H older estimates of weak solutions to degenerate chemotaxis systems with a source term. M.Marras1 F.Ragnedda2 S.Vernier-Piro3and V.Vespri4

2025-05-06 0 0 483.75KB 38 页 10玖币

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H¨older estimates of weak solutions to degenerate

chemotaxis systems with a source term.

M.Marras 1, F.Ragnedda 2, S.Vernier-Piro 3and V.Vespri 4

In memory of our friend Emmanuele DiBenedetto

Abstract

In this note we consider degenerate chemotaxis systems with porous

media type diﬀusion and a source term satisfying the Hadamard growth

condition. We prove the H¨older regularity for bounded solutions to parabolic-

parabolic as well as for elliptic-parabolic chemotaxis systems.

Keywords: Chemotaxis systems, degenerate parabolic equations, elliptic equa-

tions, H¨older regularity.

AMS Subject Classiﬁcation: 92C17, 35K65, 35J70, 35B65

1Dipartimento di Matematica e Informatica, Universit`a di Cagliari, via Ospedale 72, 09124

Cagliari (Italy), mmarras@unica.it

2Facolt`a di Ingegneria e Architettura, Universit`a di Cagliari, Viale Merello 92, 09123

Cagliari, ragnedda@unica.it

3Facolt`a di Ingegneria e Architettura, Universit`a di Cagliari, Viale Merello 92, 09123

Cagliari (Italy), svernier@unica.it

4Dipartimento di Matematica ed Informatica ”U. Dini”, Universit`a di Firenze, viale Mor-

gagni 67/a, 50134 Firenze (Italy) vincenzo.vespri@uniﬁ.it

arXiv:2210.00565v1 [math.AP] 2 Oct 2022

1 Introduction

Let us consider the following class of degenerate chemotaxis systems











ut= div(∇um)−χdiv(uq−1∇v) + B(x, t, u, ∇u),in RN×(t > 0),

˜τvt= ∆v−av +u, in RN×(t > 0),

u(x, 0) = u0(x)≥0, v(x, 0) = v0(x)≥0,in RN,

(1.1)

with N≥2, m ≥1,q≥max{m+1

2,2}and a, χ > 0. The constant ˜τis taken

nonnegative. When the constant ˜τ= 0 we are in the parabolic-elliptic case and,

when ˜τ > 0, we are in the parabolic-parabolic case. In the latter case, WLOG,

we may assume ˜τ= 1 . The initial data (u0(x), v0(x)) satisfy











u0(x)≥0, u0(x)∈L∞(RN)∩L1(RN), um

0∈H1,

v0(x)≥0, v0(x)∈L1(RN)∩W1,p(RN),

(1.2)

with 1 <p<∞.

A pair (u, v) of non negative measurable functions deﬁned in RN×[0, T ], T > 0

is a local weak solution to (1.1) if

u∈L∞(0, T ;Lp(RN)), um∈L2(0, T ;H1(RN)), v ∈L∞(0, T ;H1(RN)),

and (u, v) satisﬁes (1.1) in the sense that for every compact set K ⊂ RNand

every time interval [t1, t2]⊂[0, T ] one has

uψdx

+Zt2

t1ZKh−uψt+ (∇um,∇ψ)−χuq−1(∇v, ∇ψ)idx dt

=Zt2

t1ZK

B(x, t, u, ∇u)ψ dxdt;

(1.3)

˜τvψdx

+Zt2

t1ZKh−˜τvψt+ (∇v, ∇ψ)idxdt =Zt2

t1ZK

(−av +u)ψ dxdt, (1.4)

for all locally bounded non negative testing function ψ∈W1,2

loc (0, T ;L2(K)) ∩

loc(0, T ;W1,p

0(K)).

In the last years, there was a growing interest in the chemotaxis systems. We

recall that Keller and Segel in the seminal paper [15] proposed a mathematical

model describing the aggregation process of amoebae by chemotaxis. For such a

reason, nowadays, such kind of systems are named Keller-Segel in their honour.

Recently many authors studied systems with porous medium-type diﬀusion and

with a power factor in the drift term (see [12], [13], [14], [16], [20] and the ref-

erences therein). In this note, we consider a degenerate chemotaxis model with

porous media type diﬀusion with m > 1. When m= 1,q= 2, B = 0, the

system (1.1) is reduced to the classical Keller-Segel system. In our model, the

diﬀusion of the cells (div(∇um)) depends only on own density and degenerates

when u= 0, mdenotes the intensity of diﬀusion and the exponent qin the

drift term takes in account the nonlinear aspects of the biological phenomenon.

Moreover we assume χ > 0 which means that the cells move toward the in-

creasing signal concentration (chemoattractant). For sake of simplicity, we take

χ= 1.

This model relies on the presence of the the source term Bwhich describes the

growth of the cells. Some experimental evidences (see [1]) show that Bis a non-

linear term and satisﬁes natural or Hadamard growth condition, more precisely

it satisﬁes the inequality (see [5], [7])

|B(x, t, u, ∇u)| ≤ C|∇um|2+φ(x, t), C > 0,(1.5)

with φ(x, t) in the parabolic space Lq,r

RN×(t>0) =Lr(0,∞;Lq(RN)).The presence

of this term makes more challenging the math approach of this system (see the

monograph of Giaquinta [9]).

In literature a great part of the results concerns the case B= 0 where, de-

pending upon the choice of mand q, it is possible to ﬁnd initial data for which

we have global existence and initial data for which blow up in ﬁnite time oc-

curs (see [12], [13], [14], [21] and references therein). To our knowledge, in the

more general case B6= 0, the global existence of the solutions and the blow up

phenomenon are not still studied in fully detail. Hence this argument will be

occasion of our next future studies. For the above reason, we will work in a

bounded time interval [0, T ] i.e. before the eventual blow-up time, assuming,

therefore, that the solution uremains bounded.

For the solutions to (1.1) with B= 0 and ˜τ= 1,Ishida and Yokota in [12]

proved that a weak solution exists globally when q< m +2

Nwithout restriction

on the size of initial data, improving both Sugiyama ([19]) and Sugiyama and

Kunii results ([20]) where q≤mwas assumed. In [13], the Authors established

the global existence of weak solutions with small initial data when q≥m+2

while in [22] Winkler proved that there are initial data such that if q≥m+2

the solution blows up in ﬁnite time. Moreover, in [14] uniform boundedness of

nonnegative solutions was derived assuming q< m +2

If B= 0 and ˜τ= 0,the existence in large of the solutions was proved in

the case of m > q−2

Nwithout any restrictions on initial data and in the case

1≤m≤q−2

Nonly for small initial data ([20]). For (local and global) existence

and nonexistence of solutions to diﬀerent classes of Keller-Segel type system we

refer to [1].

If B6= 0 and ˜τ= 0, m = 1, q = 2, in [17] the authors investigated blow-up

phenomena and obtained a safe time interval of existence for the solutions by

deriving a lower bound of the blow-up time.

In [16], following the De Giorgi approach, Kim and Lee proved regularity and

uniqueness results for solutions to degenerate chemotaxis parabolic-parabolic

system assuming that the source term is vanishing.

In this paper we focus our attention only on the local H¨older regularity of the

solution (u, v).More precisely, we give a unitary and more organic proof that

allows us to treat in the same framework a more general equation (with source

term) either in the parabolic-parabolic case and in the parabolic-elliptic case.

Our approach is based on suitable a-priori estimates on the function vthat solves

the second equation of (1.1) and on De Giorgi-DiBenedetto approach ( [4], [5],

[6]) for proving regularity of um. The regularity of v, either for ˜τ= 1 or ˜τ= 0,

follows in a straightforward way from classical results theory (see, for instance,

[5], [10], [18]). The proof has however some remarkable diﬀerences from the

classical approach by DiBenedetto. In this paper we focus our attention only on

the real novelties. When the modiﬁcations are based only on technicalities, we

will postpone the proof in the Appendix and when it is a structural modiﬁcation

we quote the corresponding papers by DiBenedetto.

Our main result is

Theorem 1.1. (Regularity)

Let ube a locally bounded local weak solution of (1.1) and let Bsatisfy (1.5),

then (x, t)→u(x, t)is H¨older continuous in RN×(0, T )and there exists αo∈

(0,1) such that, for every ε > 0, there exists a constant γ(ε)>0such that

|um(x1, t1)−um(x2, t2)| ≤ γ(ε)(|x1−x2|αo+|t1−t2|αo

2),

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摘要：

Holderestimatesofweaksolutionstodegeneratechemotaxissystemswithasourceterm.M.Marras1,F.Ragnedda2,S.Vernier-Piro3andV.Vespri4InmemoryofourfriendEmmanueleDiBenedettoAbstractInthisnoteweconsiderdegeneratechemotaxissystemswithporousmediatypediusionandasourcetermsatisfyingtheHadamardgrowthcondition.Wep...

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H older estimates of weak solutions to degenerate chemotaxis systems with a source term. M.Marras1 F.Ragnedda2 S.Vernier-Piro3and V.Vespri4

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