A refined criterion and lower bounds for the blow--up time in a parabolic--elliptic chemotaxis system with nonlinear diffusion
aa r X i v : . [ m a t h . A P ] A p r A refined criterion and lower bounds for the blow–up time in aparabolic–elliptic chemotaxis system with nonlinear diffusion
Monica Marras , Teruto Nishino and Giuseppe Viglialoro ,♯ Dipartimento di Matematica e InformaticaUniversità di Cagliari, Viale Merello 92, 09123 (Italy)
Abstract
This paper deals with unbounded solutions to the following zero–flux chemotaxis system ( u t = ∇ · [( u + α ) m − ∇ u − χu ( u + α ) m − ∇ v ] ( x, t ) ∈ Ω × (0 , T max ) , v − M + u ( x, t ) ∈ Ω × (0 , T max ) , ( )where α > , Ω is a smooth and bounded domain of R n , with n ≥ , t ∈ (0 , T max ) , where T max theblow-up time, and m , m real numbers. Given a sufficiently smooth initial data u := u ( x, ≥ and set M := | Ω | R Ω u ( x ) dx , from the literature it is known that under a proper interplay betweenthe above parameters m , m and the extra condition R Ω v ( x, t ) dx = 0 , system ( ) possesses for any χ > a unique classical solution which becomes unbounded at t ր T max . In this investigation we firstshow that for p > n ( m − m ) any blowing up classical solution in L ∞ (Ω) –norm blows up also in L p (Ω) –norm. Then we estimate the blow–up time T max providing a lower bound T .
1. Introduction and motivation
In this paper we study properties of given solutions which classically solve this chemotaxis problem u t = ∇ · [( u + α ) m − ∇ u − χu ( u + α ) m − ∇ v ] ( x, t ) ∈ Ω × (0 , ∞ ) , v − M + u ( x, t ) ∈ Ω × (0 , ∞ ) ,u ν = v ν = 0 ( x, t ) ∈ ∂ Ω × (0 , ∞ ) ,u ( x,
0) = u ( x ) x ∈ Ω , Z Ω v ( x, t ) dx = 0 t ∈ (0 , ∞ ) , (1.1)where α, χ > , the spatial variable x is a vector of R n , with n ≥ , belonging to a smooth and boundeddomain Ω and t is the time variable. Further m , m are proper real numbers, ν is the outward normalvector to ∂ Ω and the initial data u := u ( x ) , supposed to be nonnegative and sufficiently regular, definesalso the constant M through the relation M = | Ω | R Ω u ( x ) dx .In the framework of self organization mechanisms for biological populations, and similarly to manyvariants of the well–known Keller–Segel models (see the celebrated papers [10, 11, 12]), system (1.1), whichis expressed as a particular case of a more general formulation provided in [25], represents the situation wherethe motion of a certain cell density u ( x, t ) at the position x and at the time t , living in an impenetrable(homogeneous Neumann boundary conditions) domain and initially distributed according to the law of u ( x ) , is influenced by the presence of a chemical signal concentrations, whose deviation from its spatialmean at the same position time is indicated with v ( x, t ) . Mathematics Subject Classification . Primary: 35B44, 35K51, 35K55. Secondary: 35Q92, 92C17.
Keywords and phrases : blow–up time; chemotaxis system; nonlinear diffusion; lower bound. ♯ Corresponding author: [email protected] emark 1.1. Let us precise that in this paper the mentioned deviation v is, essentially, the differencebetween the signal concentration and its mean, and that conversely to what happens to the cell and signaldensities (which are nonnegative) it changes sign. In particular, from the definition itself of v , we have thatits mean is zero (as fixed in the last assumption of problem (1.1) ), which in turn ensures the uniqueness ofthe solution for the Poisson equation under homogeneous Neumann boundary conditions. In this sense, thecorresponding compatibility condition, leads for t > to R Ω u ( x, t ) dx = M | Ω | and by virtue of R Ω u ( x, t ) dx = R Ω u ( x ) dx (coming by integrating over Ω the equation for u ), the choice M = | Ω | R Ω u ( x ) dx remainsjustified. Finally, and in line with what we said, we advise the reader that in the literature, and also indifferent places of this section, v stands for the chemical signal concentration itself and not for its deviation;nevertheless, we understand that in view of what we specified above it is not necessary to introduce a furthersymbol for the deviation, since it will be very clear from the context to which one of these quantities we arereferring to. A natural and singular situation possible appearing also in more general cellular processes than thatintroduced in model (1.1), but also idealized by two partial differential equations (one for the cell distributionand one for the chemical), is the chemotaxis collapse , when an uncontrolled gathering of cells at certainspatial locations is perceived as time evolves; essentially, u , in a particular instant of time (the blow-uptime), becomes unbounded in one or more points of its domain. This degeneration of the cell movementsinto aggregation is, above all, justified by the presence of the destabilizing effect in the coupled term (cross-diffusion term in the evolutive equation for u : in our case the expression χu ( u + α ) m − ∇ v in system (1.1));in turn the strength of such a destabilizing factor depends on the evolutive equation of v (in our case, ofcourse the second one in system (1.1)).The pioneer Keller–Segel system [11], already cited, is obtained from (1.1) when m = 1 , m = 2 andwith second equation given by τ v t = ∆ v − v + u , with τ ∈ { , } , where in this case v is the chemicalsignal concentration (and not its deviation). For positive chemical and cell distributions the expression − v + u manifests how an increase of the cells favors a production of the signal. For this case a verycomprehensive and extensive theory on existence and properties of global, uniformly bounded or blow-upsolutions, especially in terms of the size of the initial data, is available; for a complete picture, we suggestthe introduction of [5] for the parabolic-parabolic case (i.e., τ = 1 ), [9] and [17] for the parabolic-ellipticcase (i.e., τ = 0 ) and in addition the survey by [4, Hillen and Painter] where, inter alia, reviews of variousmodels about Keller-Segel-type systems are discussed.Besides the size of the initial data, there is another aspect related to the existence of both bounded orunbounded solutions to chemotaxis–systems; this is the mutual interplay between the weight of diffusion A ( u, v ) and that of the chemotactic sensitivity B ( u, v ) , which in our context are ( u + α ) m − and χ ( u + α ) m − , respectively. (To the readers interested to numerical simulations indicating the influence of theparameters m and m on solutions to a Keller–Segel system similar to that studied in the current researchwe suggest [24, § v t = ∆ v − v + u . In [1], [2] and [23] it is essentially established that the relation m < m + n is a necessary and sufficient condition to ensure global existence and boundedness of solutionseven emanating from large initial data. This is a generalization of [5, Theorems 4.1 and 6.1], where m = 1 (see also [23] and [6]). Even more, in [7] a parabolic–parabolic degenerate chemotaxis system ( α = 0 in(1.1)) is discussed: resorting to the natural concept of weak solutions, it is shown that for m < m + n and Ω = R n the problem possesses global bounded solutions (we refer also to [8] for a discussion onthe super–critical case m ≥ m + n ). Let us note that the reciprocal iteration involving m , m and n somehow establishes that the destabilizing effect of the chemo-sensitivity B ( u, v ) is weaker than that fromthe diffusion A ( u, v ) , which conversely tends to provide equilibrium to the model.Motivated by the above discussion, aim of the present research is expanding the theory of the mathe-matical analysis of problem (1.1) studied in [25], which, so far we are aware, covers the following situations:(i) for m ≤ , m < m + n , any sufficiently regular initial data emanates solutions which are global anduniformly bounded; (ii) for m ≤ , m > , m > m + n and Ω a ball of R n there exist initial data u which emanates unbounded solutions at some finite time T max .In light with this, we are interested in deriving a lower bound T for the blow–up time T max of theunbounded solutions to (1.1), so to essentially obtain a safe interval of existence [0 , T ) where such solutionsexist. We will achieve this result according to the steps specified in the next section.2 . Some premises and preparatory tools: plan of the paper For these coming reasons, we want to observe that there is no automatic connection between theoccurrence of blow-up for solutions to (1.1) in the classical L ∞ (Ω) –norm and that in L p (Ω) –norm ( p > ).Indeed, once it is assumed that Ω is a bounded domain, we only can conclude that k u ( · , t ) k L p (Ω) ≤ | Ω | p k u ( · , t ) k L ∞ (Ω) , so that if a solution blows up in L p (Ω) –norm, it does in L ∞ (Ω) –norm; conversely, if a solution becomesunbounded in L ∞ (Ω) –norm at some finite time T max , R Ω u p might also remain bounded in a neighborhoodof T max . In particular, since for a classical solution ( u, v ) to system (1.1) the u –component is continuous in I = [0 , T max ) , the function R Ω u p enjoys this same property on I and if lim sup R Ω u p is finite as t ց T max , R Ω u p can even be continuously prolonged up to the boundary T max .On the other hand, the evolution in time for the function t R Ω u p is more amenable to be analyzedthan that for t
7→ k u ( · , t ) k L ∞ (Ω) , so that it is preferable to use this to derive lower bounds for T max . Inthis way, in order to avoid the gap between the analysis of the blow–up time T max in the two differentmentioned norms will move toward a twofold action:• to detect proper L p –norms, for suitable p depending on n, m and m , ensuring that the unboundedsolutions in L ∞ (Ω) –norm also blow up in these L p (Ω) –norms;• to provide lower bounds for the blow–up time of unbounded solutions in these L p (Ω) –norms.To be more precise, we invoke the results in [25] in order to frame scenarios where local classical solutions ( u, v ) to system (1.1) are detected ( § m and m any local solution to system (1.1)which blows up at finite time T max in L ∞ (Ω) –norm also does in L p (Ω) –norm (Theorem 3.3 of §
3, provedin § Φ( t ) = p R Ω ( u + α ) p , for some p > , defined for all t ∈ (0 , T max ) and associated tothe local solution ( u, v ) ; these estimates are derived in §
5. (In § §
6, it is established that the same Φ( t ) satisfies a first order differential inequality(ODI) of the type Φ ′ ( t ) ≤ Ψ(Φ( t )) on (0 , T max ) . In particular, for any τ > the function Ψ( τ ) obeys theOsgood criterion ([19]), Z ∞ τ dτ Ψ( τ ) < ∞ with τ > , (2.1)so that an integration on (0 , T max ) of the mentioned ODI implies T max ≥ Z ∞ Φ(0) d ΦΨ(Φ) := T, thereby yielding the desired lower bound T for the blow–up time T max . (This is Theorem 3.4 of §
3, whoseproof is presented in §
3. Starting point and presentation of the main theorems
From the above considerations, let us give the following proposition, which represents the starting pointof our work and that we claim according to our purposes.First, we fix these mutual blow–up restrictions on the parameters m , m , since in the light of the resultspresented in § m > m + 2 n , m ≤ , m > . ( BU ) Proposition 3.1.
Let Ω be a bounded and smooth domain of R n , with n ≥ , α, χ > and ≤ u ∈ C κ ( ¯Ω) ,for some κ > , a nontrivial initial data with M = | Ω | R Ω u ( x ) dx . Additionally, let m , m ∈ R complywith the blow–up restrictions ( BU ) . Then, there exist a finite time T max > and a unique local classicalsolution ( u, v ) ∈ C (cid:0) ¯Ω × [0 , T max ) (cid:1) ∩ C , (cid:0) ¯Ω × (0 , T max ) (cid:1) × C , (cid:0) ¯Ω × (0 , T max ) (cid:1) o system (1.1) which blows up at T max in the sense that lim sup t ր T max k u ( · , t ) k L ∞ (Ω) = ∞ . (3.1) Proof.
See [25, Theorem 4.5].
Remark 3.2.
For the sake of scientific information, [25, Theorem 4.5] is proved in a ball of R n andmoreover under additional restrictions on the data u , as in particular some assumptions on its supportwhich rule out the choice of constant initial data. (Indeed, ( u, v ) = ( constant, is a bounded globalsolutions to system (1.1) ; this is the reason why we exclude trivial u in Proposition 3.1 and throughoutall the paper.) Despite that, since in the present investigation we are mostly interested in the derivation oflower bounds for the blow–up time T max to unbounded solutions to system (1.1) , we understand that themore general claim proposed in Proposition 3.1 does not mislead and is consistent with our overall aim. Theorem 3.3.
Let Ω be a bounded and smooth domain of R n , with n ≥ , α, χ > and ≤ u ∈ C κ ( ¯Ω) ,for some κ > , a nontrivial initial data. Then, for m , m ∈ R complying with the blow–up restrictions ( BU ) and M = | Ω | R Ω u ( x ) dx , the blow–up classical solution ( u, v ) to system (1.1) provided by Proposition3.1 is such that for all p > n ( m − m )lim sup t ր T max k u ( · , t ) k L p (Ω) = ∞ . Theorem 3.4.
Let Ω be a bounded and smooth domain of R n , with n ≥ , α, χ > and ≤ u ∈ C κ ( ¯Ω) ,for some κ > , a nontrivial initial data. Then, for m , m ∈ R complying with the blow–up restrictions ( BU ) and M = | Ω | R Ω u ( x ) dx , it is possible to find ¯ p > and E , E , E > as well as γ, δ > , dependingon ¯ p , such that the blow–up time T max of the unbounded classical solution ( u, v ) to system (1.1) providedby Proposition 3.1 satisfies T max ≥ Z ∞ Φ(0) dτE τ γ + E τ δ + E , (3.2) where Φ(0) = p R Ω ( u + α ) ¯ p . Remark 3.5.
In the absence of the result of Theorem 3.3, and taking in mind what discussed at thebeginning of §
2, the current formulation of Theorem 3.4 might fail without adding the extra hypothesis that lim sup t → T max Φ( t ) = ∞ . In fact, the above ODI Φ ′ ( t ) ≤ Ψ(Φ( t )) would infer, by integration on (0 , T max ) as well, that T max ≥ Z Φ( T max )Φ(0) dτ Ψ( τ ) , which does not produce any lower bound if no additional assumption on Φ( T max ) is given. Thereafter,even though in the literature there are several papers concerning estimates for lower bounds of blow-up timefor solutions to general evolutive problems whose formulation relies on the hypothesis on the divergenceof certain energy functions (see, for instance, [13, Theorem 1 and Theorem 2], [15, Theorem 2.4 andTheorem 2.7] and [22, Theorem 1 and Theorem 2] for contributions in the frame of chemotaxis models or[20, Theorem 2.1] and [21, Theorem 1 and Theorem 4] for others in different areas), the inspiring paper[3] represents a cornerstone that allows us to avoid this hypothesis, precisely thanks to the implication“A solution to (1.1) which blows up in L ∞ (Ω) –norm automatically does in L p (Ω) –norm”given in Theorem 3.3. (An equivalent approach is employed in [18] for unbounded solutions to the samefully parabolic chemotaxis problem analyzed in [3].)
4. Fixing some parameters and functional inequalities
In the following lemma, we fix the value of an important parameter, used to quantify certain constantsappearing throughout our logical steps, essentially by adjusting the data m , m and n defining problem(1.1). This parameter will be set in a such a way that the employments of some crucial inequalities belowwill be straightforwardly justified. 4 emma 4.1. For n ∈ N , let m , m satisfy the assumptions in ( BU ) and p > n ( m − m ) . Additionally,for any q > n + 2 , q > ( n + 2) / , let ¯ p := max p − m − m − m − n p − m + 1 q q ( m − − m n +1) q − ( n +2) q − ( n +2) − m − nn +2 q q − + 1 . (4.1) Then for all p ≥ ¯ p these relations hold p > n − m ) (4.2a) < a := ( p + m − (1 − p ) ( p + m − + n − < (4.2b) < β := p + m − p − p + m − p + n − < (4.2c) k > − n (4.2d) < a := p + m − p − kp + m − p + n − < (4.2e) < a := p + m − p − kp + m − p + n − < (4.2f) < σ < and γ > δ > , (4.2g) where k := 2( p + m − p + m − , δ = p + m − p , σ = ka , γ = p + m − p − a − σ . (4.3) Proof.
The assumptions done in ( BU ) and the definition of ¯ p , in conjunction with the restriction on p ,imply ¯ p > n (1 − m ) , i.e. (4.2a), so that in turn we have p + m − ≥ ¯ p + m − > ¯ p + m − p and that ¯ p + m − p > n − n ; therefore − n n p + m − > and, thus, also (4.2b) is attained. Again from the assumption on p , we also have, recalling again ( BU ),that (cid:0) − n (cid:1) p + 1 − m > , so that relation p > p − m + 1 and the definition of k also easily give (4.2c)and (4.2d), this last one also used to show (4.2e) and (4.2f). The remaining inequalities come from p + m − p + 1 n − > , k > and − σ = 1 − ka < (1 − a ) . emark 4.2. As to the definition of the parameter ¯ p in (4.1) , we desire to point out that the expressionof ¯ p is precisely fixed in that way exactly to avoid to have to enlarge a general p > up to some suitablevalues which are used in our derivations. Moreover, the addition “+1” in the same definition is not strictlynecessary but, undoubtedly, its presence will allow us to uniquely establish the magnitudes of some constants,many of these taking part, inter alia, in the quantitative calculations of the lower bound T for T max ofTheorem 3.4. As announced, let us now recall the Gagliardo–Nirenberg inequality, which throughout this paper willbe used in a less common version:
Lemma 4.3.
Let Ω be a bounded and smooth domain of R n , with n ≥ , and m , m ∈ R complying withthe blow–up restrictions ( BU ) . Additionally, for p = ¯ p given in (4.1) , let q , s ∈ [ p + m − , pp + m − ] , p ∈ [ pp + m − , p + m − p + m − ] . Then there exists a uniquely determined positive constant C GN = C GN ( n, m , m , Ω) such that k w k L p (Ω) ≤ C GN (cid:16) k∇ w k aL (Ω) k w k − aL q (Ω) + k w k L s (Ω) (cid:17) for all w ∈ W , (Ω) ∩ L q (Ω) , where a := q − p q + n − ∈ (0 , .Proof. This is an adaptation and a specific case of [14, Lemma 2.3] with r = 2 . Given for ¯ p as in (4.1), for p and q as in our assumptions, we see that < q ≤ p ≤ ∞ and, by virtue of (4.2a), r ≤ n + p , so that theclaim straightforwardly follows by [14, Lemma 2.3], taking as C GN the maximum value of the constant c therein used under the constraints on p , q , r and s herein assumed.
5. The energy function Φ( t ) := p R Ω ( u + α ) p : some a priori estimates Having ensured existence of unbounded solutions ( u, v ) to system (1.1) we can now turn our attentionto the evolution in time of the energy function Φ( t ) := p R Ω ( u + α ) p , with p > . (In particular, havingproperly fixed the parameter ¯ p in Lemma 4.1, and in light of Remark 4.2, we will analyze Φ( t ) for p = ¯ p. )Apparently, this analysis will provide crucial information for both the proofs of Theorems 3.3 and 3.4. Lemma 5.1.
Under the assumptions of Proposition 3.1, let ( u, v ) be the local solution to system (1.1) which blows up at finite time T max in the sense of (3.1) , and Φ( t ) the energy function defined for p = ¯ p asin (4.1) by Φ( t ) := 1 p Z Ω ( u + α ) p on (0 , T max ) . Then there exist E , E , E > such that Φ ′ ( t ) ≤ − E Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + E Z Ω ( u + α ) p + m − + E for all t ∈ (0 , T max ) . (5.1) Proof.
The first equation of (1.1) enables us to see p ddt Z Ω ( u + α ) p = − Z Ω ∇ ( u + α ) p − · (cid:2) ( u + α ) m − ∇ u − χu ( u + α ) m − ∇ v (cid:3) =: − I + I for all t ∈ (0 , T max ) , (5.2)where I := Z Ω ∇ ( u + α ) p − · ( u + α ) m − ∇ u for all t ∈ (0 , T max ) ,I := Z Ω ∇ ( u + α ) p − · χu ( u + α ) m − ∇ v for all t ∈ (0 , T max ) .
6s to the addendum I , from Lemma 4.1 we have p > − m , so that we can write I = Z Ω ∇ ( u + α ) p − · ( u + α ) m − ∇ u = ( p − Z Ω ( u + α ) p + m − |∇ u | = E Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) for all t ∈ (0 , T max ) , (5.3)where E = p − p + m − . Similarly, in order to control I , we start to write I = Z Ω ∇ ( u + α ) p − · χu ( u + α ) m − ∇ v = χ ( p − Z Ω u ( u + α ) p + m − ∇ u · ∇ v on (0 , T max ) . Setting F ( u ) := Z u τ ( τ + α ) p + m − dτ, we explicitly see from problem (1.1) that I = χ ( p − Z Ω ∇ F ( u ) · ∇ v = − χ ( p − Z Ω F ( u )∆ v = − χ ( p − Z Ω F ( u )( M − u )= − χ ( p − M Z Ω F ( u ) + χ ( p − Z Ω F ( u ) u on (0 , T max ) . (5.4)In view again of Lemma 4.1, we have p > − m so we can calculate F ( u ) as F ( u ) = Z u τ ( τ + α ) p + m − dτ = 1 p + m − (cid:16) u ( u + α ) p + m − − Z u ( τ + α ) p + m − dτ (cid:17) . Additionally, from the relation Z u ( τ + α ) p + m − dτ = 1 p + m − u + α ) p + m − − p + m − α p + m − , with some manipulations and using u ( u + α ) p + m − = ( u + α ) p + m − − α ( u + α ) p + m − , we infer F ( u ) = ( u + α ) p + m − p + m − − α ( u + α ) p + m − p + m − α p + m − ( p + m − p + m − . (5.5)Henceforth, (5.4)-(5.5) and u ( u + α ) p + m − = ( u + α ) p + m − − α ( u + α ) p + m − now produce on (0 , T max ) I = C C Z Ω u ( u + α ) p + m − − C C Z Ω u ( u + α ) p + m − − C C Z Ω ( u + α ) p + m − + C C Z Ω ( u + α ) p + m − − C C | Ω | + C C M | Ω | = C C Z Ω ( u + α ) p + m − − ( αC C + C C + C C ) Z Ω ( u + α ) p + m − (5.6) + ( αC C + C C ) Z Ω ( u + α ) p + m − − C C | Ω | + C C M | Ω | = E Z Ω ( u + α ) p + m − − E Z Ω ( u + α ) p + m − + E Z Ω ( u + α ) p + m − + E , (5.7)7here we have set C = χ ( p − M, C = χ ( p − ,C = p + m − , C = αp + m − , C = α p + m − ( p + m − p + m − ,E = C C , E = αC C + C C + C C ,E = αC C + C C , E = − C C | Ω | + C C M | Ω | . On the other hand, since a combination of relations (5.3) and (5.7) yields the following identity Φ ′ ( t ) = − E Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + E Z Ω ( u + α ) p + m − − E Z Ω ( u + α ) p + m − + E Z Ω ( u + α ) p + m − + E for all t ∈ (0 , T max ) , (5.8)we can estimate the forth term on the right–hand side of (5.8) by the Young inequality, so to have for any δ > E Z Ω ( u + α ) p + m − ≤ δ Z Ω ( u + α ) p + m − + D ( δ ) on (0 , T max ) , (5.9)with D ( δ ) = 1 p + m − (cid:16) δ E − p + m − p + m − p + m − p + m − (cid:17) − ( p + m − | Ω | . Subsequently, from (5.8) and (5.9) is achieved that Φ ′ ( t ) ≤ − E Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + E Z Ω ( u + α ) p + m − − ( E − δ ) Z Ω ( u + α ) p + m − + E + D ( δ ) on (0 , T max ) , so that, by taking δ = E and considering that from Lemma 5.1 the constant E might be negative, wefinally conclude posing E := | E | + D ( E ) and obtaining Φ ′ ( t ) ≤ − E Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + E Z Ω ( u + α ) p + m − + E for all t ∈ (0 , T max ) . The coming lemma includes the details used to control the terms R Ω |∇ ( u + α ) p + m − | and R Ω ( u + α ) p + m − . Lemma 5.2.
Under the assumptions of Proposition 3.1, let ( u, v ) be the local solution to system (1.1) ,which blows up at finite time T max in the sense of (3.1) , and Φ( t ) the energy function defined in Lemma5.1. Then there exist positive constants E and λ such that − Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) ≤ − E Φ λ ( t ) + 1 for all t ∈ (0 , T max ) . (5.10) If, additionally, for some p > n ( m − m ) it is known that k u ( · , t ) k L p (Ω) ≤ L for all t ∈ (0 , T max ) , (5.11) then we can find E > such that for all ε > it holds Z Ω ( u + α ) p + m − ≤ ε Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + E on (0 , T max ) . (5.12)8 roof. Thanks to Lemma 4.1 we can set q = s = 2 p + m − , p = 2 pp + m − , and make use of the Gagliardo–Nirenberg inequality in Lemma 4.3. We achieve Z Ω ( u + α ) p = (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) pp + m − L pp + m − (Ω) ≤ C GN (cid:13)(cid:13)(cid:13) ∇ ( u + α ) p + m − (cid:13)(cid:13)(cid:13) pa p + m − L (Ω) (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) p (1 − a p + m − L p + m − (Ω) + C GN (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) pp + m − L p + m − (Ω) ≤ c (cid:18)(cid:13)(cid:13)(cid:13) ∇ ( u + α ) p + m − (cid:13)(cid:13)(cid:13) L (Ω) + 1 (cid:19) pa p + m − ≤ c (cid:18)Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) (cid:19) λ on (0 , T max ) , (5.13)for some c > , λ := p + m − pa > , and where a := p + m − (1 − p ) p + m − + n − belongs to (0 , in view of relation (4.2b). Subsequently for E := ( pc ) λ and through the definition R Ω ( u + α ) p = p Φ( t ) , we have that this relation is satisfied − Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) ≤ − (cid:18) c Z Ω ( u + α ) p (cid:19) λ + 1 = − E Φ λ ( t ) + 1 on (0 , T max ) , (5.14)so that the first part of this lemma is shown.As to the second claim, we will proceed in a similar way to deal with the term R Ω ( u + α ) p + m − . Withthe aid of bound (5.11), for k := p + m − p + m − and Lemma 4.1, if we set q = 2 p p + m − , p = k, s = 2 p + m − , the Gagliardo–Nirenberg inequality given in Lemma 4.3 yields constants c > , a := p + m − p − kp + m − p + n − ∈ (0 ,
1) ( recall (4.2e) ) , and β = ka p + m − p − p + m − p + n − ∈ (0 ,
1) ( recall (4.2c) ) , (5.15)with the property that Z Ω ( u + α ) p + m − = (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) kL k (Ω) ≤ C GN (cid:13)(cid:13)(cid:13) ∇ ( u + α ) p + m − (cid:13)(cid:13)(cid:13) ka L (Ω) (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) k (1 − a ) L p p + m − (Ω) + C GN (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) p + m − p + m − L p + m − (Ω) ≤ c " (cid:18)Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) (cid:19) β on (0 , T max ) . (5.16)9pplying to the gradient term appearing in (5.16) the Young inequality, supported with the introductionof an arbitrary positive constant ε , we can write c (cid:18)Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) (cid:19) β ≤ ε Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + D ( ε ) on (0 , T max ) , (5.17)with some D ( ε ) > , so as a consequence bound (5.16) is reduced to Z Ω ( u + α ) p + m − ≤ ε Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + E for all t ∈ (0 , T max ) , (5.18)where E := c + D ( ε ) .This following result will be the last step toward the proof of Theorem 3.3. Lemma 5.3.
Under the assumptions of Proposition 3.1, let ( u, v ) be the local solution to system (1.1) ,which blows up at finite time T max in the sense of (3.1) , and Φ( t ) the energy function defined in Lemma5.1. If, additionally, for some p > n ( m − m ) it is known that for some L > k u ( · , t ) k L p (Ω) ≤ L for all t ∈ (0 , T max ) , (5.19) then there exists K > with this property: k u ( · , t ) k L p (Ω) ≤ K for all t ∈ (0 , T max ) , (5.20) and for any q > n + 2 it holds that (cid:13)(cid:13) u ( · , t )( u ( · , t ) + α ) m − ∇ v ( · , t ) (cid:13)(cid:13) L q (Ω) ≤ K for all t ∈ (0 , T max ) . (5.21) Proof.
With the results of lemmata 5.1, 5.2 and 5.3 in hour hands, we have by plugging (5.18) into (5.1) Φ ′ ( t ) ≤ − ( E − E ε ) Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + E E + E for all t ∈ (0 , T max ) . (5.22)Moreover, in order to have strictly positivity of the first term on the right-hand side of this gained inequality,we choose ε small enough as to satisfy E − E ε > . In this way, taking into consideration (5.14), relation(5.22) reads Φ ′ ( t ) ≤ − ( E − E ε ) (cid:0) E Φ λ ( t ) + 1 (cid:1) + E E + E ≤ − J Φ λ ( t ) + J on (0 , T max ) , (5.23)where J := ( E − E ε ) E and J := E E + E . Subsequently, we arrive at this initial problem ( Φ ′ ( t ) ≤ J − J Φ λ ( t ) t ∈ (0 , T max ) , Φ(0) = p R Ω ( u + α ) p , so to have, by an application of a comparison principle, Φ( t ) ≤ max ( Φ(0) , (cid:18) J J (cid:19) λ ) =: L for all t ∈ (0 , T max ) . (5.24)On the other hand, from this bound, elliptic regularity results applied to the second equation of system (1.1),i.e. − ∆ v = u − M , imply v ∈ L ∞ ((0 , T max ); W ,p (Ω)) and, hence, ∇ v ∈ L ∞ ((0 , T max ); W ,p (Ω)) . In par-ticular, the Sobolev embeddings (from Lemma 4.1 is p = ¯ p > q > n + 2 ) infer ∇ v ∈ L ∞ ((0 , T max ); L ∞ (Ω)) .Consequently, through the Hölder inequality with exponents q ( m − /p and − q ( m − /p (againLemma 4.1 ensures that p > q ( m − ), we have on (0 , T max ) Z Ω | u ( u + α ) m − ∇ v | q ≤ Z Ω ( u + α ) q ( m − |∇ v | q ≤ k∇ v ( · , t ) k q L ∞ (Ω) | Ω | p − q m − p (cid:18)Z Ω ( u + α ) p (cid:19) q m − p . Therefore, in view of estimate (5.24) we also get Z Ω | u ( u + α ) m − ∇ v | q ≤ k∇ v ( · , t ) k q L ∞ (Ω) | Ω | p − q m − p L q m − p =: L with q > n + 2 , so that (5.20) and (5.21) are attained posing K = max { ( L p ) p , ( L ) q } .10 roof of Theorem 3.3 Proposition 3.1 provides the unique local classical solution ( u, v ) to system(1.1) which blows up at finite time T max > . By reduction to the absurd, let ( u, v ) such that for all p > n ( m − m ) it holds that lim sup t → T max k u ( · , t ) k L p (Ω) < ∞ ; then, for some L > we get k u ( · , t ) k L p (Ω) ≤ L for all t ∈ (0 , T max ) . Now, for for p = ¯ p given in (4.1), Lemma 5.3 ensures that ( u ∈ L ∞ ((0 , T max ); L p (Ω)) ( for p = ¯ p ) ,u ( u + α ) m − ∇ v ∈ L ∞ ((0 , T max ); L q (Ω)) for all q > n + 2 . (5.25)Hereafter, with the same nomenclature used by Tao and Winkler, u also classically solves in Ω × (0 , T max ) problem (A.1) of [23, Appendix A] for D ( x, t, u ) = ( u + α ) m − , f ( x, t ) = χu ( u + α ) m − ∇ v, g ( x, t ) ≡ . In particular, from the boundary condition on v , we see that (A.2)–(A.5) and the second inclusion of (A.6)for any choice of q are complied. Moreover, always from the definition of ¯ p , relations (A.8), (A.9) and(A.10) of [23, Lemma A.1.] are also valid, so we have through this lemma that for some C > k u ( · , t ) k L ∞ (Ω) ≤ C for all t ∈ (0 , T max ) , which is in contradiction to the fact that the solution ( u, v ) blows up at finite time T max .
6. The ordinary differential inequality for Φ( t ) : derivation of lower bounds In preparation to the last proof, let us now use some of the above derivations to obtain an ODI for theenergy function Φ( t ) := p R Ω ( u + α ) ¯ p . This ODI, actually, is satisfied by Φ( t ) both if such energy functionis associated to a local or a global solution ( u, v ) to system (1.1); despite this, since we will make use ofthis ODI to estimate the blow–up time for T max , we also confine the forthcoming lemma to the case ofunbounded solutions. Lemma 6.1.
Under the assumptions of Proposition 3.1, let ( u, v ) be the local solution to system (1.1) ,which blows up at finite time T max in the sense of (3.1) , and Φ( t ) the energy function defined in Lemma5.1. Then there exist E , E , E > such that Φ( t ) satisfies this ODI Φ ′ ( t ) ≤ E Φ γ ( t ) + E Φ δ ( t ) + E on (0 , T max ) , (6.1) being γ, δ > as in (4.3) .Proof. We start from Lemma 5.1 and use the Gagliardo-Niremberg inequality to estimate the last term onthe right hand side of (5.1). For k := p + m − p + m − as in Lemma 4.1 and a defined in (4.2f), if we set q = s = 2 pp + m − , p = k, Lemma 4.3 yields Z Ω ( u + α ) p + m − = (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) kL k (Ω) (6.2) ≤ C GN (cid:13)(cid:13)(cid:13) ∇ ( u + α ) p + m − (cid:13)(cid:13)(cid:13) ka L (Ω) (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) k (1 − a ) L pp + m − (Ω) + C GN (cid:13)(cid:13)(cid:13) ( u + α ) p + m − (cid:13)(cid:13)(cid:13) kL pp + m − (Ω) on (0 , T max ) . σ = ka and − σ leads to Z Ω ( u + α ) p + m − ≤ E E Z Ω (cid:12)(cid:12)(cid:12) ∇ ( u + α ) p + m − (cid:12)(cid:12)(cid:12) + c Φ γ + c Φ δ for all t ∈ (0 , T max ) , (6.3)with c = p γ C GN (1 − σ ) (cid:16) E E C GN σ (cid:17) − σ − σ and c = p δ C GN . (6.4)Then, by inserting (6.3) into (5.1) gives the claimed ordinary differential inequality Φ ′ ( t ) ≤ E Φ γ ( t ) + E Φ δ ( t ) + E for all t ∈ (0 , T max ) , (6.5)with E = c E , E = c E . Proof of Theorem 3.4
For n ∈ N and m , m ∈ R complying with the blow–up restrictions ( BU ),let p = ¯ p be the number given in Lemma 4.1 and T max the finite blow–up time, in L ∞ (Ω) –norm, of thelocal solution ( u, v ) to system (1.1) provided by Proposition 3.1. Since p = ¯ p > p , from Theorem 3.3 weknow that lim sup t → T max p R Ω ( u + α ) p = ∞ . On the other hand, Lemma 6.1 ensures that u satisfies the ODI(6.1) for any < t < T max , where in particular it is seen that the function Ψ( ξ ) = E ξ γ + E ξ δ + E obeysthe Osgood criterion (2.1), where E = E (¯ p ) , E = E (¯ p ) , E = E (¯ p ) have been computed in lemmata5.1 and 6.1 and γ = γ (¯ p ) > , δ = δ (¯ p ) > defined in (4.2g). Thereafter, by integrating (6.1) between and T max , we obtain estimate (3.2), and the proof is completed. Remark 6.2.
We observe that, conversely to what happens with relation (3.2) , it is possible to obtainan explicit expression for the lower bound T by reducing (6.5) as follows: from the definition of M , i.e. M = | Ω | R Ω u ( x ) dx , and the Hölder inequality we can estimate E in relation (6.1) as E = E M | Ω | Z Ω u ≤ E M | Ω | Z Ω ( u + α ) ≤ E Φ p , (6.6) with E = E M (cid:16) p | Ω | (cid:17) p , so that (6.1) can be rewritten in this form: Φ ′ ( t ) ≤ E Φ γ ( t ) + E Φ δ ( t ) + E Φ( t ) p on (0 , T max ) . (6.7) Now, similarly to what done in [16], since Φ blows up at finite time T max there exists a time t ∈ [0 , T max ) such that Φ( t ) ≥ Φ(0) for all t ≥ t ∈ [0 , T max ) . From γ > δ > p (recall (4.2g) ), we can estimate the second and third terms of (6.7) by means of Φ γ : Φ δ ( t ) ≤ Φ(0) δ − γ Φ γ ( t ) and Φ p ( t ) ≤ Φ(0) p − γ Φ γ ( t ) for all t ≥ t ∈ [0 , T max ) . (6.8) By plugging expressions (6.8) into (6.7) we obtain for H = E + E Φ(0) δ − γ + E Φ(0) p − γ , Φ ′ ( t ) ≤ H Φ γ ( t ) for all t ≥ t ∈ [0 , T max ) , (6.9) so that an integration of (6.9) on ( t , T max ) yields this explicit lower bound for T max : Φ(0) − γ H ( γ −
1) = Z ∞ Φ(0) dτHτ γ ≤ Z T max t dτ ≤ Z T max dτ = T max . cknowledgments The authors would like to express their sincere gratitude to Professor Stella Vernier Piro and ProfessorTomomi Yokota for giving them the precious opportunity of a joint study and their encouragement. GV andMM are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and are partially supported by theresearch project
Integro–differential Equations and Non–Local Problems , funded by Fondazione di Sardegna(2017).
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