Thm. 37,I]) and a Sobolev embedding, we can find c > such that (cid:107)∇ v ( u ) (cid:107) C ([0 ,T ]; L ∞ (Ω)) ≤ c (cid:107) u (cid:107) C ([0 ,T ]; L p (Ω)) for all u ∈ X T . According to [23, Thm. VI.39], for each v ( u ) , u ∈ X M,T , the problem u t = ∇ · ( ∇ u − u ∇ v ( u )) in Ω × (0 , T ) , ( ∇ u − u ∇ v ( u )) · ν = 0 on ∂ Ω × (0 , T ) , u ( · ,
0) = u in Ω , has a unique solution u ∈ V = (cid:8) u ∈ L ∞ ((0 , T ); L (Ω)) | ∇ u ∈ L (Ω × (0 , T )) (cid:9) which is nonnegativeand bounded by some c ( M ) in Ω × [0 , T ] ([23, Thm. VI.40]) and Hölder-continuous in Ω × (0 , T ) ([31,Thm. 1.3 and Remark 1.3]). We denote this solution by Φ( u ) , thus defining a mapping Φ : X M,T → X T .For arbitrary t ∈ (0 , T ) , h ∈ (0 , T − t ) , h ∈ (0 , T − t − h ) , we let ψ ≡ on [0 , t ) , ψ ≡ on ( t + h , T ) and linearly interpolated between t and t + h . Given u , u ∈ X M,T , we then let ϕ ( x, τ ) := 1 h (cid:90) τ + h τ (Φ( u ) − Φ( u )) p − ( x, s ) ds · ψ ( τ ) , x ∈ Ω , τ ∈ (0 , T ) , and use this regularized version of (Φ( u ) − Φ( u )) p − as test function in the difference of the definitionsof weak solutions (cf. [23, p. 136]) for Φ( u ) and Φ( u ) . After successively taking h → and h → and several applications of Young’s inequality we find that with some c > , p (cid:90) Ω ((Φ( u ) − Φ( u ))( t )) p ≤ c (1 + M p ) (cid:90) t (cid:90) Ω (Φ( u ) − Φ( u )) p + c c p ( M ) (cid:90) t (cid:90) Ω |∇ ( v − v ) | p holds for every t ∈ (0 , T ) , u , u ∈ X M,T . Therefore, by a Grönwall-type argument we find that withsome c > , (cid:107) Φ( u )( t ) − Φ( u )( t ) (cid:107) pL p (Ω) ≤ c ( e c t − (cid:107)∇ v −∇ v (cid:107) pL ∞ ((0 ,T ); L p (Ω) ≤ c c ( e c T − (cid:107) u − u (cid:107) pL ∞ ((0 ,T ); L p (Ω) is satisfied for all u , u ∈ X M,T and all t ∈ (0 , T ) . Upon suitably small choice of T , the map Φ : X M,T → X M,T becomes a contraction. Banach’s theorem hence entails the existence of a fixed7oint u = Φ( u ) , unique within X M,T , whose further regularity follows from successive applications of[13, Thm. 6.6], [22, Thm 1.1] and [19, Thm. IV.5.3]. The extensibility criterion (2.2) is a consequenceof the exclusive dependence of T on M , and hence on (cid:107) u (cid:107) L ∞ (Ω) , whereas (2.3) is obvious in view of(1.1) and (1.2). (cid:3) The following observation on boundedness enforced by suitably small data generalizes knowledge onsimilar properties in related Keller-Segel type systems ([10]), and will be of importance in our derivationboth of Theorem 1.1 and of Theorem 1.3. For simplicity in presentation, we confine ourselves here toan argument based on uniform smallness of the initial data, but we at least note that, in fact, at thecost of additional technical expense the norm appearing in (2.4) could be replaced by that in L n (Ω) . Lemma 2.2
Let n ≥ and Ω ⊂ R n be a bounded domain with smooth boundary, and let k ≥ . Thenthere exists δ > with the property that whenever u ∈ C (Ω) is nonnegative with (cid:107) u (cid:107) L ∞ (Ω) < δ, (2.4) the solution ( u, v ) of (1.1) - (1.2) is global and satisfies (1.3) with some C > . Proof.
In view of a known result from parabolic regularity theory ([23, Theorem VI.40]), it issufficient to find δ > such that whenever (2.4) holds, we have sup t ∈ (0 ,T max ) (cid:107)∇ v ( · , t ) (cid:107) L ∞ (Ω) < ∞ . (2.5)To achieve this, we fix any p > n and then invoke standard elliptic regularity ([12, Thm. 19.1]) toobtain c > such that (cid:107)∇ ϕ (cid:107) L ∞ (Ω) ≤ c (cid:107) ∆ ϕ + kϕ (cid:107) L p (Ω) for all ϕ ∈ W ,p (Ω) ∩ W ,p (Ω) , (2.6)while according to a Poincaré inequality ([17, Cor. 9.1.4], [20, Lemma 9.1]) we can pick c > fulfilling (cid:90) Ω ϕ ≤ c (cid:90) Ω |∇ ϕ | for all ϕ ∈ W , (Ω) such that (cid:12)(cid:12) { ϕ = 0 } (cid:12)(cid:12) ≥ | Ω | . (2.7)We then abbreviate c := 2( p − pc , c := 2 p + p +1 p ( p − c and c := 2 p − p ( p − c · (2 p | Ω | − p ) p +2 p , and let δ := min (cid:26)(cid:16) c y c (cid:17) p +2 , (cid:16) y | Ω | (cid:17) p (cid:27) (2.8)with y := (cid:16) c c (cid:17) p , observing that the first restriction in (2.8) guarantees that c y − c y p +2 p − c δ p +2 = c y · (cid:16) − c c y p (cid:17) + c · (cid:16) y − c c δ p +2 (cid:17) ≥ . (2.9)8ow assuming u ∈ C (Ω) to be nonnegative and such that (2.4) holds, we may use that p > n ≥ , andthat writing a := | Ω | (cid:82) Ω u we thus know that ≤ ξ (cid:55)→ ( ξ − a ) p + ∈ C ([0 , ∞ )) , to see relying on (1.1),Young’s inequality, and (2.6) that y ( t ) := (cid:82) Ω ( u ( · , t ) − a ) p + , t ∈ [0 , T max ) , belongs to C ([0 , T max )) ∩ C ((0 , T max )) with y (cid:48) ( t ) + 2( p − p (cid:90) Ω (cid:12)(cid:12)(cid:12) ∇ ( u − a ) p + (cid:12)(cid:12)(cid:12) = − p ( p − (cid:90) Ω ( u − a ) p − |∇ u | + p ( p − (cid:90) Ω u ( u − a ) p − ∇ u · ∇ v ≤ p ( p − (cid:90) Ω u ( u − a ) p − |∇ v | ≤ p ( p − c (cid:107) u (cid:107) L p (Ω) (cid:90) Ω u ( u − a ) p − ≤ p ( p − c (cid:107) u (cid:107) p +2 L p (Ω) for all t ∈ (0 , T max ) . (2.10)Since (2.3) ensures that m = (cid:82) Ω u ≥ a · |{ u > a }| and thus |{ u ≤ a }| ≥ | Ω | for all t ∈ (0 , T max ) according to our choice of a , we may hence utilize (2.7) to estimate p − p (cid:90) Ω (cid:12)(cid:12)(cid:12) ∇ ( u − a ) p + (cid:12)(cid:12)(cid:12) ≥ p − pc (cid:90) Ω ( u − a ) p + = c y ( t ) for all t ∈ (0 , T max ) , whereas noting that a ≤ δ | Ω | by (2.4) we obtain the inequality p ( p − c (cid:107) u (cid:107) p +2 L p (Ω) = p ( p − c · (cid:26) (cid:90) { u ≥ a } u p + (cid:90) { u< a } u p (cid:27) p +2 p ≤ p ( p − c · (cid:26) p (cid:90) { u ≥ a } ( u − a ) p + (2 a ) p | Ω | (cid:27) p +2 p ≤ p ( p − c · (cid:110) p y ( t ) + 2 p | Ω | − p δ p (cid:111) p +2 p ≤ p − p ( p − c · (cid:110) (2 p y ( t )) p +2 p + (2 p | Ω | − p δ p ) p +2 p (cid:111) = c y p +2 p ( t ) + c δ p +2 for all t ∈ (0 , T max ) . Therefore, (2.10) implies that y (cid:48) ( t ) + c y ( t ) − c y p +2 p ( t ) − c δ p +2 ≤ for all t ∈ (0 , T max ) , so that since (2.4) along with the second requirement on δ in (2.8) guarantees that y (0) = (cid:90) Ω ( u − a ) p + ≤ δ p | Ω | ≤ y, a comparison argument on the basis of (2.9) asserts that y ( t ) ≤ y for all t ∈ (0 , T max ) . As thus sup t ∈ (0 ,T max ) (cid:107) u ( · , t ) (cid:107) L p (Ω) is finite, once again relying on (2.6) we obtain (2.5) and conclude as intended. (cid:3) Mass bounds for steady states. Proofs of Theorem 1.1 and of Corol-lary 1.2
Our strategy toward proving Theorem 1.1 will be based on the link between solutions to (1.1)-(1.2)and solutions of the corresponding stationary problem ∇ uu − ∇ v = 0 , x ∈ Ω , ∆ v − kv + u = 0 , x ∈ Ω ,v = 0 , x ∈ ∂ Ω , (3.1)as established through an energy-based argument in the following. Lemma 3.1
Let n ≥ and Ω ⊂ R n be a bounded domain with smooth boundary, and let k ≥ and ≤ u ∈ C (Ω) be such that the solution ( u, v ) of (1.1) - (1.2) from Proposition 2.1 is global in time andbounded in the sense that u ∈ L ∞ (Ω × (0 , ∞ )) . Then there exist ( t j ) j ∈ N ⊂ (1 , ∞ ) and functions u ∞ and v ∞ from C (Ω) such that u ∞ > and v ∞ ≥ in Ω , that t j → ∞ , u ( · , t j ) → u ∞ and v ( · , t j ) → v ∞ in C (Ω) as j → ∞ , and that ( u ∞ , v ∞ ) solves (3.1) with (cid:82) Ω u ∞ = (cid:82) Ω u . Proof.
Using that u > in Ω × (0 , ∞ ) by the strong maximum principle, by means of a standardcomputation we obtain the identity F ( t ) + (cid:90) t D ( τ ) dτ = F (1) for all t > , (3.2)where we have set F ( t ) := (cid:82) Ω |∇ v ( · , t ) | + k (cid:82) Ω v ( · , t ) − (cid:82) Ω u ( · , t ) v ( · , t )+ (cid:82) Ω u ( · , t ) ln u ( · , t ) and D ( t ) := (cid:82) Ω | ∇ (cid:112) u ( · , t ) − (cid:112) u ( · , t ) ∇ v ( · , t ) | for t > . Now since u is bounded and nonnegative, it readily followsthat inf t> F ( t ) > −∞ , by (3.2) meaning that (cid:82) ∞ D ( τ ) dτ is finite, so that we can pick ( t j ) j ∈ N ⊂ (1 , ∞ ) such that t j → ∞ and ∇ (cid:113) u ( · , t j ) − (cid:113) u ( · , t j ) ∇ v ( · , t j ) → a.e. in Ω (3.3)as j → ∞ . Once more due to the boundedness of u , we may next invoke elliptic regularity theory([13]) to see that also ∇ v is bounded in Ω × (0 , ∞ ) , and that thus we may employ a standard result onHölder continuity in parabolic equations under no-flux boundary conditions ([31]) to obtain θ ∈ (0 , such that ( u ( · , t )) t> is bounded in C θ (Ω) . Again by elliptic estimates, this entails boundedness of ( v ( · , t )) t> even in C θ (Ω) , whence the Arzelà–Ascoli theorem provides a subsequence of ( t j ) j ∈ N , forconvenience again denoted by ( t j ) j ∈ N , such that u ( · , t j ) → u ∞ in C θ (Ω) and v ( · , t j ) → v ∞ in C (Ω) as j → ∞ with θ := θ and some nonnegative limit functions u ∞ ∈ C θ (Ω) and v ∞ ∈ C (Ω) for whichusing (1.1) and (1.2) we can easily verify that − ∆ v ∞ + kv ∞ = u ∞ in Ω with v ∞ = 0 on ∂ Ω , and that (cid:82) Ω u ∞ = (cid:82) Ω u . Moreover, along with (3.3) this entails that as j → ∞ we have ∇ (cid:113) u ( · , t j ) → √ u ∞ ∇ v ∞ in C θ (Ω) for some θ ∈ (0 , . Therefore, (cid:112) u ( · , t j ) → √ u ∞ in C θ (Ω) as j → ∞ and ∇√ u ∞ ≡ √ u ∞ ∇ v ∞ in Ω , which in particular means that if we pick x ∈ Ω such that u ∞ ( x ) = (cid:107) u ∞ (cid:107) L ∞ (Ω) ≥ | Ω | (cid:82) Ω u > ,10hen in the connected component C of { x ∈ Ω | u ∞ ( x ) > } containing x we have ∇ (ln u ∞ − v ∞ ) ≡ and hence can find c > such that ln u ∞ ≡ v ∞ + c in C . As ln ξ → −∞ as ξ (cid:38) , however, thisensures that actually C = Ω and that thus u ∞ ≡ e v ∞ + c is positive in Ω and belongs to C (Ω) , andthat also the first equation in (3.1) holds throughout Ω . (cid:3) Now a crucial observation, generalizing and quantitatively sharpening a statement from [4] concen-trating on radial solutions in a disk, rules out large-mass steady states in strictly star-shaped two- orhigher-dimensional domains:
Lemma 3.2
Let n ≥ and Ω ⊂ R n be a bounded domain with smooth boundary such that γ := min x ∈ ∂ Ω x · ν ( x ) > , (3.4) and suppose that k ≥ . Then whenever u ∈ C (Ω) ∩ C (Ω) and v ∈ C (Ω) ∩ C (Ω) are such that u > and v ≥ in Ω and that ( u, v ) solves (3.1) , we necessarily have (cid:90) Ω u ≤ n | ∂ Ω | γ + 2 k | Ω | . (3.5) Proof.
We firstly integrate the second equation in (3.1) to see that (cid:90) Ω u = k (cid:90) Ω v − (cid:90) ∂ Ω ∂v∂ν , (3.6)and in order to estimate both summands on the right-hand side herein appropriately, we next use x ·∇ v as a test function for the second equation in (3.1) to find the identity (cid:90) Ω ∆ v ( x · ∇ v ) − k (cid:90) Ω v ( x · ∇ v ) = − (cid:90) Ω u ( x · ∇ v ) . (3.7)Here following a well-known observation ([32]), twice integrating by parts and using our definition of γ we obtain that (cid:90) Ω ∆ v ( x · ∇ v ) = − (cid:90) Ω |∇ v | − (cid:90) Ω x · ∇|∇ v | + (cid:90) ∂ Ω ∂v∂ν ( x · ∇ v )= n − (cid:90) Ω |∇ v | − (cid:90) ∂ Ω ( x · ν ) |∇ v | + (cid:90) ∂ Ω ∂v∂ν ( x · ∇ v )= n − (cid:90) Ω |∇ v | + 12 (cid:90) ∂ Ω ( x · ν ) |∇ v | ≥ γ (cid:90) ∂ Ω |∇ v | , (3.8)because n ≥ , and because the properties v | ∂ Ω = 0 and v ≥ in Ω imply that on ∂ Ω we have ∇ v = −|∇ v | ν and hence ∂v∂ν ( x · ∇ v ) = ( x · ν ) |∇ v | .Apart from this, again due to the homogeneous Dirichlet boundary conditions satisfied by v . anotherintegration by parts yields − k (cid:90) Ω v ( x · ∇ v ) = − k (cid:90) Ω x · ∇ v = k (cid:90) Ω ( ∇ · x ) v = nk (cid:90) Ω v , (3.9)11nd using that u ∇ v = ∇ u by (3.1) we infer from a final integration by parts that − (cid:90) Ω u ( x · ∇ v ) = − (cid:90) Ω x · ∇ u = (cid:90) Ω ( ∇ · x ) u − (cid:90) ∂ Ω ( x · ν ) u ≤ n (cid:90) Ω u, (3.10)once more because x · ν ≥ by (3.4).Now a combination of (3.7) with (3.8)-(3.10) reveals that γ (cid:90) ∂ Ω |∇ v | + nk (cid:90) Ω v ≤ n (cid:90) Ω u and that hence, by Young’s inequality, k (cid:90) Ω v − (cid:90) ∂ Ω ∂v∂ν ≤ k (cid:90) Ω v + (cid:90) ∂ Ω |∇ v |≤ (cid:26) k (cid:90) Ω v + k | Ω | (cid:27) + (cid:26) γ n (cid:90) ∂ Ω |∇ v | + n | ∂ Ω | γ (cid:27) ≤ (cid:90) Ω u + k | Ω | + n | ∂ Ω | γ . In conjunction with (3.6), this entails (3.5). (cid:3)
A combination of the latter two statements readily yields the first part of our main results:
Proof of Theorem 1.1. Thanks to (1.6), from Lemma 3.2 when combined with Lemma 3.1 andProposition 2.1 it immediately follows that the set in (1.7) is not empty and hence M (cid:63) (Ω , k ) a well-defined nonnegative number which moreover satisfies the upper estimates in (1.8) and (1.9), respec-tively. The left inequality in (1.8) is obvious from Corollary B, whereas in the case n ≥ , positivityof M (cid:63) (Ω , k ) is an evident by-product of Lemma 2.2. (cid:3) Proof of Corollary 1.2. Since for each x ∈ ∂ Ω we have ν ( x ) = x | x | and hence x · ν ( x ) = R , allstatements are obvious from Theorem 1.1. (cid:3) In view of Corollary C, Corollary 1.2, and Lemma 2.2, verifying the occurrence of a genuinely secondarycritical mass phenomenon in the flavor of Theorem 1.3 amounts to making sure that whenever thedegradation parameter k in (1.1) is positive, in any planar disk we can find global bounded radialsolutions at some mass level larger than π . To accomplish this, for such radial solutions ( u, v ) =( u ( r, t ) , v ( r, t )) , r ∈ [0 , R ] , of (1.1)-(1.2) in Ω = B R (0) ⊂ R with R > , again maximally extended upto T max ∈ (0 , ∞ ] in the style of Proposition 2.1, we follow the idea of [16] and [6] and introduce thecumulated quantities w ( s, t ) := (cid:90) √ s ρu ( ρ, t ) dρ s ∈ [0 , R ] , t ∈ [0 , T max ) , (4.1)12nd z ( s, t ) := k (cid:90) √ s ρv ( ρ, t ) dρ s ∈ [0 , R ] , t ∈ [0 , T max ) , (4.2)as well as w ( s ) := (cid:90) √ s ρu ( ρ ) dρ, s ∈ [0 , R ] . (4.3)Then from the nonnegativity of u , and from (1.1) as well as (1.2), it follows that w s ≥ in [0 , R ] × [0 , T max ) and w t = 4 sw ss + 2 ww s − zw s , s ∈ (0 , R ) , t ∈ (0 , T max ) ,w (0 , t ) = 0 , w ( R , t ) = π · (cid:82) Ω u , t ∈ (0 , T max ) ,w ( s,
0) = w ( s ) , s ∈ (0 , R ) , (4.4)and the core of our strategy will consist in appropriately making use of the rightmost absorptivecontribution to the first equation herein in order to ensure that some of these solutions remain boundedin C ([0 , R ]) even though satisfying w | s = R > . This will be achieved by means of a paraboliccomparison with stationary supersolutions, to be constructed in Lemma 4.5, on the basis of a pointwiselower estimate for the function z which plays a central role in this additional dissipative part, but whichthrough (1.1)-(1.2) and (4.1) is linked to w in a nonlocal manner.As a first step toward adequately coping with this, to be completed in Lemma 4.4, let us invoke acomparison argument to derive a fairly rough but useful lower bound for w . Lemma 4.1
Let
R > and Ω = B R (0) ⊂ R , let k > , and suppose that u ∈ C rad (Ω) is nonnegativeand such that w as in (4.3) satisfies w ( s ) ≥ δs β for all s ∈ (0 , R ) (4.5) with some δ > and some β ≥ π · (cid:90) Ω u . (4.6) Then w ( s, t ) ≥ δs β for all s ∈ (0 , R ) and t ∈ (0 , T max ) . (4.7) Proof.
We abbreviate m := (cid:82) Ω u and first observe that since k (cid:90) Ω v = (cid:90) Ω u + (cid:90) ∂ Ω ∂v∂ν ≤ m for all t ∈ (0 , T max ) according to the second equation in (1.1) and (2.3), the function z from (4.2) satisfies z ( s, t ) ≤ k (cid:90) R ρv ( ρ, t ) dρ ≤ m π for all s ∈ (0 , R ) and any t ∈ (0 , T max ) . Therefore, writing w ( s, t ) := δs β , s ∈ [0 , R ] , t ≥ ,
13y nonnegativity of w and w s we can estimate sw ss + 2 ww s − z ( s, t ) w s ≥ sw ss − mπ w s = 4 β ( β − δs β − − mπ · βδs β − ≥ for all s ∈ (0 , R ) and t ∈ (0 , T max ) , (4.8)because (4.6) asserts that β ( β − ≥ mπ β . Since (4.5) implies that w ( s, ≤ w ( s ) for all s ∈ (0 , R ) ,and that necessarily also w ( R , t ) ≤ w ( R ) = w ( R , t ) for all t ∈ (0 , T max ) by (2.3), noting that w (0 , t ) = 0 for all t ∈ (0 , T max ) we infer from the comparison principle in Lemma 7.1 from theappendix that due to (4.8) indeed w ≥ w in (0 , R ) × (0 , T max ) . (cid:3) As a consequence, we obtain the following statement on lower control of the mass accumulated in thedisk B R (0) throughout evolution, uniform with respect to mass levels within any fixed interval. Corollary 4.2
Let
Ω = B R (0) ⊂ R with some R > , and let k > , m > , and M ≥ m . Then thereexists C > such that for all nonnegative u ∈ C rad (Ω) fulfilling m ≤ (cid:90) Ω u ≤ M (4.9) as well as − (cid:90) B r (0) u ≥ − (cid:90) B R (0) u for all r ∈ (0 , R ) , (4.10) the solution ( u, v ) of (1.1) - (1.2) satisfies (cid:90) B R (0) u ( · , t ) ≥ C for all t ∈ (0 , T max ) . (4.11) Proof.
In order to apply Lemma 4.1 to β := 1 + M π and δ := m πR β , we note that when rewrittenin the variables w , z and s from (4.1) and (4.3), (4.10) together with (4.9) guarantees that w ( s ) ≥ ms πR for all s ∈ (0 , R ) . As β > , namely, this entails that w ( s ) δs β ≥ m πδR s β − ≥ m πδR · ( R ) β − = m πδR β = 1 for all s ∈ (0 , R ) , whence Lemma 4.1 ensures that for w as in (4.1) we have w ( s, t ) ≥ δs β for all s ∈ (0 , R ) and any t ∈ (0 , T max ) .
14s a particular consequence, this implies that (cid:90) B R (0) u ( · , t ) = 2 π · w (cid:16) R , t (cid:17) ≥ π · δ (cid:16) R (cid:17) β for all t ∈ (0 , T max ) and thereby proves (4.11). (cid:3) This lemma will be combined with the following well-known result on positivity of the kernel associatedwith the solution operator for the Helmholtz problem solved by v : Lemma 4.3
Let
Ω = B R (0) ⊂ R with some R > , and for k > let G k denote Green’s function of − ∆ + k under homogeneous Dirichlet boundary conditions in Ω . Then G k ( x, y ) ≥ for all x ∈ Ω and y ∈ Ω \ { x } , and there exists C > such that G k ( x, y ) ≥ C for all ( x, y ) ∈ (cid:16) B R (0) × B R (0) (cid:17) \ (cid:110) (˜ x, ˜ y ) ∈ B R (0) × B R (0) (cid:12)(cid:12)(cid:12) ˜ x = ˜ y (cid:111) . Proof.
This can be found in [40, Section 4.9]. (cid:3)
In fact, by means of a corresponding integral representation the function v can be estimated frombelow in such a way that its cumulated version satisfies a linear lower bound in the following sense: Lemma 4.4
Let
Ω = B R (0) ⊂ R with some R > , and suppose that k > , m > , and M ≥ m .Then there exists C > such that whenever u ∈ C rad (Ω) is nonnegative and satisfies (4.9) as well as(4.10), the function z given by (4.2) fulfils z ( s, t ) ≥ C · s for all s ∈ (0 , R ) and each t ∈ (0 , T max ) . (4.12) Proof.
According to Corollary 4.2, we can pick c > such that for any choice of u with theindicated properties we have (cid:90) B R (0) u ( · , t ) ≥ c for all t ∈ (0 , T max ) . Thus, if relying on Lemma 4.3 we fix c > such that Green’s function G k of − ∆+ k under homogeneousDirichlet conditions in Ω satisfies G k ( x, y ) ≥ c whenever x ∈ B R (0) and y ∈ B R (0) \ { x } , due to(1.1)-(1.2) and the nonnegativity of G k and u we can estimate v ( x, t ) = (cid:90) Ω G k ( x, y ) u ( y, t ) dy ≥ (cid:90) B R (0) G k ( x, y ) u ( y, t ) dy ≥ c (cid:90) B R (0) u ( y, t ) dy ≥ c c for all x ∈ B R (0) and t ∈ (0 , T max ) .
15y definition of z , this entails that z ( s, t ) = k π (cid:90) B √ s (0) v ( x, t ) dx ≥ k π · c c · | B √ s (0) | = c c k · s for all s ∈ (cid:16) , R (cid:17) and t ∈ (0 , T max ) . As z ( · , t ) is nondecreasing on (0 , R ) thanks to the nonnegativity of v , this moreover entails that z ( s, t ) s ≥ c c k · R R = c c k for all s ∈ (cid:104) R , R (cid:17) and t ∈ (0 , T max ) , and that thus (4.12) holds with C := c c k . (cid:3) The key step in our derivation of Theorem 1.3 can now be found in the following essentially explicitconstruction of a stationary supersolution to (4.4) that corresponds to a mass level exceeding the value π . Lemma 4.5
Let
Ω = B R (0) ⊂ R with some R > , and let k > . Then there exist m = m ( R, k ) > π and a function w ∈ W , ∞ ((0 , R )) such that w (0) = 0 (4.13) in addition to w ( R ) = m π (4.14) and w ( s ) > ms πR for all s ∈ (0 , R ) , (4.15) and such that whenever u ∈ C rad (Ω) is a nonnegative function for which w from (4.3) satisfies sR ≤ w ( s ) ≤ w ( s ) for all s ∈ (0 , R ) , (4.16) the solution of (1.1) - (1.2) has the property that w ( s, t ) ≤ w ( s ) for all s ∈ (0 , R ) and t ∈ (0 , T max ) (4.17) with w as defined in (4.1) , so that sup ( s,t ) ∈ (0 ,R ) × (0 ,T max ) w ( s, t ) s < ∞ . (4.18) Proof.
Given
R > and k > , upon application of Lemma 4.4 to m := 8 π and M := 10 π weobtain c > such that for arbitrary nonnegative u ∈ C rad (Ω) fulfilling (4.9) and (4.10), the function z in (4.2) satisfies z ( s, t ) ≥ c s for all s ∈ (0 , R ) and t ∈ (0 , T max ) , (4.19)16here without loss of generality we may assume that c ≤ R . (4.20)We next use that ln s → + ∞ as s (cid:38) to fix s ∈ (0 , R ) sufficiently small to ensure that c · ln R s > c R , (4.21)noting that the latter implies that s · (cid:90) R s σ − e c ( σ − s ) dσ > s . (4.22)Indeed, using that e c ξ ≥ c ξ for ξ ≥ shows that s · (cid:90) R s σ − e c ( σ − s ) dσ ≥ s · (cid:90) R s σ − · (cid:110) c σ − s ) (cid:111) dσ = s · (cid:16) − c s (cid:17) · (cid:16) s − R (cid:17) + c s · ln R s = 1 − (cid:16) c R (cid:17) · s + c R s + c s · ln R s > s · (cid:26) c · ln R s − (cid:16) c R (cid:17)(cid:27) > by (4.21). Now (4.22) enables us to pick b > small enough such that s · (cid:90) R s σ − e c ( σ − s ) dσ > s + b, which in turn warrants the existence of ε ∈ (0 , such that still s ε · (cid:90) R s σ − ε e c ( σ − s ) dσ > s + b + ε b · ( s + b ) . (4.23)Observing that ϕ ( ξ ) := s ε · (cid:90) R s σ − ε e ξ ( σ − s ) dσ, ξ > , in the limit ξ (cid:38) satisfies ϕ ( ξ ) → s ε · (cid:90) R s σ − ε dσ = 22 + ε · s ε · (cid:16) s − ε − R − − ε (cid:17) <
22 + ε · s < s + b + ε b · ( s + b ) c ∈ (0 , c ] such that the precise equality s ε · (cid:90) R s σ − ε e c ( σ − s ) dσ = s + b + ε b · ( s + b ) (4.24)holds.Upon these choices, we now let w ( s ) := (cid:40) w in ( s ) if s ∈ [0 , s ] ,w out ( s ) if s ∈ ( s , R ] , (4.25)where w in ( s ) := 4 ss + b , s ∈ [0 , s ] , (4.26)which already ensures (4.13), and where w out denotes the solution of the initial-value problem (cid:40) s∂ s w out + 2(4 + ε ) ∂ s w out − c s · ∂ s w out = 0 , s ∈ ( s , R ) ,w out ( s ) = w in ( s ) , ∂ s w out ( s ) = ∂ s w in ( s ) . (4.27)Then w evidently belongs to C ([0 , R ]) ∩ C ([0 , s ]) ∩ C ([ s , R ]) , and hence clearly also to W , ∞ ((0 , R )) ,with w s ( s ) = 4 b ( s + b ) and w ss ( s ) = − b ( s + b ) for all s ∈ (0 , s ) , (4.28)and with an explicit integration of (4.27) showing that w s ( s ) = w s ( s ) · exp (cid:26) (cid:90) ss (cid:16) − ε · σ + c (cid:17) dσ (cid:27) = 4 b ( s + b ) · (cid:16) s s (cid:17) ε e c ( s − s ) for all s ∈ ( s , R ] (4.29)as well as w ( s ) = w ( s ) + 4 b ( s + b ) · (cid:90) ss (cid:16) s σ (cid:17) ε e c ( σ − s ) dσ = 4 s s + b + 4 b ( s + b ) · s ε · (cid:90) ss σ − ε e c ( σ − s ) dσ for all s ∈ ( s , R ] . (4.30)In particular, (4.28) and (4.30) guarantee that thanks to (4.24), w ( s ) ≤ w ( R )= 4 s s + b + 4 b ( s + b ) · s ε · (cid:90) ss σ − ε e c ( σ − s ) dσ = 4 s s + b + 4 b ( s + b ) · (cid:110) s + b + ε b · ( s + b ) (cid:111) = 4 + ε for all s ∈ [0 , R ] , (4.31)18hile recalling the inequality c ≤ c and (4.20) we directly obtain from (4.27) and (4.29) that sw ss ( s ) = − (4 + ε − c s ) w s ( s ) ≤ − (4 + ε − c R ) w s ( s ) ≤ − (4 − c R ) w s ( s ) < for all s ∈ ( s , R ) and that hence, by (4.28), w ss ( s ) < for all s ∈ (0 , R ) \ { s } . In conjunction with (4.31), the latter concavity property in particular implies that indeed both (4.14)and (4.15) hold if we let m := 2 π · (4 + ε ) , where we note that our restriction ε < warrants that m ≤ π = M . As obviously also m ≥ π = m , assuming henceforth that u ∈ C rad (Ω) is nonnegativeand such that (4.16) is valid, we firstly observe that (4.19) in fact applies to the function z thereupondefined through (4.2), whence again using that c ≤ c we may infer from (4.30), (4.19), and (4.27)that w t − sw ss − ww s + 2 zw s = − sw ss − ww s + 2 zw s ≥ − sw ss − ε ) w s + 2 c sw s = 0 for all s ∈ ( s , R ) and t ∈ (0 , T max ) , whereas, simply by nonnegativity of z and w s , (4.28) ensures that w t − sw ss − ww s + 2 zw s ≥ − sw ss − ww s = 0 for all s ∈ (0 , s ) and t ∈ (0 , T max ) . Since clearly w (0 , t ) = w (0 , t ) = 0 and w ( R , t ) = w ( R , t ) = 4 + ε for all t ∈ (0 , T max ) , we may employthe comparison principle from Lemma 7.1 to conclude that indeed (4.17) holds. Finally, (4.18) followsfrom (4.13) together with boundedness of w s and (4.17). (cid:3) In order to prepare an appropriate conclusion on boundedness of w s from this, let us add the followingobservation on a linear upper bound for z . Lemma 4.6
Let n = 2 , R > , Ω = B R (0) ⊂ R , and k > and let u ∈ C rad (Ω) be nonnegative andsuch that w taken from (4.1) satisfies sup ( s,t ) ∈ (0 ,R ) × (0 ,T max ) w ( s, t ) s < ∞ . (4.32) Then there exists
C > such that z ( s, t ) ≤ Cs for all s ∈ (0 , R ) and t ∈ (0 , T max ) , (4.33) where z is as in (4.2) . roof. Utilizing (4.32), let us define c > such that w ( s,t ) s ≤ c for all s ∈ (0 , R ) and t ∈ (0 , T max ) .Then since z s ( R , t ) = v ( R, t ) = 0 for all t ∈ (0 , T max ) due to the Dirichlet condition on v in (1.2), and since by (1.1) we moreover have sz ss ( s, t ) = k ( z ( s, t ) − w ( s, t )) ≥ − kw ( s, t ) for all s ∈ (0 , R ) and t ∈ (0 , T max ) due to the nonnegativity of z , on integration we infer that z s ( s, t ) = 0 − (cid:90) R s z ss ( σ, t ) dσ ≤ k (cid:90) R s w ( σ, t ) σ dσ ≤ c kR c for all t ∈ (0 , T max ) and any s ∈ (0 , R ) . After one more integration, in view of the fact that z (0 , t ) = 0 for all t ∈ (0 , T max ) this shows that z ( s, t ) ≤ c s for all t ∈ (0 , T max ) and s ∈ (0 , R ) and thereby readily entails (4.33). (cid:3) Now employing a Bernstein-type argument in the style of [41, Lemma 4.1], we can indeed turn theoutcome of Lemma 4.5 into an L ∞ bound for u by means of the following implication. Lemma 4.7
Let n = 2 , R > , Ω = B R (0) ⊂ R , and k > and let (cid:54)≡ u ∈ C rad (Ω) be nonnegativeand such that w from (4.1) satisfies (4.32). Then there exists C > such that (cid:107) u ( · , t ) (cid:107) L ∞ (Ω) ≤ C for all t ∈ (0 , T max ) . (4.34) Proof.
In accordance with (4.32) and (4.33), we first fix c > and c > such that w ( s, t ) ≤ c s and z ( s, t ) ≤ c s for all ( s, t ) ∈ (0 , R ) × (0 , T max ) . With τ := min { , T max } , the continuity properties of u stated in Proposition 2.1 enable us to find c > satisfying w s ( s, t ) = u ( √ s, t ) ≤ c for all s ∈ [0 , R ] , t ∈ [0 , τ ] , (4.35)and positivity of u ( · , τ ) in Ω , as ensured by the strong maximum principle, warrants the existence of c > such that c ≤ u ( √ s, τ ) = w s ( s, τ ) for all s ∈ [0 , R ] . If for c := min { c c , π (cid:107) u (cid:107) L (Ω) } · exp( − c R ) we let w ( s, t ) := c (exp( c s ) − , s ∈ [0 , R ] , t ∈ [ τ, T max ) , then w ( s, τ ) ≤ c s for s ∈ [0 , R ] , w ( R , t ) ≤ π (cid:107) u (cid:107) L (Ω) = w ( R , t ) for all t ∈ [ τ, T max ) ,and, furthermore, w ( s, t ) ≥ c s for all ( s, t ) ∈ [0 , R ] × [ τ, T max ) with c := c c . Since w t − sw ss − ww s + 2 zw s ≤ − sc (cid:16) c (cid:17) e c s + 0 + 2 c sc c e c s = 0 in (0 , R ) × ( τ, T max ) ,
20 first comparison argument thus shows that w ( s, t ) ≥ w ( s, t ) ≥ c s for all ( s, t ) ∈ (0 , R ) × [ τ, T max ) . (4.36)To conclude our series of selections, we note that boundedness of w and non-degeneracy of (4.4) in ( R , R ) × (0 , T max ) allows us to invoke parabolic Schauder theory in the form of [19, Thm. IV.10.1]so as to obtain c > fulfilling w s ( R , t ) ≤ c for all t ∈ [ τ, T max ) . (4.37)For α > , we now let y α ( s, t ) := s α w s ( s, t ) w ( s, t ) , ( s, t ) ∈ (0 , R ] × [ τ, T max ) , and observe that then (4.36) ensures that letting y α (0 , t ) = 0 for t ∈ [ τ, T max ) extends y α so asto become continuous in all of [0 , R ] × [ τ, T max ) . Moreover, from (4.36) and (4.37) we know that y α ( R , t ) ≤ R α − c c for all t ∈ [ τ, T max ) , while combining (4.35) with (4.36) warrants that y α ( s, τ ) ≤ R α − c c for all s ∈ (0 , R ] . In the following, we fix T ∈ ( τ, T max ) and let ( s , t ) be any point atwhich the restriction of y = y α to (0 , R ) × ( τ, T ] attains its maximum. Then y s = αs α − w s w + 2 s α w s w ss w − s α w s w at ( s , t ) (4.38)and ≥ y ss = α ( α − s α − w s w + 4 αs α − w s w ss w − αs α − w s w − s α w s w ss w + 2 s α w ss w + 2 s α w s w sss w + 2 s α w s w at ( s , t ) (4.39)as well as ≤ y t = 2 s α w s w w st − s α w s w w t = 2 s α w s w (4 w ss + 4 s w sss + 2 w s + 2 ww ss − z s w s − zw ss ) − s α w s w (4 s w ss + 2 ww s − zw s )= 4 s · s α w s w sss w + 8 s α w s w w ss − zs α w s w w ss − s α +10 w s w w ss + 4 s α w s w ss + 2 s α w s w − s α z s w s w + 2 zs α w s w at ( s , t ) . (4.40)Here we note that, evidently, (4.38) entails that w ss = w s (cid:18) w s w − αs (cid:19) at ( s , t ) , s α w s w sss w ≤ − s α w s w (cid:18) w s w − αs (cid:19) − αs α − w s w (cid:18) w s w − αs (cid:19) + 52 s α w s w (cid:18) w s w − αs (cid:19) − α ( α − s α − w s w + 2 αs α − w s w − s α w s w = (cid:18) −
12 + 52 − (cid:19) s α w s w + (cid:18) − −
52 + 2 (cid:19) αs α − w s w + (cid:16) − α α − ( α − (cid:17) αs α − w s w = − αs α − w s w + α (cid:16) α (cid:17) s α − w s w at ( s , t ) . Inserting these latter two pieces of information into (4.40), we obtain ≤ s (cid:18) − αs α − w s w + α (cid:16) α (cid:17) s α − w s w (cid:19) + 4 s α w s w (cid:18) w s w − αs (cid:19) − zs α w s w (cid:18) w s w − αs (cid:19) − s α +10 w s w (cid:18) w s w − αs (cid:19) + 2 s α w s (cid:18) w s w − αs (cid:19) + 2 s α w s w − s α z s w s w + 2 zs α w s w = − s α +10 w s w + w s w ( − αs α + 4 s α + 2 αs α ) + w s w (2 s α + 2 s α )+ w s w (2 α (2 + α ) s α − − αs α − ) − αs α − w s + 2 αzs α − w s w − s α z s w s w ≤ − s α +10 w s w + 4 s α w s w + 2 α s α − w s w + 2 αzs α − w s w ≤ − s α +10 w s w + 4 s α − w s w + 2 α s α − w s w + 2 αzs α − w s w in ( s , t ) , so that finally y = 1 s w w s · s α +10 w s w ≤ s w w s (cid:18) s α − w s w + 2 α s α − w s w + 2 αzs α − w s w (cid:19) = 4 s α − w + 2 α s α − w + 2 αs α − wz ≤ c s α +10 + 2 α c s α − + 2 αc c s α at ( s , t ) . This entails that y α ( s, t ) ≤ max (cid:26) R α − c c , R α − c c , c s α +10 + 2 α c s α − + 2 αc c s α (cid:27) for all ( s, t ) ∈ [0 , R ] × [ τ, T max ) , whence letting α (cid:38) we conclude that sup s ∈ (0 ,R ) ,t ∈ ( τ,T max ) sw ( s, t ) w s ( s, t ) ≤ max (cid:26) c c , c c , c R + 2 c + 2 c c R (cid:27) ,
22o that boundedness of w s in (0 , R ) × [ τ, T max ) , and thus of u in Ω × [ τ, T max ) , results from (4.32).Together with (4.35), this concludes the proof. (cid:3) The second of our main results has thereby actually been achieved already:
Proof of Theorem 1.3. The first identities in (1.12), (1.14) and (1.15) have precisely been statedin Corollary C already. Both inequalities in (1.12) are obvious by definition, and in view of Corollary1.2, (1.12) directly implies (1.13).Finally, the strict inequality in (1.15) can be verified by once more employing Lemma 2.2, whereasthat in (1.14) can be seen as follows: Given
R > and k > , we take m ( R, k ) from Lemma 4.5 anduse that m ( R, k ) > π in choosing any m > π such that m < m ( R, k ) . Then simply defining u ( x ) := mπR , x ∈ Ω , we see on applying Lemma 4.5 in conjunction with Lemma 4.7 and Proposition 2.1 that the corre-sponding maximally extended solution ( u, v ) of (1.1)-(1.2) indeed is global in time and bounded in thesense that (1.3) holds. In particular, this entails that indeed we must have m (cid:63) (2 , R, k ) ≥ m > π forany such R and k . (cid:3) Corollary 5.1
Let
Ω = B R (0) ⊂ R with some R > , and let k > . Then for all m < m (cid:63) (2 , R, k ) ,there exists at least one pair ( u, v ) ∈ ( C (Ω)) of radial functions with u > and v ≥ in Ω whichsatisfy (cid:82) Ω u = m and solve the stationary problem (3.1) in the classical sense. Proof.
This is an evident consequence of Theorem 1.3 when combined with Lemma 3.1. (cid:3)
In fact, simulations suggest the following
Conjecture 5.2
For
Ω = B R (0) ⊂ R with R > and k > ,(i) there is a unique steady state with (cid:82) Ω u = m for each m ∈ [0 , m (cid:63) (2 , R, k )] ,(ii) there are two steady states with (cid:82) Ω u = m for each m ∈ ( m (cid:63) (2 , R, k ) , m (cid:63) (2 , R, k )) , and(iii) there is a unique steady state with (cid:82) Ω u = m (cid:63) (2 , R, k ) . As detailed in [4], these steady states form a continuum and can be parametrized by (cid:107) u (cid:107) ∞ L . In figure1, the curves of steady states in the m - Λ plane are shown where Λ is the Lagrange multiplier enteringproblem (3.1) with k = 1 upon integrating the first equation to u = Λ exp( v ) and plugging this intothe second equation to obtain (cid:40) − ∆ v + v = Λ e v , x ∈ Ω ,v = 0 , x ∈ ∂ Ω (5.1)As the curves are traced from the origin to the point (8 π, , the norm (cid:107) u (cid:107) L ∞ increases, and the23 = 1 Λ Maximal total mass at steady state m m a x - π R k = 0.5 k = 1k = 2k = 3k = 4k = 60.5 π R Figure 1:
Left:
Curves of steady states as solutions of 5.1; shown is the Lagrange multiplier Λ plotted againstthe total mass m = (cid:82) Ω u for k = 1 and disks B R (0) ∈ R of radii R = 1 , R = 2 , and R = 4 , respectively.Note the more pronounced tilt to the right for increasing R and the common end points (0 , and (8 π, forall curves. Right:
Log-log plot of the maximal value of m = (cid:82) Ω u , corrected for π , in numerically found steadystate solutions in Ω = B R (0) ⊂ R depending on R for different values of k . The data points are the valuesdetermined from simulation, the dashed lines correspond to the curves m − π = kπR . solution becomes more strongly concentrated near the origin. The limit point (8 π, would representthe singular Dirac-solution u = 8 πδ . The observed maximal values of m = (cid:82) Ω u for which steady statesare found, depend quadratically on the radius and hence linearly on the domain size as predicted bythe upper bound on M (cid:63) ( B R (0) , k ) for n = 2 from corollary 1.2 and behave approximately as m max (2 , R, k ) (cid:39) π + kπR . (5.2)Indeed, the steady state solution maximizing the total mass for large R exhibits a small peak at theFigure 2: Solutions v max (as function of r ) with maximal total mass m = Λ (cid:82) B e v in B R (0) for k = 1 and increasing values of R = 10 , , (left to right).origin, a wide plateau with the value v plat ≈ , and decreases to zero in a thin annulus given by r (cid:47) R .This behavior becomes obvious from the radially symmetric form ˜ v (cid:48)(cid:48) + ˜ v (cid:48) r − k ˜ v = − Λ exp(˜ v ) , < r < R, ˜ v (cid:48) (0) = 0 , ˜ v ( R ) = 0 (5.3)24f the steady state problem (5.1). For large R , the maximal value Λ c of Λ allowing a solution approaches k e − , meaning that the solutions v ± of kv = Λ e v are close to for Λ close to Λ c . Since v ± are thevalues of v satisfying the differential equation in (5.3) as constants, we can expect plateaus in thesolution at v ≈ . As we moreover observe that for large R the maximal total mass is attained at Λ max (cid:47) Λ c it is not surprising that the maximal mass behaves like m max = 2 π Λ (cid:90) R r exp(˜ v ( r )) dr ≈ πke (cid:90) R re dr + small contributions for r (cid:39) and r ≈ kπR (5.4)where the small contributions of the peak near r = 0 and the boundary layer near r = R contributewith opposite signs.Figure 5 illustrates the shape of the mass maximizing solutions for different values of R . The plateauand lack of a pronounced peak at the origin are clearly visible for large R = 250 . Having found three distinct solvability behaviors for (1.1)-(1.2) in two dimensions, viz. global solutionsfor any initial conditions with m = (cid:82) Ω u < M (cid:63) (Ω , k ) , unconditional blow up in finite time for m >M (cid:63) (Ω , k ) , and the coexistence of both global and blowing up solutions for M (cid:63) (Ω , k ) < m < M (cid:63) (Ω , k ) ,we shall now briefly discuss what these results mean for the cytoskeleton of a hypothetical cell.As described in [24], increased myosin activity – corresponding to larger values of m – can result inthe total disruption of cells. This may be interpreted as the solution to the free boundary problemassociated with (1.1)-(1.2) (cf. [4]) breaking down due to Ω becoming disconnected. This kind ofdomain blow up – breakdown of the solution accompanied by singularities in domain shape – has alsobeen discussed by [33] in one dimension where blow up in our sense – that is, (cid:107) u (cid:107) ∞ → ∞ in a stationarydomain – can be ruled out. Our results show that in two dimensions, the appropriate setting for akeratocyte fragment or a thin amoeboid cell on a flat substrate, classical blow-up is to be expected aswell. This may be viewed as strong concentration of myosin in small regions of the cell, thereby locallydisrupting the actomyosin meshwork. Clearly, upon this disruption the model will not appropriatelydescribe the cytoskeleton anymore and would have to be replaced by another one.Acoording to this view, the regime m < M (cid:63) will be thought of as describing a cell comfortably comingto rest on its (very sticky) substrate, and the solutions will be expected to approach the unique steadystate solution with well defined distributions of myosin u and the stress v . Increasing m into theintermediate region M (cid:63) < m < M (cid:63) allows different fates, depending on the precise shape of theinitial conditions. A cell with initial strongly concentrated myosin distribution u will be expected tosuffer disruption of its cytoskeleton while moderately concentrated u may allow for a global solutionapproaching the presumably stable, weakly concentrated steady state. Further increasing m beyond M (cid:63) should then lead to disruption, no matter how myosin is initially distributed inside the cell.That the difference between m (cid:63) (2 , R, k ) and m (cid:63) (2 , R, k ) increases with R , as suggested by figure 1has a physical interpretation as well. Recall that, given k , the cell size R is measured in multiples of √ k L with L being the viscous length of the actin gel. For small R , any locally generated stress willbe felt throughout the cell, while for large R , stresses generated at one place in the cell have littleimpact at places far away. The stress v is supposed to vanish at the boundary, and the lower branch25f the two steady state solutions indicated in figure 1 for m (cid:63) < m < m (cid:63) comprises solutions whichare monotone in r but not concave down. These solutions rather feature a peak at the center of thecell, at r = 0 , where myosin is concentrated and the stress is high, a plateau at intermediate r withalmost constant stress and u ≈ kv , and a region of further decreasing stress at the boundary. If thecell is large compared to the viscous length, a peak in the center can easily be established without thelocally high stress being felt at the boundary, and a wider range of this type of steady states can beimagined. Recall that these steady states are expected to be unstable, and starting close to these, thesolution to the time dependent problem should be expected to blow up in finite time or to relax to thesupposedly stable steady state on the upper branch.It should be noted that the above discussion refers to an immobilized cell that cannot undergo shapechanges or the bifurcation to a traveling wave solution. This switch from rest to steady motion occursat even lower values m < M (cid:63) in the free boundary problem, and it cannot be ruled out that travelingwave solutions survive as global solutions for m > M (cid:63) . In fact, the local disruption of the actomyosinmeshwork has been implicated in the very symmetry breaking initiating cell motion [42]. Still, evenhigher values of m may destroy this mode of motion and lead to physical disruption of the cell asindicated above [24]. (4.4) Let us finally extract from [3] the following comparison principle for problems of type (4.4), forminga reduced version of an actually more comprehensive statement involving more general degenerateparabolic operators.
Lemma 7.1
Let
L > and T > , and suppose that w and w are two functions which belong to C ([0 , L ] × [0 , T )) and satisfy w s ( s, t ) > and w ( s, t ) > for all s ∈ (0 , L ) and t ∈ (0 , T ) as well as w ( · , t ) ∈ W , ∞ loc ((0 , L )) and w ( · , t ) ∈ W , ∞ loc ((0 , L )) for all t ∈ (0 , T ) . If for some a ≥ and some uniformly continuous b = b ( s, t, ξ ) : (0 , L ) × (0 , T ) × [0 , ∞ ) , Lipschitzcontinuous with respect to ξ ∈ [0 , ξ ] in (0 , L ) × (0 , T ) × [0 , ξ ] for any ξ > , we have w t ≤ asw ss + b ( s, t, w ) w s and w t ≥ asw ss + b ( s, t, w ) w s for all t ∈ (0 , T ) and a.e. s ∈ (0 , L ) ,and if moreover w ( s, ≤ w ( s, for all s ∈ (0 , L ) as well as w (0 , t ) ≤ w (0 , t ) and w ( L, t ) ≤ w ( L, t ) for all t ∈ (0 , T ) , then w ( s, t ) ≤ w ( s, t ) for all s ∈ [0 , L ] and t ∈ [0 , T ) . roof. This immediately results from [3, Lemma 5.1]. (cid:3)
Acknowledgement.
The third author acknowledges support of the
Deutsche Forschungsgemein-schaft within the project
Emergence of structures and advantages in cross-diffusion systems , projectnumber 411007140.The work presented here was initiated at the Mini Workshop
PDE models of motility and invasion inactive biosystems at the Mathematical Research Institute Oberwolfach in 2017.
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