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

0.35 MB·

by  Johannes Lankeit

A generalized solution concept for the Keller-Segel system with
7
1 logarithmic sensitivity: Global solvability for large nonradial data
0
2

n Johannes Lankeit∗ Michael Winkler#
a
Institut für Mathematik, Universität Paderborn, Institut für Mathematik, Universität Paderborn,
J
5 33098 Paderborn, Germany 33098 Paderborn, Germany
2
January 26, 2017

]
P
A
.
h
t Abstract
a
m The chemotaxis system
[
ut =∆u χ (uv v),
1 − ∇· ∇
v ( vt =∆v v+u,
−
1
9 is considered in a bounded domain Ω Rn with smooth boundary, where χ>0.
⊂
3
Anapparentlynoveltypeofgeneralizedsolutionframeworkisintroducedwithinwhichanextension
7
0 of previously known ranges for the key parameter χ with regard to global solvability is achieved.
. In particular, it is shown that under the hypothesis that
1
0
if n=2,
7
∞
1 χ< √8 if n=3,

:
v  n if n 4,
n−2
i ≥
X
forallinitialdatasatisfyingsuitableassumptionsonregularityandpositivity,anassociatedno-flux
r
a initial-boundary value problem admits a globally defined generalized solution. This solution inter
alia has the property that
1
u L (Ω [0, )).
loc
∈ × ∞
Key words: chemotaxis; logarithmic sensitivity; global existence; generalized solution
Math Subject Classification (2010): 35K55, 35D99, 92C17
∗[email protected]
#[email protected]
1
1 Introduction
WeconsidertheKeller-Segel systemwithlogarithmic sensitivity, as givenbytheinitial-boundary value
problem
u = ∆u χ u v , x Ω, t > 0,
t − ∇· v∇ ∈
 ∂u = 0, (cid:16) (cid:17) x ∂Ω, t > 0, (1.1)
∂ν ∈

 u(x,0) = u (x), x Ω,
0
∈
coupled to the parabolic problem

v = ∆v v+u, x Ω, t > 0,
t
− ∈
∂v = 0, x ∂Ω, t > 0, (1.2)
 ∂ν ∈
 v(x,0) = v (x), x Ω,
0
∈
where Ω is a bounded domain in Rn, n 2, with smooth boundary, χ is a positive parameter and the
≥
given initial data u and v satisfy suitable regularity and positivity assumptions.
0 0
This system can be viewed as a prototypical parabolic model for self-enhanced chemotactic migration
processesin which cross-diffusionoccurs in accordance with the Weber-Fechner law of stimulus percep-
tion ([9], [15]), and accordingly a considerable literature is concerned with its mathematical analysis.
However, up to now it seems yet unclear to which extent the particular mechanism of taxis inhibition
at large signal densities in (1.1) is sufficient to prevent phenomena of blow-up, known as the probably
most striking qualitative feature of the classical Keller-Segel system: Indeed, in its fully parabolic
version, as determined by the choice τ := 1 in
u = ∆u χ (u v),
t
− ∇· ∇ (1.3)
( τvt = ∆v v+u,
−
the latter admits solutions blowing up in finite time for any choice of χ > 0 whenever n 2 ([8],
≥
[22]), and in the simplified parabolic-elliptic case obtained on choosing τ := 0 it is even known that
some radial solutions to an associated Cauchy problem in the whole plane collapse into a persistent
Dirac-type singularity in the sense that a globally defined measure-valued solution exists which has a
singular part beyond some finite time and asymptotically approaches a Dirac measure (cf. e.g. [19] or
also [12]).
As opposed to this, the literature has identified various circumstances under which phenomena of
this type are ruled out in (1.1)-(1.2): For instance, when χ < χ (n) with some χ (2) > 1.015 and
0 0
χ (n) := 2 for n 3, global bounded classical solutions exist for all reasonably regular positive
0 n ≥
initial datqa ([11], [2], [4], [24], [13], [21]); in the corresponding parabolic-elliptic analogue, the same
conclusion holds with χ (2) = ([5]) and with χ (n) := 2 when n 3 and the spatial setting is
0 ∞ 0 n−2 ≥
radially symmetric ([14], cf. also [6] for a related result addressing a variant with its second equation
being τv = ∆v v +u for small τ > 0), whereas it is known that some exploding solutions exist if
t
−
n 3 and χ > 2n ([14]). As for larger values of χ in the fully parabolic problem (1.1)-(1.2), in some
≥ n−2
cases at least certain global generalized solutions can be found which satisfy
u L1 (Ω [0, )) (1.4)
loc
∈ × ∞
2
and thereby indicate the absence of strong singularity formation of the flavor described above. Such
constructions are possible in the context of natural weak solution concepts if
n+2
χ < (1.5)
3n 4
r −
([21]) and within a slightly more generalized framework if merely
n
χ < (1.6)
n 2
r −
butinadditionthesolutionsaresupposedtoberadiallysymmetric([17]). Tothebestofourknowledge,
however, the question how far (1.5) is optimal with respect to the existence of not necessarily radial
solutions fulfilling (1.4) is yet unsolved; in particular, it appears to be unknown whether in nonradial
planar settings such solutions do exist also beyond the range χ < √2 determined by (1.5).
Main results. The purpose of this work is to design a novel concept of generalized solvability
which is yet suitably strong so as to require (1.4), but which on the other hand is mild enough so
that it enables us to construct corresponding global solutions without any symmetry hypotheses and
under conditions somewhat weaker than (1.5) and actually also than (1.6). More precisely, considering
(1.1)-(1.2) under the assumptions that
u C0(Ω) is such that u 0 in Ω and u 0, and that
0 ∈ 0 ≥ 0 6≡ (1.7)
v W1,∞(Ω) satisfies v > 0 in Ω,
0 0
(cid:26) ∈
we can state our main results as follows.
Theorem 1.1. Let n 2 and Ω Rn be a bounded domain with smooth boundary, and let χ > 0 be
≥ ⊂
such that
if n = 2,
∞
χ < √8 if n = 3, (1.8)

 n if n 4.
n−2 ≥
Then for any u and v fulfilling (1.7), theproblem (1.1)-(1.2) possesses at least one global generalized
0 0 
solution (u,v) in the sense of Definition 2.4 below. In particular, this solution satisfies (1.4), and
moreover we have
u(,t) = u for a.e. t > 0. (1.9)
0
·
ZΩ ZΩ
Plan of the paper. Our approach will be based on the essentially well-known fact that the
functional upvq enjoys certain quasi-entropy features along trajectories of (1.1)-(1.2), provided that
Ω
the crucial positive parameter p therein satisfies p < 1 (cf. Section 4 for a corresponding observation
R χ2
addressingglobalsmoothsolutionstotheregularizedproblems(3.1)below). Thechallengenowconsists
in taking appropriate advantage of correspondingly implied a priori estimates obtained in Sections 5, 6
and 7, which inter alia seem far from sufficient to warrant L1 bounds for the cross-diffusive flux χu v
v∇
especially in cases when χ is large and hence p needs to be chosen small.
InthepreparatorySection3,wewillthereforeresorttoasolutionframeworkinvolvingcertainsublinear
powers of u rather than u itself, thus reminiscent of the celebrated concept of renormalized solutions
3
[3]). This idea has partially been adapted to the present context in [17] already, but in the present
work we shall further weaken the requirements on solutions to a considerable extent: Namely, for the
crucial first sub-problem (1.1) to be solved we shall only require that the coupled quantity upvq, with
certain positive p and q, satisfies a parabolic inequality associated with (1.1)-(1.2) in a weak form,
and that moreover u(,t) u for a.e. t > 0; a key observation, to be made in Lemma 2.5, will
Ω · ≤ Ω 0
reveal that if we furthermore assume the component v to fulfill (1.2) in a natural weak sense, then
R R
we indeed obtain a concept consistent with that of classical solvability in (1.1)-(1.2) for all suitably
smooth functions.
As seen in Section 8 by means of appropriate compactness arguments, the previously gained estimates
in fact enable us to construct a global solution within this framework.
2 A concept of generalized solvability
In specifying the subsequently pursued concept of weak solvability, we first require certain products
upvq to satisfy an inequality which can be viewed as generalizing a classical supersolution property of
this quantity with regard to (1.1)-(1.2).
Definition 2.1. Let p (0,1) and q (0,1), and suppose that u and v are measurable functions on
∈ ∈
Ω (0, ) such that u > 0 and v > 0 a.e. in Ω (0, ), that
× ∞ × ∞
upvq L1 (Ω [0, )) and up+1vq−1 L1 (Ω [0, )), (2.1)
loc loc
∈ × ∞ ∈ × ∞
and that ∇up2 and ∇vq2 belong to L1loc(Ω×(0,∞)) and are such that
v2q up2 L2loc(Ω [0, )) and up2 v2q L2loc(Ω [0, )). (2.2)
∇ ∈ × ∞ ∇ ∈ × ∞
Then (u,v) will be called a global weak (p,q)-supersolution of (1.1) if
∞
upvqϕ upvqϕ(,0)
− t − 0 0 ·
Z0 ZΩ ZΩ
4(1−p)q−4q2−p(1−p)2χ2 ∞ vq up2 2ϕ
≥ pq(pχ+1 q) |∇ |
− Z0 ZΩ
4(pχ+1 q) ∞ p q (1 p)χ+2q q p 2
+ − u2 v2 − v2 u2 ϕ
q ∇ − 2(pχ+1 q) ∇
2pχ ∞ up2Zv0q ZuΩp2(cid:12)(cid:12)(cid:12) ϕ − (cid:12)(cid:12)(cid:12)
− q ∇ ·∇
Z0 ZΩ
pχ ∞
+ 1 upvq∆ϕ
− q
(cid:16) ∞ (cid:17)Z0 ZΩ ∞
q upvqϕ+q up+1vq−1ϕ (2.3)
−
Z0 ZΩ Z0 ZΩ
for all nonnegative ϕ C∞(Ω [0, )) such that ∂ϕ = 0 on ∂Ω (0, ) and if moreover
∈ 0 × ∞ ∂ν × ∞
upvq > 0 a.e. on ∂Ω (0, ). (2.4)
× ∞
4
Remark 2.2. (i) Observing that (2.1) in particular ensures that up2vq2 ∈ L2loc(Ω×[0,∞)), and that
hence (2.1) and (2.2) warrant that
up2vq up2 = (up2vq2) (v2q up2) L1loc(Ω [0, ))
∇ · ∇ ∈ × ∞
and similarly upvq2∇vq2 ∈ L1loc(Ω×[0,∞)), it follows that under the above requirements all integrals in
(2.3) are indeed well-defined.
(ii) According to (2.1) and (2.2), for a.e. t > 0, up2(,t)vq2(,t) W1,2(Ω) so that up2vq2(,t)
· · ∈ · ∂Ω ∈
L2(∂Ω) exists in the sense of traces, giving meaning to the positivity requirement in (2.4).
(cid:12)
(cid:12)
Apartfrom that, wewill require thesecond problem (1.2)to besatisfied inthe following rather natural
weak sense.
Definition 2.3. A pair (u,v) of functions
u L1 (Ω [0, )),
∈ loc × ∞ (2.5)
( v ∈ L1loc([0,∞);W1,1(Ω))
will be named a global weak solution of (1.2) if
∞ ∞ ∞ ∞
vϕ v ϕ(,0) = v ϕ vϕ+ uϕ (2.6)
t 0
− − · − ∇ ·∇ −
Z0 ZΩ ZΩ Z0 ZΩ Z0 ZΩ Z0 ZΩ
for all ϕ C∞(Ω [0, )).
∈ 0 × ∞
Following an approach already pursued in [23] in a considerably less involved related context, in order
tocompleteoursolutionconceptwewillcomplementtheabovetworequirementsbymerelypostulating
an upper bound for the mass functional u in terms of u :
Ω Ω 0
Definition 2.4. A couple of nonnegativRe measurable fuRnctions u and v defined on Ω (0, ) will
× ∞
be said to be a global generalized solution of (1.1)-(1.2) if (u,v) is a global weak solution of (1.2)
according to Definition 2.3, if there exist p (0,1) and q (0,1) such that (u,v) is a global weak
∈ ∈
(p,q)-supersolution of (1.1) in the sense of Definition 2.1, and if moreover
u(,t) u for a.e. t > 0. (2.7)
0
· ≤
ZΩ ZΩ
This is indeed consistent with the concept of classical solvability in the following sense.
Lemma 2.5. Let χ > 0, and suppse that (u,v) (C0(Ω [0, )) C2,1(Ω (0, )))2 is such that
∈ × ∞ ∩ × ∞
(u,v) is a global generalized solution of (1.1)-(1.2) in the sense of Definition 2.4. Then (u,v) satisfies
(1.1)-(1.2) classically in Ω (0, ).
× ∞
Proof. By means of standard arguments relying on the assumed regularity properties of v, it can easily
be verified that v solves (1.2) classically. According to the maximum principle, v hence is strictly
positive in Ω [0, ) and vq−1 is uniformly bounded in every set Ω [0,T) for T (0, ). Positivity
× ∞ × ∈ ∞
ofv ensures thatby (2.4)u > 0onadensesubsetof∂Ω (0, ) which moreover is openin ∂Ω (0, )
× ∞ × ∞
by continuity of u.
5
For arbitrary ψ C∞(Ω) with ψ 0 and ∂ψ = 0, testing (2.3) by ϕ(x,t) := ψ(x)(1 1t) , ε
∈ ≥ ∂ν ∂Ω − ε + ∈
(0,1), which is permissible by Weierstrass’ theorem, and invoking Lebesgue’s dominated convergence
(cid:12)
theorem and continuity of t up(,t)vq(,t)(cid:12) at t = 0 in taking ε 0 we readily achieve
7→ Ω · · ց
R ∂ψ
up(,0)vq(,0)ψ upvqψ for all ψ C∞(Ω),ψ 0, = 0,
· · ≥ 0 0 ∈ ≥ ∂ν ∂Ω
ZΩ ZΩ
(cid:12)
showing that up(,0)vq(,0) upvq throughout Ω. Because of v(,0) = v > 0 an(cid:12)d the monotonicity of
· · ≥ 0 0 · 0
1
()p we obtain u(,0) u0 in Ω and from continuity of u and (2.7) we can conclude that u(,0) = u0
· · ≥ ·
in Ω.
In the first two integrals on the right of (2.3) straightforward computations yield
4(1−p)vq up2 2 4(1−p) +8 up2vq2 up2 vq2 + 4(pχ+1−q)up vq2 2
p |∇ | − q ∇ ∇ q |∇ |
(cid:16) (cid:17)
= 4(pχ+1−q) up vq2 2 (1−p)χ+2qup2vq2 u2p vq2 + (χ−pχ+2q)2vq up2 2
q |∇ | − pχ+1 q ∇ ∇ 4(pχ+1 q)2 |∇ |
(cid:26) − − (cid:27)
+ 4(1−p) ((1−p)χ+2q)2 vq up2 2
p − q(pχ+1 q) |∇ |
(cid:26) − (cid:27)
= 4(pχ+1−q) up2 v2q (1−p)χ+2qvq2 up2 2+ 4(1−p)q−4q2−p(1−p)2χ2vq up2 2,
q ∇ − 2(pχ+1 q) ∇ pq(pχ+1 q) |∇ |
(cid:12) − (cid:12) −
(cid:12) (cid:12) (2.8)
(cid:12) (cid:12)
(cid:12) (cid:12)
since
4(1 p) ((1 p)χ+2q)2 4(1 p)q(pχ+1 q) ((1 p)χ+2q)2p
− − = − − − −
p − q(pχ+1 q) pq(pχ+1 q)
− −
4(1 p)q 4q2 p(1 p)2χ2
= − − − − .
pq(pχ+1 q)
−
In preparation of the following calculations we also note that for each positive function w C2(Ω) and
∈
any r > 0, we have the pointwise identities
wr2∆w2r = wr2 rwr−22 w = wr2 r(r−2)wr−24 w 2 + rwr−22∆w
∇· 2 ∇ 4 |∇ | 2
r(r 2(cid:16)) (cid:17) r (cid:16) r 2 r (cid:17)
= − wr−2 w 2+ wr−1∆w = − wr2 2+ wr−1∆w (2.9)
4 |∇ | 2 r |∇ | 2
and
4(r 1)
∆wr = (rwr−1 w) = r(r 1)wr−2 w 2+rwr−1∆w = − w2r 2+rwr−1∆w (2.10)
∇· ∇ − |∇ | r |∇ |
The positivity requirement on w in (2.9) and (2.10) prompts us to perform the following calculations
only for test functions ϕ compactly supported in u > 0 := (x,t) Ω [0, ) : u(x,t) > 0 ,
{ } { ∈ × ∞ }
ensuring strict positivity of u and boundedness of up−1 on suppϕ.
6
Accordingly, for all nonnegative ϕ C∞(Ω (0, )) with suppϕ u > 0 and ∂ϕ = 0, by
∈ 0 × ∞ ⊂ { } ∂ν ∂Ω×(0,∞)
(2.10) applied to u and p, an integration by parts in the integral in (2.3) containing ϕ yields
(cid:12)∇
(cid:12)
2pχ ∞ up2vq u2p ϕ = 2pχ ∞ vq up2 2ϕ+ 4pχ ∞ up2vq2 v2q u2pϕ
− q ∇ ∇ q |∇ | q ∇ ∇
Z0 ZΩ Z0 ZΩ Z0 ZΩ
+ 2pχ ∞up2vq∆u2pϕ 2pχ ∞ up2vq∂up2ϕ
q − q ∂ν
Z0 Z0 Z∂Ω
= 2pχ ∞ vq up2 2ϕ+ 4pχ ∞ up2v2q vq2 up2ϕ
q |∇ | q ∇ ∇
Z0 ZΩ Z0 ZΩ
+ 2(p−2)χ ∞ vq up2 2ϕ+ p2χ ∞ up−1vq∆uϕ 2pχ ∞ up2vq∂up2ϕ, (2.11)
q |∇ | q − q ∂ν
Z0 ZΩ Z0 ZΩ Z0 Z∂Ω
whereas integrating by parts twice in the integral containing ∆ϕ in (2.3), by (2.10) applied to u,p and
v,q, respectively, leads to
pχ ∞ pχ ∞ pχ ∞
1 upvq∆ϕ = 1 vq∆(up)ϕ+ 1 up∆(vq)ϕ
− q − q − q
(cid:16) (cid:17)Z0 ZΩ (cid:16) pχ(cid:17)Z0 ∞ZΩ p p q(cid:16) q (cid:17)Z0 ZΩ
+2 1 2u2 u22v2 v2ϕ
− q ∇ ∇
(cid:16)1 pχ (cid:17)Z∞0 ZΩ2up2∂up2vqϕ 2 1 pχ ∞ upv2q ∂vq2ϕ
− − q ∂ν − − q ∂ν
= 4(p−1) 1(cid:16) pχ (cid:17)∞Z0 vZq∂Ω up2 2ϕ+ 4(q−1(cid:16)) 1 pχ(cid:17)Z0 ∞Z∂Ω up v2q 2ϕ
p − q |∇ | q − q |∇ |
(cid:16) (cid:17)pχZ0 ZΩ∞ (cid:16) pχ (cid:17)Z∞0 ZΩ
+ 1 p vqup−1∆uϕ+ 1 q upvq−1∆vϕ
− q − q
(cid:16) pχ(cid:17) Z0∞ZΩ p q p q(cid:16) (cid:17) Z0 ZΩ
+8 1 u2v2 u2 v2ϕ
− q ∇ ∇
(cid:16)1 pχ (cid:17)Z∞0 ZΩ2up2∂up2vqϕ (2.12)
− − q ∂ν
(cid:16) (cid:17)Z0 Z∂Ω
for any such ϕ, for we already know that ∂v = 0 on ∂Ω (0, ). If we combine (2.3) with (2.8), (2.11)
∂ν × ∞
and (2.12), we obtain
∞ (upvq)tϕ 4(1−p) + 2pχ + 4(p−1) 1 pχ + 2(p−2)χ ∞ vq up2 2ϕ
≥ p q p − q q |∇ |
Z0 ZΩ (cid:26) (cid:16) (cid:17) (cid:27)Z0 ZΩ
+ 4(pχ+1−q) + 4(q−1) 1 pχ ∞ up vq2 2ϕ
q q − q |∇ |
(cid:26) (cid:16) (cid:17)(cid:27)Z0 ZΩ
4(1 p)χ 4pχ 8pχ ∞ p q p q
+ − 8+ +8 u2v2 u2 v2ϕ
− q − q − q ∇ ∇
(cid:26) (cid:27)Z0 ZΩ
p2χ pχ ∞
+ + 1 p vqup−1∆uϕ
q − q
(cid:26) pχ(cid:16) ∞ (cid:17) (cid:27)Z0 ZΩ ∞ ∞
+ 1 q upvq−1∆vϕ q upvqϕ+q up+1vq−1ϕ
− q −
(cid:16) (cid:17) Z0 ZΩ Z0 ZΩ Z0 ZΩ
7
2pχ ∞ up2vq∂up2ϕ 1 pχ ∞ 2up2∂up2 vqϕ
− q ∂ν − − q ∂ν
Z0 Z∂Ω (cid:18) (cid:19)Z0 Z∂Ω
= ∞ 4pχup vq2 2 4χup2vq2 up2 v2q +pvqup−1∆u pχupvq−1∆v ϕ
q2 |∇ | − q ∇ ∇ −
Z0 ZΩ(cid:26) (cid:27)
∞
+q upvq−1 ∆v v+u ϕ
{ − }
Z0 ZΩ
2 ∞ up2∂up2vqϕ (2.13)
− ∂ν
Z0 Z∂Ω
for every ϕ C∞(Ω (0, )) satisfying ϕ 0 throughout Ω (0, ) and ∂ϕ = 0 as well as
∈ 0 × ∞ ≥ × ∞ ∂ν ∂Ω×(0,∞)
suppϕ u > 0 .
(cid:12)
⊂ { }
The observations that (cid:12)
u
pup−1vq ∆u χ ( v) = pup−1vq∆u pχup−1vq−1 u v+pχupvq−2 v 2 pχup−1vq−1∆v
− ∇· v∇ − ∇ ∇ |∇ | −
(cid:16) (cid:17) = pup−1vq∆u 4χup2vq2 up2 v2q + 4pχup vq2 2 pχup−1vq−1∆v,
− q ∇ ∇ q2 |∇ | −
and that v solves (1.2), now turn (2.13) into
∞ ∞ u ∞ ∂u
p up−1vqu ϕ p up−1vq ∆u χ v ϕ p up−1vq ϕ (2.14)
t
≥ − ∇· v∇ − ∂ν
Z0 ZΩ Z0 ZΩ n (cid:16) (cid:17)o Z0 Z∂Ω
for all nonnegative ϕ C∞(Ω (0, )) with suppϕ u > 0 and ∂ϕ = 0.
∈ 0 × ∞ ⊂ { } ∂ν ∂Ω×(0,∞)
Specializing this to nonnegative ϕ C∞(Ω (0, ) u > 0 ) by a Du Bois-Reymond lemma type
∈ 0 × ∞ ∩{ } (cid:12)
argument we conclude (cid:12)
u
u ∆u χ v in u > 0 . (2.15)
t
≥ − ∇· v∇ { }
(cid:16) (cid:17)
Density of u > 0 in Ω (0, ), obtained from the assumption that u > 0 a.e., and continuity show
{ } × ∞
that (2.15) actually holds on all of Ω (0, ).
We pick t > 0 and some nonnegativ×e ψ ∞C1(Ω) with ∂ψ = 0 such that suppψ u(,t ) >
0 ∈ ∂ν ∂Ω ⊂ { · 0
0 := x Ω : u(x,t ) > 0 . Then by continuity of u we can find some τ > 0 such that suppψ
} { ∈ 0 } (cid:12) ⊂
u(,t) > 0 . Applying (2.14) to functions of th(cid:12)e form ϕ(x,t) = ζ(t)ψ(x), ζ C∞((t
∩t∈(t0−τ,t0+τ){ · } ∈ 0 0 −
τ,t + τ)) by once more invoking a Weierstrass type density argument and the Du Bois-Reymond
0
lemma, we see that
u ∂u(,t)
up−1vqu (,t)ψ up−1vq ∆u χ v (,t)ψ up−1vq · ψ
t
· ≥ − ∇· v∇ · − ∂ν
ZΩ ZΩ n (cid:16) (cid:17)o Z∂Ω
for every nonnegative ψ C1(Ω) such that ∂ψ = 0, suppψ u(,t ) > 0 and for almost every
∈ ∂ν ∂Ω ⊂ { · 0 }
t (t τ,t + τ) – and due to continuity especially for t = t . In particular inserting ψ (x) :=
∈ 0 − 0 (cid:12) 0 ε
(1 1dist(x,∂Ω)) ψ and Lebesgue’s theorem(cid:12) show that for every t > 0, ψ C1(Ω), ψ 0 with
− ε + · ∈ ≥
∂ψ = 0 and suppψ ∂Ω u(,t) > 0 ,
∂ν ∂Ω ∩ ⊂ { · }
(cid:12) ∂u
(cid:12) up−1vq ψ 0. (2.16)
∂ν ≥
Z∂Ω
8
Since the integral only depends on ψ and not on the values of ψ inside Ω, it can be seen that (2.16)
∂Ω
actually holds for any t > 0 and any nonnegative ψ C0(Ω) such that suppψ ∂Ω u(,t) > 0 .
(cid:12) ∈ ∩ ⊂ { · }
If ∂u(x ,t ) < 0 for some (x ,t ) (cid:12)∂Ω (0, ) with u(x ,t ) > 0, we pick ψ C0(∂Ω) such that
∂ν 0 0 0 0 ∈ × ∞ 0 0 1 ∈
ψ (x ) > 0, ψ 0, and suppψ x ∂Ω : ∂u(x,t ) < 0,u(x,t ) > 0 =: M. Moreover, we let
1 0 1 ≥ 1 ⊂ { ∈ ∂ν 0 0 }
d := dist(suppψ ,∂Ω M) (or d = 1 if ∂Ω M = ) and let ψ be the solution to ∆ψ = 0 in Ω,
1 2 2
\ \ ∅ −
ψ = u1−p(,t )v−q(,t )ψ ∂u on ∂Ω. Then, thanks to the choice of suppψ , ψ is nonnegative on
2 − · 0 · 0 1∂ν 1 2
the boundary and hence by the maximum principle in Ω. Defining ψ (x) := ψ (x)(1 2dist(x,M))
3 2 − d +
we obtain a nonnegative continuous function on Ω whose support intersects the boundary only in
x ∂Ω : u(x,t) > 0 and which hence is a permissible test function in (2.16). We conclude that
{ ∈ }
0 up−1vq∂uψ = ∂u 2ψ and hence in particular ∂u(x ,t ) = 0, which is a contradiction.
≤ ∂Ω ∂ν 3 − ∂Ω|∂ν| 1 ∂ν 0 0
We conclude that ∂u 0 on ∂Ω (0, ) u > 0 and hence, by continuity of ∂u and density of this
R ∂ν ≥ R × ∞ ∩{ } ∂ν
set that ∂u 0 on ∂Ω (0, ).
∂ν ≥ × ∞
Finally integrating (2.15) over Ω (0,t) and taking (2.7) into consideration, we see that
×
t t u t ∂u
u u(,t) u + ∆u χ v = u +
0 0 0
≥ · ≥ − ∇· v∇ ∂ν
ZΩ ZΩ ZΩ Z0 ZΩ Z0 ZΩ (cid:16) (cid:17) ZΩ Z0 Z∂Ω
by Gauss’ theorem and ∂v = 0, which firstly shows that ∂u = 0 on ∂Ω (0, ) and secondly that
∂ν ∂ν × ∞
(2.15) actually is an equality. (cid:3)
3 Global smooth solutions to approximate problems
Now in order to approximate solutions by means of a convenient regularization of (1.1)-(1.2), for
ε (0,1) we consider
∈
u = ∆u χ uε v , x Ω, t > 0,
εt ε− ∇· vε∇ ε ∈
 vεt = ∆vε−vε+ 1(cid:16)+uεεuε, (cid:17) x ∈ Ω, t > 0, (3.1)
 ∂uε = ∂vε = 0, x ∂Ω, t > 0,
 ∂ν ∂ν ∈
u (x,0) = u (x), v (x,0) =v (x), x Ω,
ε 0 ε 0
 ∈



and then first obtainthe following.
Lemma 3.1. For all ε (0,1), the problem (3.1) admits a global classical solution (u ,v ) (C0(Ω
ε ε
∈ ∈ ×
[0, )) C2,1(Ω (0, )))2 for which u > 0 in Ω (0, ) and v > 0 in Ω [0, ).
ε ε
∞ ∩ × ∞ × ∞ × ∞
Proof. The local existence of a solution can be obtained in a standard manner (cf. [1, Lemma 3.1] for
a related general setting). Boundedness of the source term uε 1 of the second equation in (3.1)
1+εuε ≤ ε
translates into a bound on t vε(,t) W1,∞(Ω). Combined with the strict positivity property of vε
7→ k · k
on Ω (0,T) for any finite T – to be made more precise in Lemma 3.3 below – this serves to provide a
×
uniform bound on u on Ω (0,T), in light of the extensibility criterion [1, (3.3)] thus ensuring global
ε
×
existence of the solution. Positivity of u follows from a classical strong maximum principle. (cid:3)
ε
These approximate solutions clearly preserve mass:
9
Lemma 3.2. Let ε (0,1). Then
∈
u (,t) = u for all t > 0. (3.2)
ε 0
·
ZΩ ZΩ
Proof. This directly results on integrating the first equation in (3.1). (cid:3)
Moreover, the assumed positivity of v enables us to control v from below at least locally in time:
0 ε
Lemma 3.3. For each ε (0,1), we have
∈
v (x,t) inf v (x) e−t for all x Ω and t > 0.
ε 0
≥ x∈Ω · ∈
(cid:18) (cid:19)
Proof. As u is nonnegative, this is a straightforward consequence of a comparison argument applied
ε
to the second equation in (3.1). (cid:3)
By means of well-known smoothing estimates of the heat semigroup, the mass conservation property
(3.2) readily implies some basic regularity features of the second component.
Lemma 3.4. Let r 1 and s 1 be such that r < n and s < n . Then there exists C > 0 such
≥ ≥ n−2 n−1
that for each ε (0,1),
∈
vr(,t) C for all t > 0 (3.3)
ε
· ≤
ZΩ
and
v (,t)s C for all t > 0. (3.4)
ε
|∇ · | ≤
ZΩ
Proof. The representation of v as
ε
t u (,s)
v (,t) = et(∆−1)v + e(t−s)(∆−1) ε · ds
ε 0
· 1+εu (,s)
Z0 ε ·
makes it possible to apply well-known estimates for the Neumann heat-semigroup (cf. [20, Lemma
1.3]), which provide positive constants c ,c ,c and c such that
1 2 3 4
t u (,s)
kvε(·,t)kLr(Ω) ≤ c1kv0kLr(Ω)+c2Z0 (1+(t−s)−n2(1−1r)e−(t−s)(cid:13)1+εεu·ε(·,s)(cid:13)L1(Ω)ds
(cid:13) (cid:13)
and (cid:13) (cid:13)
t u (,s)
k∇vε(·,t)kLs(Ω) ≤ c3kv0kW1,∞(Ω)+c4Z0 (1+(t−s)−12−n2(1−1s))e−(t−s)(cid:13)1+εεu·ε(·,s)(cid:13)L1(Ω)ds
(cid:13) (cid:13)
for all t > 0, so that the estimate uε u (,t) = u (cid:13) due to Lem(cid:13)ma 3.2 and
k1+εuεkL1(Ω) ≤ k ε · kL1(Ω) k 0kL1(Ω)
finiteness of ∞(1+τ−n2(1−1r))e−τdτ and ∞(1+τ−21−n2(1−1s))e−τdτ due to the conditions on r and s
0 0
prove the lemma. (cid:3)
R R
10