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Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type

  • Michael Winkler EMAIL logo
Published/Copyright: May 11, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

The Cauchy problem in R n , n ≥ 2 , for

u t = Δ u − ∇ ⋅ ( u S ⋅ ∇ v ) , 0 = Δ v + u , ( ⋆ )

is considered for general matrices S ∈ R n × n . A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to BUC ( R n ) ∩ L p ( R n ) with some p ∈ [ 1 , n ) , there exist T max ∈ ( 0 , ∞ ] and a uniquely determined u ∈ C 0 ( [ 0 , T max ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T max ) ; L p ( R n ) ) ∩ C ∞ ( R n × ( 0 , T max ) ) such that with v ≔ Γ ⋆ u , and with Γ denoting the Newtonian kernel on R n , the pair ( u , v ) forms a classical solution of ( ⋆ ) in R n × ( 0 , T max ) , which has the property that

if T max < ∞ , then both limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ and limsup t ↗ T max ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ .

An exemplary application of this provides a result on global classical solvability in cases when ∣ S + 1 ∣ is sufficiently small, where 1 = diag ( 1 , … , 1 ) .

Keywords: chemotaxis; self-interacting particles; local existence; extensibility
MSC 2010: 35K55; 35B65; 35Q92; 92C17

1 Introduction

The Cauchy problem for the Keller-Segel system,

(1.1) u t = Δ u − ∇ ⋅ ( u S ⋅ ∇ v ) , x ∈ R n , t > 0 , 0 = Δ v + u , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ R n ,

has been at the core of interest in numerous studies concerned with fundamental principles of self-organization and spontaneous aggregation both in biology and in astrophysics: in the case when the cross-diffusion mechanism in (1.1) represents complete attraction in the sense that the tensor S ∈ R n × n coincides with the unit matrix 1 = diag ( 1 , … , 1 ) ∈ R n × n , this problem has been used since the 1940s to describe self-interaction of particles, mediated, e.g., through gravitational or electrical potentials [1–4]. As very similar types of interplay have been found responsible for striking experimental observations on chemotactically induced spontaneous aggregation in microbial populations, from the 1970s on system (1.1) has received considerable additional attention [2,5–8], with the choice S = 1 again being of predominant interest initially, but with more general classes of S appearing especially in the literature of the past few years, quite in line with experimental and refined modeling-oriented literature reflecting rotational parts in cross-diffusive movement [9].

According to these roles of (1.1) in application contexts, a predominant focus of corresponding analytical studies has ever been on phenomena related to blow-up of solutions, as widely being understood as a mathematical expression of spontaneous aggregation [6]. In fact, one of the striking technical advantages of the markedly slim parabolic–elliptic setting of the Cauchy problem (1.1) consists in the circumstance that the destabilizing action of cross-diffusion can formally be traced by methods significantly more elementary than those that appear necessary in more complex frameworks of related boundary value problems [10,11], or fully parabolic variants [12–16]: nonexistence of global solutions to (1.1) with presupposed suitably favorable smoothness and integrability features, namely, can already be conjectured by means of fairly simple virial-type arguments based on moment evolution [4,17–19].

For rigorous confirmations of such reasonings, having available not only a handy theory of local classical solvability, but especially also corresponding conveniently applicable criteria for extensibility, seems of considerable relevance; to the best of our knowledge, however, these issues have not been addressed comprehensively in the literature so far. Indeed, basic existence theories for (1.1) with S = 1 are fairly well developed both in frameworks of weak solutions enjoying certain additional properties related to the evolution of moments and energies [17,20,21], and within concepts of mild solutions that involve various types of requirements on spatial regularity, operating inter alia in spaces of pseudomeasures or in Morrey spaces [2,22–25]. Unlike in relatives of (1.1) posed in bounded domains [26], however, settings of genuinely classical solutions seem to have not been explicitly covered in the literature.

Main results. The purpose of the present note is to briefly describe an approach toward local-in-time existence and uniqueness in (1.1) that operates in frameworks of classical solutions, and that artlessly provides some easy-to-handle basic information on behavior near maximal existence times in cases when these are finite. This will be achieved on the basis of an essentially well-established argument involving Banach’s fixed point theorem, in contrast to previous related literature [2] now arranged in a setting of solutions ( u , v ) for which u ( ⋅ , t ) belongs to BUC ( R n ) ∩ L p ( R n ) for fixed times t . Here, as usual, BUC ( R n ) denotes the Banach space of bounded and uniformly continuous functions on R n , and the summability exponent p ≥ 1 will be chosen small enough such that the second equation in (1.1) can be fulfilled in a standard manner by defining v through convolution with the corresponding Newtonian kernel. By construction, at this first stage, we will obtain solvability up to some maximal T max ≤ ∞ , which, if being finite, satisfies ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) + ‖ u ( ⋅ , t ) ‖ L p ( R n ) → ∞ as T ↗ T max (Lemma 2.2). A refined analysis relying on the local regularity properties of this solution asserted by this first step, however, will afterward enable us to make sure that actually the spatial L ∞ norm of any non-global solution must become unbounded, and to hence exclude any situation of blow-up due to a loss of sufficient spatial decay at t = T max , occurring exclusively with respect to L p norms with finite p (Lemma 3.1); apart from that, an independent additional argument will ensure that also ∇ v cannot remain bounded near any finite blow-up time (Lemma 3.2).

In summary, this will lead to the following first result of this manuscript, where as usual we let B R ( 0 ) ≔ { x ∈ R n ∣ ∣ x ∣ < R } for n ≥ 2 and R > 0 .

Proposition 1.1

Let n ≥ 2 and S ∈ R n × n , and suppose that with some p ∈ [ 1 , n ) , the function u 0 ∈ BUC ( R n ) ∩ L p ( R n ) is non-negative. Then, there exist T max ∈ ( 0 , ∞ ] and a non-negative

(1.2) u ∈ C 0 ( [ 0 , T max ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T max ) ; L p ( R n ) ) ∩ C ∞ ( R n × ( 0 , T max ) )

such that writing v ( ⋅ , t ) ≔ Γ ∗ u ( ⋅ , t ) , t ∈ ( 0 , T max ) , with

(1.3) Γ ( z ) ≔ − 1 2 π ln ∣ z ∣ , z ∈ R n \ { 0 } , i f n = 2 , 1 n ( n − 2 ) ∣ B 1 ( 0 ) ∣ ∣ z ∣ 2 − n z ∈ R n \ { 0 } , i f n ≥ 3 ,

we obtain v ∈ C ∞ ( R n × ( 0 , T max ) ) such that

(1.4) ∇ v ∈ L loc ∞ ( [ 0 , T max ) ; L ∞ ( R n ; R n ) ) ,

that ( u , v ) forms a classical solution of (1.1) in R n × ( 0 , T max ) , and that

(1.5) i f T max < ∞ , t h e n b o t h limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ a n d limsup t ↗ T max ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ .

This solution is uniquely determined in the sense that if  T ∈ ( 0 , T max ) , and if  ( u ^ , v ^ ) is a classical solution of (1.1) in R n × ( 0 , T ) fulfilling u ^ ∈ C 0 ( [ 0 , T ] ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T ] ; L p ( R n ) ) ∩ C 2 , 1 ( R n × ( 0 , T ) ) and v ^ ∈ C 2 , 0 ( R n × ( 0 , T ) ) as well as ∇ v ^ ∈ L ∞ ( R n × ( 0 , T ) ; R n ) , then u ^ ≡ u in R n × ( 0 , T ) .

Moreover, in the case when p = 1 , we additionally have

(1.6) ∫ R n u ( ⋅ , t ) = ∫ R n u 0 f o r a l l t ∈ ( 0 , T max ) .

Proposition 1.1 can be used to provide alternative approaches toward blow-up detections in (1.1)[2,4,17,18], under appropriate assumptions on initial concentratedness now yielding solutions known to be bounded and continuous up to some T max ∈ ( 0 , ∞ ) , and to satisfy limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ due to (1.5); some exemplary applications in this direction can be found detailed in [27]. Moreover, as will also be seen in [27], the second part of the blow-up characterization in (1.5) can be used to preclude the occurrence of explosions especially in some radially symmetric situations, in which suitable conditions on smallness of u 0 ensure L ∞ bounds for the functions ( r , t ) ↦ r ⨍ B r ( 0 ) u ( ⋅ , t ) .

In the second part of the present manuscript, we shall next apply the above general theory in order to derive a result on global solvability in (1.1), unconditional with respect to the size of the initial data, in cases when the matrix S represents cross-diffusion, which is suitably close to repulsion. In planar situations, this will rigorously confirm part of the properties presented and formally addressed in [19]; in fact, no additional requirements on finiteness of second moments, and hence on spatial decay of u 0 , will be needed here, and furthermore, the obtained solutions will be smooth and bounded. In three- or higher-dimensional frameworks, this even seems to be the first statement on global solvability in (1.1) when rotational flux components are involved. Indeed, by making essential use of the refined extensibility criterion in (1.5) we will see in Section 4 that the solutions from Proposition 1.1 actually are global and bounded when ∣ S + 1 ∣ is suitably small:

Proposition 1.2

Let n ≥ 2 . Then, there exists δ > 0 with the property that if

(1.7) S = − 1 + S 1 ,

with some S 1 ∈ R n × n fulfilling

(1.8) ∣ S 1 ∣ ≤ δ ,

then for any non-negative u 0 ∈ BUC ( R n ) ∩ L 1 ( R n ) , problem (1.1) admits a global classical solution ( u , v ) with

(1.9) u ∈ C 0 ( [ 0 , ∞ ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , ∞ ) ; L 1 ( R n ) ) ∩ C ∞ ( R n × ( 0 , ∞ ) )

and v ( ⋅ , t ) = Γ ∗ u ( ⋅ , t ) , t > 0 , uniquely determined in the sense specified in Proposition 1.1. This solution is bounded in the sense that there exists C > 0 satisfying

(1.10) ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ C f o r a l l t > 0 .

2 Local existence, uniqueness, non-negativity and mass conservation

As a starting point for our analysis, let us state a basic observation on Newtonian potentials.

Lemma 2.1

Let n ≥ 2 , and for z ∈ R n \ { 0 } , let Γ be as in (1.3). Then for each p ∈ [ 1 , n ) , there exists γ = γ ( p ) > 0 such that

(2.1) ‖ ∇ ( Γ ∗ φ ) ‖ L ∞ ( R n ) ≤ γ ∣ ∣ ∣ φ ∣ ∣ ∣ p f o r a l l φ ∈ L ∞ ( R n ) ∩ L p ( R n ) ,

where we have set

(2.2) ∣ ∣ ∣ φ ∣ ∣ ∣ p ≔ ‖ φ ‖ L ∞ ( R n ) + ‖ φ ‖ L p ( R n ) for φ ∈ L ∞ ( R n ) ∩ L p ( R n ) .

Proof

We decompose

(2.3) ∇ ( Γ ∗ φ ) ( x ) = ∫ ∣ x − y ∣ ≤ 1 ( ∇ Γ ) ( x − y ) φ ( y ) d y + ∫ ∣ x − y ∣ > 1 ( ∇ Γ ) ( x − y ) φ ( y ) d y , x ∈ R n ,

and observe that since

∣ ( ∇ Γ ) ( z ) ∣ ≤ c 1 ∣ z ∣ 1 − n for all z ∈ R n \ { 0 } , with c 1 ≔ 1 2 π if n = 2 , 1 n ∣ B 1 ( 0 ) ∣ if n ≥ 3 ,

we have

∫ ∣ x − y ∣ ≤ 1 ( ∇ Γ ) ( x − y ) φ ( y ) d y ≤ c 1 ‖ φ ‖ L ∞ ( R n ) ∫ ∣ z ∣ ≤ 1 ∣ z ∣ 1 − n d z for all x ∈ R n .

Moreover, if p > 1 , then by the Hölder inequality,

∫ ∣ x − y ∣ > 1 ( ∇ Γ ) ( x − y ) φ ( y ) d y ≤ c 1 ‖ φ ‖ L p ( R n ) ⋅ ∫ ∣ z ∣ > 1 ∣ z ∣ ( 1 − n ) ⋅ p p − 1 d z p − 1 p = c 1 ⋅ ( p − 1 ) ⋅ n ∣ B 1 ( 0 ) ∣ n − p p − 1 p ⋅ ‖ φ ‖ L p ( R n ) for all x ∈ R n ,

while in the case p = 1 , we can simply estimate

∫ ∣ x − y ∣ > 1 ( ∇ Γ ) ( x − y ) φ ( y ) d y ≤ c 1 ‖ φ ‖ L 1 ( R n ) for all x ∈ R n .

Therefore, (2.1) results from (2.3).□

The inequality in (2.1) will form a crucial ingredient in an otherwise fairly straightforward derivation of local solvability in (1.1) on the basis of a contraction mapping argument.

Lemma 2.2

Let n ≥ 2 and S ∈ R n × n , let p ∈ [ 1 , n ) , and let u 0 ∈ BUC ( R n ) ∩ L p ( R n ) be non-negative. Then, there exist T max ∈ ( 0 , ∞ ] as well as a function

(2.4) u ∈ C 0 ( [ 0 , T max ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T max ) ; L p ( R n ) ) ∩ C ∞ ( R n × ( 0 , T max ) ) ,

such that letting

(2.5) v ( ⋅ , t ) ≔ Γ ∗ u ( ⋅ , t ) t ∈ ( 0 , T max ) ,

defines a function v ∈ C ∞ ( R n × ( 0 , T max ) ) , which satisfies

(2.6) ∇ v ∈ L loc ∞ ( [ 0 , T max ) ; L ∞ ( R n ; R n ) ) ,

which is such that ( u , v ) solves (1.1) in the classical sense in R n × ( 0 , T max ) , and that

(2.7) i f T max < ∞ , t h e n ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) + ‖ u ( ⋅ , t ) ‖ L p ( R n ) → ∞ a s t ↗ T max .

Proof

With T ∈ ( 0 , 1 ) to be specified below, in the Banach space

X ≔ C 0 ( [ 0 , T ] ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T ] ; L p ( R n ) ) ,

we fix the closed subset

ℳ ≔ { u ∈ X ∣ ‖ u ‖ X ≤ R + 1 } ,

where ‖ u ‖ X ≔ sup t ∈ ( 0 , T ) ∣ ∣ ∣ u ( ⋅ , t ) ∣ ∣ ∣ p for u ∈ X , and where R ≔ ∣ ∣ ∣ u 0 ∣ ∣ ∣ p . Given u ∈ ℳ , with Γ as in (1.3), we then let v ( ⋅ , t ) ≔ Γ ∗ u ( ⋅ , t ) for t ∈ ( 0 , T ) and define

( F u ) ( ⋅ , t ) ≔ e t Δ u 0 − ∫ 0 t ∇ ⋅ e ( t − s ) Δ { u ( ⋅ , t ) S ⋅ ∇ v ( ⋅ , s ) } d s , t ∈ [ 0 , T ] ,

where ( e t Δ ) t ≥ 0 denotes the heat semigroup on R n . Then according to known smoothing properties thereof ([28, Sect. 48.2]), by using Lemma 2.1, it can readily be verified that F u belongs to X for any such u and that there exist positive constants c 1 , c 2 and c 3 such that whenever u ∈ ℳ ,

∥ ( F u ) ( ⋅ , t ) ∥ L ∞ ( R n ) ≤ ‖ u 0 ‖ L ∞ ( R n ) + c 1 ∫ 0 t ( t − s ) − 1 2 ∥ u ( ⋅ , s ) S ⋅ ∇ v ( ⋅ , s ) ∥ L ∞ ( R n ) d s ≤ ‖ u 0 ‖ L ∞ ( R n ) + c 2 ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) ‖ ∇ v ( ⋅ , s ) ‖ L ∞ ( R n ) d s ≤ ‖ u 0 ‖ L ∞ ( R n ) + c 3 ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) ∣ ∣ ∣ u ( ⋅ , s ) ∣ ∣ ∣ p d s ≤ ‖ u 0 ‖ L ∞ ( R n ) + c 3 ∫ 0 t ( t − s ) − 1 2 ∣ ∣ ∣ u ( ⋅ , s ) ∣ ∣ ∣ p 2 d s ≤ ‖ u 0 ‖ L ∞ ( R n ) + 2 c 3 ( R + 1 ) 2 T 1 2 for all t ∈ [ 0 , T ] .

As we can similarly find c i > 0 , i ∈ { 4 , 5 , 6 } , such that for any choice of u ∈ ℳ , we have

∥ ( F u ) ( ⋅ , t ) ∥ L p ( R n ) ≤ ‖ u 0 ‖ L p ( R n ) + c 4 ∫ 0 t ( t − s ) − 1 2 ∥ u ( ⋅ , s ) S ⋅ ∇ v ( ⋅ , s ) ∥ L p ( R n ) d s ≤ ‖ u 0 ‖ L p ( R n ) + c 5 ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L p ( R n ) ‖ ∇ v ( ⋅ , s ) ‖ L ∞ ( R n ) d s ≤ ‖ u 0 ‖ L p ( R n ) + c 6 ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L p ( R n ) ∣ ∣ ∣ u ( ⋅ , s ) ∣ ∣ ∣ p d s ≤ ‖ u 0 ‖ L p ( R n ) + c 6 ∫ 0 t ( t − s ) − 1 2 ∣ ∣ ∣ u ( ⋅ , s ) ∣ ∣ ∣ p 2 d s ≤ ‖ u 0 ‖ L p ( R n ) + 2 c 6 ( R + 1 ) 2 T 1 2 for all t ∈ [ 0 , T ] ,

we infer that if 2 ( c 3 + c 6 ) ( R + 1 ) 2 T 1 2 ≤ 1 , then

∣ ∣ ∣ ( F u ) ( ⋅ , t ) ∣ ∣ ∣ p ≤ ‖ u 0 ‖ L ∞ ( R n ) + ‖ u 0 ‖ L p ( R n ) + 1 = R + 1 for all t ∈ [ 0 , T ] ,

meaning that F ℳ ⊂ ℳ . Since by straightforward adaptation of the above argument it can be shown that if T is further diminished, then also

‖ ( F u − F u ¯ ) ( ⋅ , t ) ‖ L ∞ ( R n ) + ‖ ( F u − F u ¯ ) ( ⋅ , t ) ‖ L p ( R n ) ≤ 1 2 ‖ u − u ¯ ‖ X for all  u ∈ ℳ , u ¯ ∈ ℳ  and t ∈ [ 0 , T ] ,

it follows that if we fix some T = T ( R ) ∈ ( 0 , 1 ) suitably small such that F acts as a contractive self-map on ℳ .

For its fixed point u , correspondingly existing thanks to the Banach contraction mapping theorem, it can then readily be verified by means of a standard reasoning that the pair ( u , v ) , with v as accordingly introduced above, solves (1.1) in the natural weak sense underlying the analysis in [29], and that hence, according to the well-established bootstrap arguments involving interior parabolic and elliptic Hölder and Schauder estimates [29–31], both u and v actually belong to C ∞ ( R n × ( 0 , T ) ) and solve (1.1) classically. As the above choice of T = T ( R ) depends on R = ∣ ∣ ∣ u 0 ∣ ∣ ∣ p only, extensibility up to some maximal T max ∈ ( 0 , ∞ ] fulfilling (2.7) is evident.□

A uniqueness property in the style of that from Proposition 1.1 will be verified by means of an argument based on Duhamel-tape identities satisfied by differences between two solutions. In order to unambiguously guarantee validity of corresponding variation-of-constants representations for arbitrary solutions from the class of functions addressed above, our reasoning in this direction will be arranged in a suitably localized setting. In preparation of this and of several further related arguments to follow, let us fix ζ ( 0 ) ∈ C ∞ ( R ) such that 0 ≤ ζ ( 0 ) ≤ 1 on R , ζ ( 0 ) ≡ 1 on ( − ∞ , 0 ) and supp ζ ( 0 ) ⊂ ( − ∞ , 1 ) , and for R > 1 , we let

(2.8) ζ R ( x ) ≔ ζ ( 0 ) ( ∣ x ∣ − R ) , x ∈ R n ,

noting that then ζ R ∈ C 0 ∞ ( R n ) with

(2.9) 0 ≤ ζ R ≤ 1 on R n , ζ R ≡ 1 in  B R and supp ζ R ⊂ B R + 1 ,

where we have abbreviated B R ≔ B R ( 0 ) ⊂ R n for R > 1 . Moreover, writing K ζ ≔ ∥ ( ζ ( 0 ) ) ″ ∥ L ∞ ( R ) + n ∥ ( ζ ( 0 ) ) ′ ∥ L ∞ ( R ) , we see that

(2.10) ∣ ∇ ζ R ∣ + ∣ Δ ζ R ∣ ≤ K ζ on R n for all R > 1 .

Utilizing this family of cutoff functions, we can now safely derive a statement on uniqueness, which actually is slightly more comprehensive than that announced in Proposition 1.1:

Lemma 2.3

Let n ≥ 2 and p ∈ [ 1 , n ) , let S ∈ R n × n and 0 ≤ u 0 ∈ BUC ( R n ) ∩ L p ( R n ) , and let T max and ( u , v ) be as obtained in Lemma 2.2. Then, ( u , v ) is unique in the sense that whenever T ∈ ( 0 , T max ) , u ^ ∈ C 0 ( [ 0 , T ] ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T ] ; L p ( R n ) ) ∩ C 2 , 1 ( R n × ( 0 , T ) ) ; and v ^ ∈ C 2 , 0 ( R n × ( 0 , T ) ) are such that ∇ v ^ ∈ L ∞ ( R n × ( 0 , T ) ; R n ) and that ( u ^ , v ^ ) forms a classical solution of (1.1) in R n × ( 0 , T ) , it follows that there exists v ˜ ∈ C 2 , 0 ( R n × ( 0 , T ) ) such that Δ v ˜ ( ⋅ , t ) ≡ 0 for all t ∈ ( 0 , T ) , and that ( u ^ , v ^ ) = ( u , v + v ˜ ) in R n × ( 0 , T ) .

Proof

According to the regularity features asserted by Lemma 2.2, as well as the current hypothesis, we can find c 1 > 0 and c 2 > 0 such that

(2.11) ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ c 1 for all t ∈ ( 0 , T ) ,

and that

(2.12) ∣ ∣ ∣ u ^ ( ⋅ , t ) ∣ ∣ ∣ p ≤ c 2 for all t ∈ ( 0 , T ) ,

and in order to make sure that these inequalities imply that

(2.13) ∣ ∣ ∣ u ( ⋅ , t ) − u ^ ( ⋅ , t ) ∣ ∣ ∣ p = 0 for all t ∈ ( 0 , T ) ,

we take ( ζ R ) R > 1 from (2.8) and note that then for each R > 1 , thanks to (1.1),

∂ t { ζ R ⋅ ( u − u ^ ) } = ζ R Δ ( u − u ^ ) − ζ R ∇ ⋅ ( u S ⋅ ∇ v ) + ζ R ∇ ⋅ ( u ^ S ⋅ ∇ v ^ ) = Δ { ζ R ⋅ ( u − u ^ ) } − 2 ∇ ⋅ { ( u − u ^ ) ∇ ζ R } + ( u − u ^ ) Δ ζ R − ∇ ⋅ { ζ R ⋅ ( u − u ^ ) S ⋅ ∇ v } + ( u − u ^ ) ( S ⋅ ∇ v ) ⋅ ∇ ζ R − ∇ ⋅ { ζ R u ^ S ⋅ ∇ ( v − v ^ ) } + u ^ ( S ⋅ ∇ ( v − v ^ ) ) ⋅ ∇ ζ R

holds in the classical pointwise sense in R n × ( 0 , T ) . Since supp ζ R is bounded, we may thus rely on a corresponding variation-of-constants representation to see that again due to known regularization features of the heat semigroup ( e t Δ ) t ≥ 0 , there exists c 3 > 0 such that for both choices of q from the set { p , ∞ } ,

(2.14) ∥ ζ R ⋅ ( u ( ⋅ , t ) − u ^ ( ⋅ , t ) ) ∥ L q ( R n ) ≤ c 3 ∫ 0 t ( t − s ) − 1 2 ∥ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) ∇ ζ R ∥ L q ( R n ) d s + c 3 ∫ 0 t ∥ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) Δ ζ R ∥ L q ( R n ) d s + c 3 ∫ 0 t ( t − s ) − 1 2 ∥ ζ R ⋅ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) S ⋅ ∇ v ( ⋅ , s ) ∥ L q ( R n ) d s + c 3 ∫ 0 t ∥ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) ( S ⋅ ∇ v ( ⋅ , s ) ) ⋅ ∇ ζ R ∥ L q ( R n ) d s + c 3 ∫ 0 t ( t − s ) − 1 2 ∥ ζ R u ^ ( ⋅ , s ) S ⋅ ∇ ( v ( ⋅ , s ) − v ^ ( ⋅ , s ) ) ∥ L q ( R n ) d s + c 3 ∫ 0 t ∥ u ^ ( ⋅ , s ) ( S ⋅ ∇ ( v ( ⋅ , s ) − v ^ ( ⋅ , s ) ) ) ⋅ ∇ ζ R ∥ L q ( R n ) d s

for all t ∈ ( 0 , T ) , because u ( ⋅ , 0 ) − u ^ ( ⋅ , 0 ) ≡ 0 in R n by assumption. Here, thanks to (2.9), (2.10), and the rough estimate 1 ≤ T 1 2 ( t − s ) − 1 2 for 0 < s < t < T ,

(2.15) c 3 ∫ 0 t ( t − s ) − 1 2 ∥ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) ∇ ζ R ∥ L q ( R n ) d s + c 3 ∫ 0 t ∥ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) Δ ζ R ∥ L q ( R n ) d s ≤ c 3 K ζ ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ‖ L q ( R n ) d s + c 3 K ζ ∫ 0 t ‖ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ‖ L q ( R n ) d s ≤ c 3 K ζ ( 1 + T 1 2 ) ∫ 0 t ( t − s ) − 1 2 ∣ ∣ ∣ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ∣ ∣ ∣ p d s for all t ∈ ( 0 , T ) ,

while a combination of (2.9) and (2.10) with (2.11) shows that, similarly,

(2.16) c 3 ∫ 0 t ( t − s ) − 1 2 ∥ ζ R ⋅ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) S ⋅ ∇ v ( ⋅ , s ) ∥ L q ( R n ) d s + c 3 ∫ 0 t ∥ ( u ( ⋅ , s ) − u ^ ( ⋅ , s ) ) ( S ⋅ ∇ v ( ⋅ , s ) ) ⋅ ∇ ζ R ∥ L q ( R n ) d s ≤ c 3 ∣ S ∣ ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ‖ L q ( R n ) ‖ ∇ v ( ⋅ , s ) ‖ L ∞ ( R n ) d s + c 3 K ζ ∣ S ∣ ∫ 0 t ‖ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ‖ L q ( R n ) ‖ ∇ v ( ⋅ , s ) ‖ L ∞ ( R n ) d s ≤ c 1 c 3 ∣ S ∣ ( 1 + K ζ T 1 2 ) ∫ 0 t ( t − s ) − 1 2 ∣ ∣ ∣ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ∣ ∣ ∣ p d s for all t ∈ ( 0 , T ) .

Furthermore, using (2.9) and (2.10) along with (2.12) and Lemma 2.1 we obtain that with γ > 0 taken from the latter we have

c 3 ∫ 0 t ( t − s ) − 1 2 ∥ ζ R u ^ ( ⋅ , s ) S ⋅ ∇ ( v ( ⋅ , s ) − v ^ ( ⋅ , s ) ) ∥ L q ( R n ) d s + c 3 ∫ 0 t ∥ u ^ ( ⋅ , s ) ( S ⋅ ∇ ( v ( ⋅ , s ) − v ^ ( ⋅ , s ) ) ) ⋅ ∇ ζ R ∥ L q ( R n ) d s ≤ c 3 ∣ S ∣ ∫ 0 t ( t − s ) − 1 2 ‖ u ^ ( ⋅ , s ) ‖ L q ( R n ) ∥ ∇ ( v ( ⋅ , s ) − v ^ ( ⋅ , s ) ) ∥ L ∞ ( R n ) d s + c 3 K ζ ∣ S ∣ ∫ 0 t ‖ u ^ ( ⋅ , s ) ‖ L q ( R n ) ∥ ∇ ( v ( ⋅ , s ) − v ^ ( ⋅ , s ) ) ∥ L ∞ ( R n ) d s ≤ c 2 c 3 ∣ S ∣ ( 1 + K ζ T 1 2 ) ∫ 0 t ( t − s ) − 1 2 ∥ ∇ ( v ( ⋅ , s ) − v ^ ( ⋅ , s ) ) ∥ L ∞ ( R n ) d s ≤ c 2 c 3 γ ∣ S ∣ ( 1 + K ζ T 1 2 ) ∫ 0 t ( t − s ) − 1 2 ∣ ∣ ∣ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ∣ ∣ ∣ p d s for all t ∈ ( 0 , T ) ,

so that from (2.14)–(2.16) we infer that with some c 4 > 0 we have

(2.17) ∣ ∣ ∣ ζ R ⋅ ( u ( ⋅ , t ) − u ^ ( ⋅ , t ) ) ∣ ∣ ∣ p ≤ c 4 ∫ 0 t ( t − s ) − 1 2 ∣ ∣ ∣ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ∣ ∣ ∣ p d s for all t ∈ ( 0 , T ) .

Since ζ R → 1 a.e. in R n as R → ∞ by (2.9), in view of Fatou’s lemma this entails that

∣ ∣ ∣ u ( ⋅ , t ) − u ^ ( ⋅ , t ) ∣ ∣ ∣ p ≤ c 4 ∫ 0 t ( t − s ) − 1 2 ∣ ∣ ∣ u ( ⋅ , s ) − u ^ ( ⋅ , s ) ∣ ∣ ∣ p d s for all t ∈ ( 0 , T ) ,

and thereby, due to the continuity of [ 0 , T ] ∋ t ↦ ∣ ∣ ∣ u ( ⋅ , t ) − u ^ ( ⋅ , t ) ∣ ∣ ∣ p guaranteed by the regularity features of u and u ^ , indeed implies (2.13) upon an application of a Gronwall-type inequality [32, Exercise 4 ⋆ , p. 190].

As thus u ^ ≡ u and, in particular, Δ v ^ ≡ Δ v , the existence of v ˜ with the claimed properties is evident.□

In quite a straightforward manner, the functions from (2.8) can also be used in the course of a localized testing procedure asserting sign preservation in the first solution components:

Lemma 2.4

Let n ≥ 2 and S ∈ R n × n , and for p ∈ [ 1 , n ) and 0 ≤ u 0 ∈ BUC ( R n ) ∩ L p ( R n ) , let T max ∈ ( 0 , ∞ ] and ( u , v ) be as obtained in Lemma 2.2. Then,

(2.18) u ( x , t ) ≥ 0 f o r a l l x ∈ R n a n d t ∈ ( 0 , T max ) .

Proof

We fix T ∈ ( 0 , T max ) and can then find c 1 ( T ) > 0 and c 2 ( T ) > 0 such that

(2.19) ∣ u ( x , t ) ∣ ≤ c 1 ( T ) and ∣ ∇ v ( x , t ) ∣ ≤ c 2 ( T ) for all  x ∈ R n  and t ∈ ( 0 , T ) .

We moreover take any q > 2 such that q ≥ p , and note that then R ∋ z ↦ z − q ≔ max { − z , 0 } belongs to C 2 ( R ) , whence for each R > 1 we may use (1.1) to see that for all t ∈ ( 0 , T ) ,

(2.20) 1 q d d t ∫ R n ζ R 2 u − q = − ∫ R n ζ R 2 u − q − 1 ∇ ⋅ { ∇ u − u S ⋅ ∇ v }

(2.21) = − ( q − 1 ) ∫ R n ζ R 2 u − q − 2 ∣ ∇ u ∣ 2 + 2 ∫ R n ζ R u − q − 1 ∇ u ⋅ ∇ ζ R − ( q − 1 ) ∫ R n ζ R 2 u − q − 1 ∇ u ⋅ ( S ⋅ ∇ v ) + 2 ∫ R n ζ R u − q ( S ⋅ ∇ v ) ⋅ ∇ ζ R .

Here, by Young’s inequality, (2.9) and (2.10), with K ζ as introduced there we have

(2.22) 2 ∫ R n ζ R u − q − 1 ∇ u ⋅ ∇ ζ R ≤ q − 1 2 ∫ R n ζ R 2 u − q − 2 ∣ ∇ u ∣ 2 + 2 q − 1 ∫ R n ∣ u ∣ q ∣ ∇ ζ R ∣ 2 ≤ q − 1 2 ∫ R n ζ R 2 u − q − 2 ∣ ∇ u ∣ 2 + 2 K ζ 2 q − 1 ∫ B R + 1 \ B R ∣ u ∣ q for all t ∈ ( 0 , T ) ,

whereas combining Young’s inequality with (2.19) shows that

(2.23) − ( q − 1 ) ∫ R n ζ R 2 u − q − 1 ∇ u ⋅ ( S ⋅ ∇ v ) ≤ q − 1 2 ∫ R n ζ R 2 u − q − 2 ∣ ∇ u ∣ 2 + q − 1 2 ∫ R n ζ R 2 u − q ∣ S ⋅ ∇ v ∣ 2 ≤ q − 1 2 ∫ R n ζ R 2 u − q − 2 ∣ ∇ u ∣ 2 + q − 1 2 c 2 2 ( T ) ∣ S ∣ 2 ∫ R n ζ R 2 u − q

for all t ∈ ( 0 , T ) . As furthermore, again by (2.9), (2.10), and (2.19),

2 ∫ R n ζ R u − q ( S ⋅ ∇ v ) ⋅ ∇ ζ R ≤ 2 c 2 ( T ) ∣ S ∣ K ζ ∫ B R + 1 \ B R ∣ u ∣ q for all t ∈ ( 0 , T ) ,

from (2.20)–(2.23) we obtain that

(2.24) d d t ∫ R n ζ R 2 u − q ≤ c 3 ( T ) ∫ R n ζ R 2 u − q + c 4 ( T ) ∫ B R + 1 \ B R ∣ u ∣ q for all t ∈ ( 0 , T ) ,

where c 3 ( T ) ≔ q ( q − 1 ) 2 c 2 2 ( T ) ∣ S ∣ 2 and c 4 ( T ) ≔ 2 q K ζ 2 q − 1 + 2 q c 2 ( T ) ∣ S ∣ K ζ . Since u 0 ≥ 0 , an integration of (2.24) leads to the inequality

(2.25) ∫ R n ζ R 2 u − q ( ⋅ , t ) ≤ ∫ 0 t e c 3 ( T ) ⋅ ( t − s ) ⋅ c 4 ( T ) ⋅ ∫ B R + 1 \ B R ∣ u ( ⋅ , s ) ∣ q d s ≤ c 4 ( T ) e c 3 ( T ) ⋅ T ∫ 0 T ∫ B R + 1 \ B R ∣ u ∣ q for all t ∈ ( 0 , T ) ,

where using that q ≥ p , by means of the dominated convergence theorem we find that

∫ 0 T ∫ B R + 1 \ B R ∣ u ∣ q ≤ c 1 q − p ( T ) ∫ 0 T ∫ B R + 1 \ B R ∣ u ∣ p → 0 as R → ∞

because of the inclusion u ∈ L p ( R n × ( 0 , T ) ) implied by Lemma 2.2. Once more relying on (2.9), according to Fatou’s lemma we thus conclude from (2.25) on letting R → ∞ that

∫ R n u − q ( ⋅ , t ) = 0 for all t ∈ ( 0 , T ) ,

and that hence (2.18) results after taking T ↗ T max .□

A close relative of the above procedure leads, in its first step, to a localized counterpart of an identity describing the evolution of spatial L q norms. Besides asserting mass conservation in Lemma 2.6, this will form a crucial ingredient in our refinement of the extensibility criterion (3.1) in Lemma 3.1, and will moreover be applied later on in our analysis of the approximately repulsive version of (1.1) addressed in Proposition 1.2 (cf. Lemma 4.1).

Lemma 2.5

Let n ≥ 2 , S ∈ R n × n , and p ∈ [ 1 , n ) , let 0 ≤ u 0 ∈ BUC ( R n ) ∩ L p ( R n ) , and let T max and ( u , v ) as well as ( ζ R ) R > 1 be as in Lemma 2.2  and (2.8). Then whenever q ≥ 1 and R > 1 ,

(2.26) d d t ∫ R n ζ R 2 u q + q ( q − 1 ) ∫ R n ζ R 2 u q − 2 ∣ ∇ u ∣ 2 = − 2 q ∫ R n ζ R u q − 1 ∇ u ⋅ ∇ ζ R − ( q − 1 ) ∫ R n ζ R 2 u q ∇ ⋅ ( S ⋅ ∇ v ) + 2 ∫ R n ζ R u q ( S ⋅ ∇ v ) ⋅ ∇ ζ R f o r a l l t ∈ ( 0 , T max ) .

Proof

This can be verified upon integrating by parts in (1.1) in a straightforward manner.□

Indeed, in the case when in Lemma 2.2 we have p = 1 , constancy of spatial L 1 norms can be obtained as a straightforward consequence of this.

Lemma 2.6

Let n ≥ 2 and S ∈ R n × n , and let 0 ≤ u 0 ∈ BUC ( R n ) ∩ L 1 ( R n ) . Then, the solution ( u , v ) of (1.1) from Lemma 2.2  has the property that ∫ R n u ( ⋅ , t ) = ∫ R n u 0 for all t ∈ ( 0 , T max ) .

Proof

For T ∈ ( 0 , T max ) , we again pick c 1 ( T ) > 0 such that ∣ ∇ v ∣ ≤ c 1 ( T ) in R n × ( 0 , T ) , and then obtain that in the identity

(2.27) d d t ∫ R n ζ R 2 u = h R ( t ) ≔ − 2 ∫ R n ζ R ∇ u ⋅ ∇ ζ R + 2 ∫ R n ζ R u ( S ⋅ ∇ v ) ⋅ ∇ ζ R , t ∈ ( 0 , T ) ,

valid according to Lemma 2.5, we can again rely on (2.9) and (2.10) to verify that

(2.28) 2 ∫ R n ζ R u ( S ⋅ ∇ v ) ⋅ ∇ ζ R ≤ c 2 ( T ) ∫ B R + 1 \ B R u for all t ∈ ( 0 , T ) ,

where c 2 ( T ) ≔ 2 c 1 ( T ) K ζ ∣ S ∣ . In the first summand on the right of (2.27), we first integrate by parts once more to see that thereafter we may employ (2.9) and (2.10) in estimating

(2.29) − 2 ∫ R n ζ R ∇ u ⋅ ∇ ζ R = 2 ∫ R n ζ R u Δ ζ R + 2 ∫ R n u ∣ ∇ ζ R ∣ 2 ≤ c 3 ∫ B R + 1 \ B R u for all t ∈ ( 0 , T )

with c 3 ≔ 2 K ζ + 2 K ζ 2 . Now since u ∈ L 1 ( R n × ( 0 , T ) ) by Lemma 2.2, in view of the dominated convergence theorem the inequalities in (2.28) and (2.29) ensure that

∫ 0 T ∣ h R ( t ) ∣ d t ≤ ( c 2 ( T ) + c 3 ) ∫ 0 T ∫ B R + 1 \ B R u → 0 as R → ∞ ,

whence (2.27) implies that for each t ∈ ( 0 , T ) ,

∫ R n ζ R 2 u ( ⋅ , t ) − ∫ R n ζ R 2 u 0 = ∫ 0 t h R ( s ) d s → 0 as R → ∞ .

Since (2.9) entails that as R → ∞ , we have ∫ R n ζ R 2 u 0 → ∫ R n u 0 and ∫ R n ζ R 2 u ( ⋅ , t ) → ∫ R n u ( ⋅ , t ) for all t ∈ ( 0 , T ) , this establishes (1.6).□

3 Refined criteria for extensibility

Again on the basis of Lemma 2.5, we can now exclude the possibility that the solution from Lemma 2.2 ceases to exist in finite time, but that this is purely due to a lack of sufficient decay at spatial infinity near the blow-up time:

Lemma 3.1

Let n ≥ 2 , S ∈ R n × n , and p ∈ [ 1 , n ) , and given a non-negative u 0 ∈ BUC ( R n ) ∩ L p ( R n ) , let T max ∈ ( 0 , ∞ ] and ( u , v ) be as in Lemma 2.2. Then, in fact,

(3.1) e i t h e r T max = ∞ , o r limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ .

Proof

If the claim was false, then for some non-negative u 0 ∈ BUC ( R n ) ∩ L p ( R n ) , we would have T max < ∞ but

(3.2) ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ M for all t ∈ ( 0 , T max )

with some M > 0 . To see that this is impossible, we fix T ∈ ( 0 , T max ) and recall (2.6) to pick c 1 ( T ) > 0 such that

(3.3) ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ c 1 ( T ) for all t ∈ ( 0 , T ) .

In the context of an application of (2.26) to q ≔ p , for each R > 1 hence leading to the identity

(3.4) d d t ∫ R n ζ R 2 u p + p ( p − 1 ) ∫ R n ζ R 2 u p − 2 ∣ ∇ u ∣ 2 = − 2 p ∫ R n ζ R u p − 1 ∇ u ⋅ ∇ ζ R − ( p − 1 ) ∫ R n ζ R 2 u p ∇ ⋅ ( S ⋅ ∇ v ) + 2 ∫ R n ζ R u p ( S ⋅ ∇ v ) ⋅ ∇ ζ R , t ∈ ( 0 , T max ) ,

this enables us to estimate the rightmost summand on the basis of (2.9) and (2.10) according to

(3.5) 2 ∫ R n ζ R u p ( S ⋅ ∇ v ) ⋅ ∇ ζ R ≤ 2 c 1 ( T ) K ζ ∣ S ∣ ∫ B R + 1 \ B R u p for all t ∈ ( 0 , T ) .

Apart from that, twice employing Young’s inequality shows that for all t ∈ ( 0 , T max ) , again by (2.9) and (2.10),

(3.6) − 2 p ∫ R n ζ R u p − 1 ∇ u ⋅ ∇ ζ R ≤ p ( p − 1 ) ∫ R n ζ R 2 u p − 2 ∣ ∇ u ∣ 2 + p p − 1 ∫ R n u p ∣ ∇ ζ R ∣ 2 ≤ p ( p − 1 ) ∫ R n ζ R 2 u p − 2 ∣ ∇ u ∣ 2 + p K ζ 2 p − 1 ∫ B R + 1 \ B R u p

and

− ( p − 1 ) ∫ R n ζ R 2 u p ∇ ⋅ ( S ⋅ ∇ v ) ≤ ( p − 1 ) ∫ R n ζ R 2 u p + 1 + ( p − 1 ) ∫ R n ζ R 2 ∣ ∇ ⋅ ( S ⋅ ∇ v ) ∣ p + 1 ≤ ( p − 1 ) ∫ R n ζ R 2 u p + 1 + ( p − 1 ) ∣ S ∣ p + 1 ∫ R n ∣ D 2 v ∣ p + 1 ,

where in line with a Calderón-Zygmund estimate [33,34], with some c 2 > 0 , we have

( p − 1 ) ∣ S ∣ p + 1 ∫ R n ∣ D 2 v ∣ p + 1 ≤ c 2 ∫ R n u p + 1 for all t ∈ ( 0 , T max ) .

Since thus (3.3) guarantees that

− ( p − 1 ) ∫ R n ζ R 2 u p ∇ ⋅ ( S ⋅ ∇ v ) ≤ ( p − 1 + c 2 ) M ∫ R n u p for all t ∈ ( 0 , T max ) ,

from (3.4)-(3.6) we obtain that

d d t ∫ R n ζ R 2 u p ≤ c 3 ∫ R n u p + c 4 ( T ) ∫ B R + 1 \ B R u p for all t ∈ ( 0 , T ) ,

where c 3 ≔ ( p − 1 + c 2 ) M and c 4 ( T ) ≔ 2 c 1 ( T ) K ζ ∣ S ∣ + p K ζ 2 p − 1 . Upon integration, this implies that

(3.7) ∫ Ω ζ R 2 u p ( ⋅ , t ) ≤ ∫ R n ζ R 2 u 0 p + c 3 ∫ 0 t ∫ R n u p + c 4 ( T ) ∫ 0 t ∫ B R + 1 \ B R u p for all t ∈ ( 0 , T ) ,

where thanks to (2.9) and Beppo Levi’s theorem,

∫ R n ζ R 2 u p ( ⋅ , t ) ↗ ∫ R n u p ( ⋅ , t ) for all t ∈ ( 0 , T max ) and ∫ R n ζ R 2 u 0 p ↗ ∫ R n u 0 as R → ∞ ,

and where due to the dominated convergence theorem,

c 4 ( T ) ∫ 0 t ∫ B R + 1 \ B R u p → 0 for all t ∈ ( 0 , T ) as R → ∞ ,

because u ∈ L p ( R n × ( 0 , t ) ) for each t ∈ ( 0 , T max ) by Lemma 2.2. In the limit R → ∞ , from (3.7) we thus infer that

∫ R n u p ( ⋅ , t ) ≤ ∫ R n u 0 p + c 3 ∫ 0 t ∫ R n u p for all t ∈ ( 0 , T ) ,

whence relying on the continuity of [ 0 , T max ) ∋ t ↦ ∫ R n u p ( ⋅ , t ) , as entailed by Lemma 2.2, we may invoke a Gronwall lemma to see that

∫ R n u p ( ⋅ , t ) ≤ ∫ R n u 0 p ⋅ e c 3 t for all t ∈ ( 0 , T ) .

Noting that c 3 is independent of T , we may take T ↗ T max here to obtain a contradiction to (2.7).□

Independently of the latter, by once more employing semigroup estimates, we can finally make sure that signal gradients cannot be bounded near a finite blow-up time:

Lemma 3.2

Let n ≥ 2 and S ∈ R n × n , let p ∈ [ 1 , n ) , and u 0 ∈ BUC ( R n ) ∩ L p ( R n ) be non-negative, and let T max ∈ ( 0 , ∞ ] and ( u , v ) be as in Lemma 2.2. Then, it follows that

(3.8) i f T max < ∞ , t h e n limsup t ↗ T max ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ .

Proof

Let us assume that T max < ∞ , but that

(3.9) ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ M for all t ∈ ( 0 , T max )

with some M > 0 . Then, with ( ζ R ) R > 1 as in (2.8), from (1.1) we obtain that

∂ t ( ζ R u ) = Δ ( ζ R u ) − 2 ∇ ⋅ ( u ∇ ζ R ) + u Δ ζ R − ∇ ⋅ ( ζ R u S ⋅ ∇ v ) + u ( S ⋅ ∇ v ) ⋅ ∇ ζ R in R n × ( 0 , T max ) .

Therefore, drawing on (2.9) and (2.10) and on standard smoothing estimates for the heat semigroup, we find c 1 > 0 such that

‖ ζ R u ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ ‖ ζ R u 0 ‖ L ∞ ( R n ) + c 1 ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ∇ ζ R ‖ L ∞ ( R n ) d s + c 1 ∫ 0 t ‖ u ( ⋅ , s ) Δ ζ R ‖ L ∞ ( R n ) d s + c 1 ∫ 0 t ( t − s ) − 1 2 ‖ ζ R u ( ⋅ , s ) S ⋅ ∇ v ( ⋅ , s ) ‖ L ∞ ( R n ) d s + c 1 ∫ 0 t ∥ u ( ⋅ , s ) ( S ⋅ ∇ v ( ⋅ , s ) ) ⋅ ∇ ζ R ∥ L ∞ ( R n ) d s ≤ ‖ u 0 ‖ L ∞ ( R n ) + c 1 K ζ ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) d s + c 1 K ζ ∫ 0 t ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) d s + c 1 ∣ S ∣ M ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) d s + c 1 K ζ ∣ S ∣ M ∫ 0 t ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) d s ≤ ‖ u 0 ‖ L ∞ ( R n ) + c 2 ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) d s for all t ∈ ( 0 , T max ) ,

where c 2 ≔ c 1 K ζ + c 1 K ζ T max 1 2 + c 1 ∣ S ∣ M + c 1 K ζ ∣ S ∣ M T max 1 2 . Taking R → ∞ here shows that

‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ ‖ u 0 ‖ L ∞ ( R n ) + c 2 ∫ 0 t ( t − s ) − 1 2 ‖ u ( ⋅ , s ) ‖ L ∞ ( R n ) d s for all t ∈ ( 0 , T max ) ,

whence again utilizing the Gronwall inequality from [32, Exercise 4 ⋆ , p. 190] we obtain that

sup t ∈ ( 0 , T max ) ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) < ∞ ,

contrary to what has been asserted by Lemma 3.1.□

Our local theory of classical solvability in (1.1) has thereby been completed:

Proof of Proposition 1.1

We only need to combine Lemma 2.2 with Lemmas 2.4, 2.3, 3.1, 3.2, and 2.6.□

4 Global existence in essentially repulsive versions of (1.1)

As a favorable consequence of Proposition 1.1 and especially of (1.5), verifying Proposition 1.2 essentially reduces to a derivation of L ∞ bounds for u under an appropriate assumption on smallness of ∣ S + 1 ∣ . This will be accomplished in a two-step argument, the first part of which again relies on Lemma 2.5 to obtain L q bounds for arbitrarily large but finite q in such close-to-repulsive frameworks:

Lemma 4.1

Given n ≥ 2 and q > 1 , one can find δ 0 ( q ) > 0 with the property that if S 1 ∈ R n × n is such that ∣ S 1 ∣ ≤ δ 0 ( q ) , and if u 0 ∈ BUC ( R n ) ∩ L 1 ( R n ) is non-negative, then there exists C = C ( q , u 0 ) > 0 such that for the solution ( u , v ) of (1.1) with S = − 1 + S 1 , as obtained in Proposition 1.1, we have

(4.1) ∫ R n u q ( ⋅ , t ) ≤ C f o r a l l t ∈ ( 0 , T max ) .

Proof

Again using a Calderón-Zygmund estimate for Newtonian potentials, we fix c 1 = c 1 ( q ) > 0 such that with Γ as in Lemma 2.1,

(4.2) ∫ Ω ∣ D 2 ( Γ ∗ φ ) ∣ q + 1 ≤ c 1 ∫ Ω ∣ φ ∣ q + 1 for all φ ∈ L q + 1 ( R n ) ,

and we let

(4.3) δ 0 ( q ) ≔ ( 2 c 1 ) − 1 q + 1 .

Then, assuming that S 1 , S , u 0 , and ( u , v ) are as indicated above, we again rely on Lemma 2.5 to see that since

− ( q − 1 ) ∫ R n ζ R 2 u q ∇ ⋅ { ( − 1 + S 1 ) ⋅ ∇ v } = ( q − 1 ) ∫ R n ζ R 2 u q Δ v − ( q − 1 ) ∫ R n ζ R 2 u q ∇ ⋅ ( S 1 ⋅ ∇ v ) = − ( q − 1 ) ∫ R n ζ R 2 u q + 1 − ( q − 1 ) ∫ R n ζ R 2 u q ∇ ⋅ ( S 1 ⋅ ∇ v )

for all t ∈ ( 0 , T max ) by (1.1), we have

(4.4) d d t ∫ R n ζ R 2 u q + q ( q − 1 ) ∫ R n ζ R 2 u q − 2 ∣ ∇ u ∣ 2 + ( q − 1 ) ∫ R n ζ R 2 u q + 1 = − 2 q ∫ R n ζ R u q − 1 ∇ u ⋅ ∇ ζ R − ( q − 1 ) ∫ R n ζ R 2 u q ∇ ⋅ ( S 1 ⋅ ∇ v ) + 2 ∫ R n ζ R u q ( S ⋅ ∇ v ) ⋅ ∇ ζ R for all t ∈ ( 0 , T max ) .

Here once more by Young’s inequality, (2.9) and (2.10),

(4.5) − 2 q ∫ R n ζ R u q − 1 ∇ u ⋅ ∇ ζ R ≤ q ( q − 1 ) ∫ R n ζ R 2 u q − 2 ∣ ∇ u ∣ 2 + q K ζ 2 q − 1 ∫ B R + 1 \ B R u q for all t ∈ ( 0 , T max ) ,

and combining (2.9) with (2.10) we find that for each fixed T ∈ ( 0 , T max ) ,

(4.6) 2 ∫ R n ζ R u q ( S ⋅ ∇ v ) ⋅ ∇ ζ R ≤ 2 c 2 ( T ) K ζ ∣ S ∣ ∫ B R + 1 \ B R u q for all t ∈ ( 0 , T ) ,

where c 2 ( T ) ≔ sup t ∈ ( 0 , T ) ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) is finite by Proposition 1.1. To estimate the second summand on the right-hand side in (4.4), we now use (4.2) to see that, again due to Young’s inequality and (2.9),

(4.7) − ( q − 1 ) ∫ R n ζ R 2 u q ∇ ⋅ ( S 1 ⋅ ∇ v ) ≤ q ( q − 1 ) q + 1 ∫ R n ζ R 2 u q + 1 + q − 1 q + 1 ∫ R n ζ R 2 ∣ ∇ ⋅ ( S 1 ⋅ ∇ v ) ∣ q + 1 ≤ q ( q − 1 ) q + 1 ∫ R n u q + 1 + ( q − 1 ) ∣ S 1 ∣ q + 1 q + 1 ∫ R n ∣ D 2 v ∣ q + 1 ≤ q ( q − 1 ) q + 1 ∫ R n u q + 1 + ( q − 1 ) ∣ S 1 ∣ q + 1 q + 1 ⋅ c 1 ∫ R n u q + 1 ≤ q ( q − 1 ) q + 1 ∫ R n u q + 1 + q − 1 2 ( q + 1 ) ∫ R n u q + 1 = ( q − 1 ) ( 2 q + 1 ) 2 ( q + 1 ) ∫ R n u q + 1 for all t ∈ ( 0 , T max ) ,

because S 1 ∣ q + 1 c 1 ≤ δ 0 q + 1 ( q ) c 1 ≤ 1 2 according to (4.3).

In summary, (4.4)–(4.7) imply that if for T ∈ ( 0 , T max ) we let c 3 ( T ) ≔ q K ζ 2 q − 1 + 2 c 2 ( T ) K ζ ∣ S ∣ , then

(4.8) d d t ∫ R n ζ R 2 u q + ( q − 1 ) ∫ R n ζ R 2 u q + 1 ≤ ( q − 1 ) ( 2 q + 1 ) 2 ( q + 1 ) ∫ R n u q + 1 + c 3 ( T ) ∫ B R + 1 \ B R u q for all t ∈ ( 0 , T ) .

Here we note that c 4 ≔ ( q − 1 ) ( 2 q + 1 ) 2 ( q + 1 ) has the property that c 5 ≔ q − 1 − c 4 ≡ q − 1 2 ( q + 1 ) is positive, and that thanks to the Hölder inequality, (2.9), (1.6) and Young’s inequality, writing c 6 ≔ ∫ R n u 0 we have

∫ R n ζ R 2 u q = ‖ ζ R 2 q u ‖ L q ( R n ) q ≤ ‖ ζ R 2 q u ‖ L q + 1 ( R n ) ( q + 1 ) ( q − 1 ) q ‖ ζ R 2 q u ‖ L 1 ( R n ) 1 q ≤ ‖ ζ R 2 q + 1 u ‖ L q + 1 ( R n ) ( q + 1 ) ( q − 1 ) q ‖ u ‖ L 1 ( R n ) 1 q = c 6 1 q ⋅ ∫ R n ζ R 2 u q + 1 q − 1 q ≤ c 6 1 q ∫ R n ζ R 2 u q + 1 + c 6 1 q for all t ∈ ( 0 , T max ) ,

and hence

c 5 ∫ R n ζ R 2 u q + 1 ≥ c 7 ∫ R n ζ R 2 u q − c 5 for all t ∈ ( 0 , T max ) ,

with c 7 ≔ c 5 c 6 − 1 q . From (4.8) it therefore follows that if T ∈ ( 0 , T max ) and R > 1 , and if for t ∈ [ 0 , T ] we let

y R ( t ) ≔ ∫ R n ζ R 2 u q ( ⋅ , t ) and h R ( t ) ≔ c 5 + c 4 ∫ R n ( 1 − ζ R 2 ) u q + 1 ( ⋅ , t ) + c 3 ( T ) ∫ B R + 1 \ B R u q ( ⋅ , t ) ,

then

(4.9) y R ′ ( t ) + c 7 y R ( t ) ≤ h R ( t ) for all t ∈ ( 0 , T ) ,

so that

y R ( t ) ≤ y R ( 0 ) ⋅ e − c 7 t + ∫ 0 t e − c 7 ( t − s ) h R ( s ) d s for all t ∈ ( 0 , T ) .

Since Beppo Levi’s theorem along with (2.9) ensures that

y R ( t ) ↗ ∫ R n u q ( ⋅ , t ) for all t ∈ ( 0 , T ) and y R ( 0 ) ↗ ∫ R n u 0 q as R → ∞ ,

and that

∫ 0 t e − c 7 ( t − s ) ⋅ c 4 ∫ R n ( 1 − ζ R 2 ) u q + 1 ( ⋅ , s ) d s ↘ 0 for all t ∈ ( 0 , T ) as R → ∞ ,

and since by dominated convergence we have

∫ 0 t e − c 7 ( t − s ) ⋅ c 3 ( T ) ∫ B R + 1 \ B R u q ( ⋅ , s ) d s → 0 for all t ∈ ( 0 , T ) as R → ∞ ,

taking R → ∞ in (4.9) leads to the inequality

∫ R n u q ( ⋅ , t ) ≤ ∫ R n u 0 q ⋅ e − c 7 t + ∫ 0 t e − c 7 ( t − s ) ⋅ c 5 d s ≤ ∫ R n u 0 q + c 5 c 7 for all t ∈ ( 0 , T ) .

As T ∈ ( 0 , T max ) was arbitrary, this establishes the claim.□

Once again thanks to heat semigroup regularization, made accessible by regularization using the functions from (2.8), an application of Lemma 4.1 to suitably large q enables us to establish the second step of our reasoning:

Lemma 4.2

Let n ≥ 2 . Then, there exists δ > 0 with the following property: Whenever S 1 ∈ R n × n satisfies ∣ S 1 ∣ ≤ δ , for any non-negative u 0 ∈ BUC ( R n ) ∩ L 1 ( R n ) , the solution ( u , v ) of (1.1) with S = − 1 + S 1 is global and bounded; that is, in Proposition 1.1  we have T max = ∞ , and there exists C > 0 such that (1.10) holds.

Proof

We fix any q > n , and with δ 0 ( ⋅ ) taken from Lemma 4.1, we let δ ≔ δ 0 ( q ) . Then assuming that S = − 1 + S 1 with some S 1 ∈ R n × n fulfilling ∣ S 1 ∣ ≤ δ , and that u 0 ∈ BUC ( R n ) ∩ L 1 ( R n ) is non-negative, from Lemma 4.1 we obtain c 1 > 0 such that

(4.10) ‖ u ( ⋅ , t ) ‖ L q ( R n ) ≤ c 1 for all t ∈ ( 0 , T max ) .

Recalling from Lemma 2.1 that with γ as provided there we have ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ γ ∣ ∣ ∣ u ( ⋅ , t ) ∣ ∣ ∣ p for all t ∈ ( 0 , T max ) , and using Young’s inequality to estimate ‖ u ( ⋅ , t ) ‖ L p ( R n ) ≤ ‖ u ( ⋅ , t ) ‖ L q ( R n ) + ‖ u ( ⋅ , t ) ‖ L 1 ( R n ) for all t ∈ ( 0 , T max ) , by means of (1.6) and (4.10) we obtain c 2 > 0 such that

(4.11) ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ c 2 ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) + c 2 for all t ∈ ( 0 , T max ) .

Moreover, choosing c 3 > 0 and c 4 > 0 such that in line with known smoothing properties of the heat equation we have

(4.12) ‖ e t Δ φ ‖ L ∞ ( R n ) ≤ c 3 t − n 2 q ‖ φ ‖ L q ( R n ) for all t > 0 and each φ ∈ C 0 ∞ ( R n ) ,

and that

(4.13) ‖ e t Δ ∇ ⋅ φ ‖ L ∞ ( R n ) ≤ c 4 t − 1 2 − n 2 q ‖ φ ‖ L q ( R n ) for all  t > 0  and each φ ∈ C 0 ∞ ( R n ; R n ) ,

we use that q > n to fix τ ∈ ( 0 , 1 ] suitably small such that

(4.14) c 1 c 2 c 4 ∣ S ∣ 1 2 − n 2 q ⋅ τ 1 2 − n 2 q ≤ 1 4 and c 1 c 2 c 3 K ζ ∣ S ∣ 1 − n 2 q ⋅ τ 1 − n 2 q ≤ 1 4 .

For arbitrary T ∈ ( 0 , T max ) , we then estimate the finite number

(4.15) M ( T ) ≔ sup t ∈ ( 0 , T ) ‖ u ( ⋅ , t ) ‖ L ∞ ( R n )

by relying on a Duhamel representation associated with the identity

∂ t ( ζ R u ) = Δ ( ζ R u ) − 2 ∇ ⋅ ( u ∇ ζ R ) + u Δ ζ R − ∇ ⋅ ( ζ R u S ⋅ ∇ v ) + u ( S ⋅ ∇ v ) ⋅ ∇ ζ R ,

valid for each R > 1 according to (1.1) (cf. also the proof of Lemma 3.2). With τ as above, namely, this implies that thanks to (4.12) and (4.13),

(4.16) ‖ ζ R u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∥ e [ t − ( t − τ ) + ] Δ { ζ R u ( ⋅ , ( t − τ ) + ) } − 2 ∫ ( t − τ ) + t e ( t − s ) Δ ∇ ⋅ { u ( ⋅ , s ) ∇ ζ R } d s + ∫ ( t − τ ) + t e ( t − s ) Δ { u ( ⋅ , s ) Δ ζ R } d s − ∫ ( t − τ ) + t e ( t − s ) Δ ∇ ⋅ { ζ R u ( ⋅ , s ) S ⋅ ∇ v ( ⋅ , s ) } d s + ∫ ( t − τ ) + t e ( t − s ) Δ { u ( ⋅ , s ) ( S ⋅ ∇ v ( ⋅ , s ) ) ⋅ ∇ ζ R } d s L ∞ ( R n ) ≤ ∥ e [ t − ( t − τ ) + ] Δ { ζ R u ( ⋅ , ( t − τ ) + ) } ∥ L ∞ ( R n ) + 2 c 4 ∫ ( t − τ ) + t ( t − s ) − 1 2 − n 2 q ‖ u ( ⋅ , s ) ∇ ζ R ‖ L q ( R n ) d s + c 3 ∫ ( t − τ ) + t ( t − s ) − n 2 q ‖ u ( ⋅ , s ) Δ ζ R ‖ L q ( R n ) d s + c 4 ∫ ( t − τ ) + t ( t − s ) − 1 2 − n 2 q ‖ ζ R u ( ⋅ , s ) S ⋅ ∇ v ( ⋅ , s ) ‖ L q ( R n ) d s + c 3 ∫ ( t − τ ) + t ( t − s ) − n 2 q ∥ u ( ⋅ , s ) ( S ⋅ ∇ v ( ⋅ , s ) ) ⋅ ∇ ζ R ∥ L q ( R n ) d s for all t ∈ ( 0 , T ) .

Here if t ∈ ( τ , T ) , then combining (4.12) with (4.10) shows that

(4.17) ∥ e [ t − ( t − τ ) + ] Δ { ζ R u ( ⋅ , ( t − τ ) + ) } ∥ L ∞ ( R n ) = ∥ e τ Δ { ζ R u ( ⋅ , t − τ ) } ∥ L ∞ ( R n ) ≤ c 3 τ − n 2 q ‖ ζ R u ( ⋅ , t − τ ) ‖ L q ( R n ) ≤ c 1 c 3 τ − n 2 q ,

because ∣ ζ R ∣ ≤ 1 ; if t ∈ ( 0 , τ ] , however, then by a simple application of the maximum principle we obtain that

(4.18) ∥ e [ t − ( t − τ ) + ] Δ { ζ R u ( ⋅ , ( t − τ ) + ) } ∥ L ∞ ( R n ) = ∥ e t Δ { ζ R u 0 } ∥ L ∞ ( R n ) ≤ ‖ ζ R u 0 ‖ L ∞ ( R n ) ≤ ‖ u 0 ‖ L ∞ ( R n ) .

Apart from that, using (2.9) and (2.10) together with (4.10) and the inequalities q > n and τ ≤ 1 we can estimate

(4.19) 2 c 4 ∫ ( t − τ ) + t ( t − s ) − 1 2 − n 2 q ‖ u ( ⋅ , s ) ∇ ζ R ‖ L q ( R n ) d s ≤ 2 c 1 c 4 K ζ ∫ ( t − τ ) + t ( t − s ) − 1 2 − n 2 q d s ≤ 2 c 1 c 4 K ζ ∫ 0 1 σ − 1 2 − n 2 q d σ = 2 c 1 c 4 K ζ 1 2 − n 2 q for all t ∈ ( 0 , T )

and, similarly,

(4.20) c 3 ∫ ( t − τ ) + t ( t − s ) − n 2 q ‖ u ( ⋅ , s ) Δ ζ R ‖ L q ( R n ) d s ≤ c 1 c 3 K ζ ∫ ( t − τ ) + t ( t − s ) − n 2 q d s ≤ c 1 c 3 K ζ 1 − n 2 q for all t ∈ ( 0 , T ) .

In view of the definition of M ( T ) and our restriction in (4.14), finally, the two last summands in (4.16) can be controlled, again on the basis of (2.9), (2.10), and (4.10), and of (4.11) and (4.15), according to

(4.21) c 4 ∫ ( t − τ ) + t ( t − s ) − 1 2 − n 2 q ‖ ζ R u ( ⋅ , s ) S ⋅ ∇ v ( ⋅ , s ) ‖ L q ( R n ) d s ≤ c 4 ∣ S ∣ ∫ ( t − τ ) + t ( t − s ) − 1 2 − n 2 q ‖ u ( ⋅ , s ) ‖ L q ( R n ) ‖ ∇ v ( ⋅ , s ) ‖ L ∞ ( R n ) d s ≤ c 1 c 2 c 4 ∣ S ∣ ⋅ ( M ( T ) + 1 ) ∫ ( t − τ ) + t ( t − s ) − 1 2 − n 2 q d s ≤ c 1 c 2 c 4 ∣ S ∣ 1 2 − n 2 q ⋅ τ 1 2 − n 2 q M ( T ) + c 1 c 2 c 4 ∣ S ∣ 1 2 − n 2 q ≤ 1 4 M ( T ) + c 1 c 2 c 4 ∣ S ∣ 1 2 − n 2 q for all t ∈ ( 0 , T )

and, in much the same fashion,

(4.22) c 3 ∫ ( t − τ ) + t ( t − s ) − n 2 q ∥ u ( ⋅ , s ) ( S ⋅ ∇ v ( ⋅ , s ) ) ⋅ ∇ ζ R ∥ L q ( R n ) d s ≤ c 1 c 2 c 3 K ζ ∣ S ∣ ⋅ ( M ( T ) + 1 ) ∫ ( t − τ ) + t ( t − s ) − n 2 q d s ≤ c 1 c 2 c 3 K ζ ∣ S ∣ 1 − n 2 q ⋅ τ 1 − n 2 q M ( T ) + c 1 c 2 c 3 K ζ ∣ S ∣ 1 − n 2 q ≤ 1 4 M ( T ) + c 1 c 2 c 3 K ζ ∣ S ∣ 1 − n 2 q for all t ∈ ( 0 , T ) ,

again because τ ≤ 1 . A combination of (4.16) with (4.17)–(4.22) now shows that if we let

c 5 ≔ c 1 c 3 τ − n 2 q + ‖ u 0 ‖ L ∞ ( R n ) + 2 c 1 c 4 K ζ 1 2 − n 2 q + c 1 c 3 K ζ 1 − n 2 q + c 1 c 2 c 4 ∣ S ∣ 1 2 − n 2 q + c 1 c 2 c 3 K ζ ∣ S ∣ 1 − n 2 q ,

then whenever T ∈ ( 0 , T max ) and R > 1 ,

‖ ζ R u ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ 1 2 M ( T ) + c 5 for all t ∈ ( 0 , T ) .

In the limit R → ∞ , this yields the inequality

‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ 1 2 M ( T ) + c 5 for all t ∈ ( 0 , T ) ,

from which it follows that M ( T ) ≤ 1 2 M ( T ) + c 5 and hence

‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) ≤ 2 c 5 for all t ∈ ( 0 , T max ) ,

as T ∈ ( 0 , T max ) was arbitrary. In light of Proposition 1.1 and especially the extensibility criterion (1.5) therein, this shows that indeed ( u , v ) exists globally and possesses the claimed boundedness property.□

Our final result on global existence and boundedness in (1.1) for small perturbations S of − 1 has thereby been achieved already:

Proof of Proposition 1.2

The statement is an immediate consequence of Proposition 1.1, Lemmas 4.1 and 4.2.□

Acknowledgement

The author acknowledges support of the Deutsche Forschungsgemeinschaft (Project No. 462888149).

  1. Conflict of interest: The author states no conflict of interest.

  2. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-01-18
Revised: 2023-03-28
Accepted: 2023-03-28
Published Online: 2023-05-11

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
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  126. New fractional integral inequalities via Euler's beta function
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  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2022-0578
Keywords for this article
chemotaxis; self-interacting particles; local existence; extensibility
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