A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

0.16 MB·

by  Masaaki Mizukami

A unified method for boundedness in fully parabolic
chemotaxis systems with signal-dependent sensitivity
7 Masaaki Mizukami
1
Department of Mathematics, Tokyo University of Science
0
2 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
n [email protected]
a
J
Tomomi Yokota∗ †
1
1 Department of Mathematics, Tokyo University of Science

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
]
P [email protected]
A
.
h January 12, 2017
t
a
m
[ Abstract. This paper deals with the Keller–Segel system

1
v ut =∆u−∇·(uχ(v)∇v), x∈Ω, t>0,
7 (vt =∆v+u v, x Ω, t>0,
1 − ∈
8 where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n 2; χ is a function
02 satisfying χ(s) K(a + s)−k for some k 1 and a 0. In th≥e case that k = 1,
≤ ≥ ≥
. Fujie (J. Math. Anal. Appl.; 2015;424;675–684)established global existence of bounded
1
0 solutionsunder the conditionK < 2. Onthe otherhand,whenk >1,Winkler (Math.
n
7
Nachr.;2010;283;1664–1673)asserqtedglobalexistenceofboundedsolutionsforarbitrary
1
: K > 0. However, there is a gap in the proof. Recently, Fujie tried modifying the proof;
v
neverthelessitalsohasagap. Itseemstobedifficulttoshowglobalexistenceofbounded
i
X solutions for arbitraryK >0. Moreover,the conditionfor K when k >1 cannotconnect
r to the condition when k = 1. The purpose of the present paper is to obtain global
a
existence and boundedness under more natural and proper condition for χ and to build
a mathematical bridge between the cases k =1 and k >1.
2010Mathematics Subject Classification. Primary: 35K51;Secondary: 35B45, 35A01, 92C17.
Key words and phrases: chemotaxis; sensitivity function; global existence; boundedness.
∗Corresponding author
†Partially supported by Grant-in-Aid for Scientific Research (C), No.16K05182.
1
1. Introduction
The chemotaxis system proposed by Keller and Segel in [9] describes a part of the life
cycle of cellular slime molds with chemotaxis. After the pioneering work [9], a number of
variations of the chemotaxis system are proposed and investigated (see e.g., [1, 6, 7]).
In this paper we consider the Keller–Segel system with signal-dependent sensitivity
u = ∆u (uχ(v) v), x Ω, t > 0,
t
−∇· ∇ ∈
v = ∆v +u v, x Ω, t > 0,
t
(1.1)  − ∈

 u ν = v ν = 0, x ∂Ω, t > 0,

∇ · ∇ · ∈
u(x,0) = u (x), v(x,0) = v (x), x Ω,
0 0
 ∈



where Ω is a bounded domain in Rn (n 2) with smooth boundary ∂Ω and ν is the
≥
outward normal vector to ∂Ω. The initial data u and v are assumed to be nonnegative
0 0
functions. The unknown function u(x,t) represents the population density of species and
v(x,t)showstheconcentrationofthesubstance atplacexandtimet. Astothesensitivity
function χ, we are interested in functions generalizing
K K
χ(s) = and χ(s) = (s > 0).
s (1+s)2
In a mathematical view, the difficulty caused by the sensitivity function is to deal with
the additional term uχ′(v) v 2 which does not appear in the case that χ is a constant.
|∇ |
Moreover, when χ is the singular sensitivity function, it is delicate to derive the estimate
for 1. In the case that χ(v) = K, Winkler [14] first attained global existence of classical
v v
solutions when K < 2 and global existence of weak solutions when K < n+2 .
n 3n−4
However, the result inq[14] could not arrive at boundedness of solutions to (1.q1). To
overcome the difficulty in a singular sensitivity, Fujie [2] established the uniform-in-time
lower estimate for v, and show global existence of classical bounded solutions to (1.1) in
the case that χ(v) = K with
v
2
(1.2) K < .
n
r
As to the problem (1.1) with χ(v) = K in the 2-dimensional setting, Lankeit [10] obtained
v
global existence of classical bounded solutions when K < χ (χ > 1). Futhermore Fujie–
0 0
Senba [4] dealt with the 2-dimensional problem which was replaced v with τv in (1.1)
t t
and showed global existence and boundedness of radially symmetric solutions under the
condition that χ(s) 0 (t ) and τ is sufficiently small. On the other hand, in the
→ → ∞
case that χ(v) K (k > 1, α > 0, K > 0), Winkler [13] established global existence
≤ (1+αv)k
and boundedness in (1.1) without any restriction on K > 0. The result in [13] affected
the result in [5] which mentioned global existence of classical bounded solutions to (1.1)
when χ is the strong singular sensitivity function such that χ(v) = K for k > 1, and
vk
the method in [13] was used in Zhang–Li [15] and Zheng–Mu [16]. However, we cannot
convince the result in [13] that global existence and boundedness hold for all K > 0,
because there is a gap in the proof. Recently, Fujie tried modifying it; nevertheless it also
2
has a gap (cf. [3]). In general, it seems to be difficult to show global existence of bounded
solutions for arbitrary K > 0. Moreover, if k 1, then the condition “arbitrary K > 0”
→
cannot connect to (1.2).
The purpose of this paper is to obtain global existence and boundedness in (1.1) under
a more natural and proper condition for χ and to build a mathematical bridge between
the cases k = 1 and k > 1. We shall suppose that χ satisfies that
K
(1.3) χ(s)
≤ (a+s)k
with some k 1, a 0 and K > 0 fulfiling
≥ ≥
2
(1.4) K < k(a+η)k−1 .
n
r
Here
(1.5) η := sup min e−2τ minv (x), c u (1 e−τ) ,
0 0 0 L1(Ω)
k k −
τ>0 x∈Ω
(cid:18) (cid:26) (cid:27)(cid:19)
where c > 0isa lower boundfor thefundamental solutionofw = ∆w w withNeumann
0 t
−
boundary condition (for more detail, see Remark 2.1). We suppose that
0 < v W1,q(Ω) ( q > n) (a = 0),
0
(1.6) 0 u C(Ω) 0 and ∈ ∃
≤ 0 ∈ \{ } (0 v0 W1.q(Ω) 0 ( q > n) (a > 0).
≤ ∈ \{ } ∃
Now the main results read as follows.
Theorem 1.1. Let n 2 and let Ω Rn be a bounded domain with smooth boundary.
≥ ⊂
Assume that χ satisfies (1.3) with some k 1, a 0, K > 0 fulfiling (1.4). Then for any
≥ ≥
u ,v satisfying (1.6) with some q > n, there exists an exactly one pair (u,v) of functions
0 0
u, v C(Ω [0, )) C2,1(Ω (0, ))
∈ × ∞ ∩ × ∞
which solves (1.1). Moreover, the solution (u,v) is uniformly bounded, i.e., there exists a
constant C > 0 such that
u( ,t) + v( ,t) C for all t 0.
L∞(Ω) L∞(Ω)
k · k k · k ≤ ≥
Remark 1.1. The unified condition (1.3) with K > 0 satisfying (1.4) may be a natural
condition for χ. Indeed, when k = 1, (1.4) becomes (1.2):
2 2
K < k(a+η)k−1 = .
n n
r !(cid:12) r
(cid:12)k=1
(cid:12)
(cid:12)
The main theorem tells us the result in the typical case of singular sensitivity.
(cid:12)
Corollary 1.2. Let χ(s) = K with K < 2. Then for any u ,v satisfying (1.6) with
s n 0 0
some q > n, (1.1) has a unique global bounqded classical solution.
3
The strategy for the proof of Theorem 1.1 is to construct the estimate for up with
Ω
some p > n. One of the keys for this strategy is to derive the unified inequality
2 R
d ϕ(v)
upϕ(v)+εp(p 1) up−2ϕ(v) u 2 c upϕ(v) r up+1
dt − |∇ | ≤ − (a+v)k
ZΩ ZΩ ZΩ ZΩ
for some ε [0,1) and c > 0, where
∈
s 1
ϕ(s) := exp r dτ
− (a+s)k
(cid:26) Z0 (cid:27)
with r > 0. This function ϕ constructed in [11, 12] unifies mathematical structures in
the cases k > 1 and k = 1.
This paper is organized as follows. In Section 2 we collect basic facts which will be
used later. In Section 3 we give a unified view point in energy estimates. Section 4 is
devoted to the proof of global existence and boundedness (Theorem 1.1).
2. Preliminaries
In this section we will collect elementary results. We first recall the uniform-in-time
lower estimate for v established by Fujie [2, 3].
Lemma 2.1. Let u C(Ω) and u C(Ω [0,T)) be nonnegative functions such that
0
∈ ∈ ×
u 0 and u( ,t) = u (t [0,T)). If v C(Ω) is a positive function in Ω and
0 6≡ Ω · Ω 0 ∈ 0 ∈
v C2,1(Ω (0,T)) C(Ω [0,T)) is a classical solution of
∈ ×R ∩ R ×
v = ∆v v +u, x Ω, t (0,T),
t
− ∈ ∈
v ν = 0 x ∂Ω, t (0,T),

∇ · ∈ ∈
v(x,0) = v (x) x Ω,
0
∈
then for all t (0,T), 
∈
inf v(x,t) η > 0,
x∈Ω ≥
where η > 0 is defined as (1.5).
Remark 2.1. When Ω is a convex bounded domain, the proof of this lemma is given in
[2, Lemma 2.2] and the constant η can be explicitly represented as
τ 1 −r+(maxx,y∈4Ωr |x−y|)2
η := sup min e−τ minv (x), u e  dr .
  0 k 0kL1(Ω) · (4πr)n 
τ>0  x∈Ω Z0 2 
 
 
On the other hand, if we do not assume the convexity of Ω, then using the positivity of
the fundamental solution U(t,x;s,y) to w = ∆w w in Ω (0,T) with w ν = 0 on
t
− × ∇ ·
∂Ω (0,T) (see e.g., [8]), we have that there exists c > 0 such that for all τ (0, T),
× 0 ∈ 2
U(s+τ,x;s,y) = U(τ,x;0,y) c > 0 for all x,y Ω, s > 0.
0
≥ ∈
Then we can see the conclusion of Lemma 2.1 by a similar argument as in [2, Lemma 2.2].
4
We next recall the well-known result about local existence of solutions to (1.1) (see
[10, Theorem 2.3], [4, Proposition 2.2] and [13, Lemma 2.1]).
Lemma 2.2. Assume that χ satisfies (1.3) and the initial data u ,v fulfil (1.6) for some
0 0
q > n. Then there exist T (0, ] and exactly one pair (u,v) of nonnegative functions
max
∈ ∞
u C(Ω [0,T )) C2,1(Ω (0,T )),
max max
∈ × ∩ ×
v C(Ω [0,T )) C2,1(Ω (0,T )) L∞([0,T );W1,q(Ω))
∈ × max ∩ × max ∩ loc max
which solves (1.1) in the classical sense and satisfies the mass conservation
(2.1) u( ,t) = u for all t (0,T )
0 max
· ∈
ZΩ ZΩ
and the lower estimate
(2.2) inf v(x,t) η for all t (0,T ),
max
x∈Ω ≥ ∈
where η > 0 is defined as (1.5). Moreover, if T < , then
max
∞
lim ( u( ,t) + v( ,t) ) = .
L∞(Ω) W1,∞(Ω)
tրTmax k · k k · k ∞
At the end of this section we shall recall the result about the estimate for v in depen-
dence onboundedness featuresofuderived byastraightforwardapplicationofwell-known
smoothing estimates for the heat semigroup under homogeneous Neumann boundary con-
ditions (see [14, Lemma 2.4] and [2, Lemma 2.4]).
Lemma 2.3. Let T > 0 and 1 θ,µ . If n(1 1) < 1, then there exists C (µ,θ) > 0
≤ ≤ ∞ 2 θ−µ 1
such that
v( ,t) C (µ,θ) 1+ sup u( ,s) for all t (0,T).
Lµ(Ω) 1 Lθ(Ω)
k · k ≤ k · k ∈
s∈(0,∞) !
3. A unified view point in energy estimates
Let (u,v) be the solution of (1.1) on [0,T ) as in Lemma 2.2. For the proof of
max
Theorem 1.1 we will recall an useful fact to derive the L∞-estimate for u.
Lemma 3.1. Assume that the solution u of (1.1) given in Lemma 2.2 satisfies
(3.1) u( ,t) C(p) for all t (0,T )
Lp(Ω) max
k · k ≤ ∈
with some p > n and C(p) > 0. Then there exists a constant C′ > 0 such that
2
u( ,t) C′ for all t (0,T ).
L∞(Ω) max
k · k ≤ ∈
Proof. Combination of (2.2) and the same argument as in [14, Lemma 3.4] leads to the
conclusion of Lemma 3.1.
5
Unified test function. Thanks to Lemmas 2.2 and 3.1 we will only make sure that the
Lp-estimate for u with some p > n to show global existence and boundedness of solutions
2
to (1.1). To establish (3.1) we introduce the functions g and ϕ by
s 1
(3.2) g(s) := r dτ, ϕ(s) := exp g(s) (s η),
− (a+τ)k { } ≥
Zη
where r > 0 is a constant fixed later and η is defined as (1.5). When k > 1, by straight-
forward calculations we have
r
ϕ(s) = C exp
ϕ (k 1)(a+s)k−1
(cid:26) − (cid:27)
with C = exp r(k 1)−1(a+η)−k+1 > 0, which is a similar function used in [13]. On
ϕ
{− − }
the other hand, when k = 1, it follows that
C
ϕ
ϕ(s) =
(a+s)r
with some constant C = (a+η)r > 0, which is a similar function used in [2]. Now we
ϕ
shall prove the following unified inequality by using the test function ϕ(v).
Lemma 3.2. Assume that χ satisfies (1.3). Then for all ε [0,1) there exists c > 0 such
∈
that
d
(3.3) upϕ(v)+εp(p 1) up−2ϕ(v) u 2
dt − |∇ |
ZΩ ZΩ
ϕ(v)
upH (v)ϕ(v) v 2+c upϕ(v) r up+1 ,
≤ ε |∇ | − (a+v)k
ZΩ ZΩ ZΩ
where
(εp+1 ε)r2 εpKr kr p(p 1)K2
H (s) := − + + − .
ε
(1 ε)(p 1)(a+s)2k (1 ε)(a+s)2k − (a+s)k+1 4(1 ε)(a+s)2k
− − − −
Proof. We let p 1. From (1.1) we have
≥
d
(3.4) upϕ(v) = p up−1ϕ(v) ( u uχ(v) v)+ upϕ′(v)(∆v v +u).
dt ∇· ∇ − ∇ −
ZΩ ZΩ ZΩ
Integration by parts yields
(3.5) p up−1ϕ(v) ( u uχ(v) v)+ upϕ′(v)∆v
∇· ∇ − ∇
ZΩ ZΩ
= p (up−1ϕ(v)) ( u uχ(v) v) (upϕ′(v)) v
− ∇ · ∇ − ∇ − ∇ ·∇
ZΩ ZΩ
= p(p 1) up−2ϕ(v) u 2+ up−1(p(p 1)ϕ(v)χ(v) 2pϕ′(v)) u v
− − |∇ | − − ∇ ·∇
ZΩ ZΩ
+ up( ϕ′′(v)+pϕ′(v)χ(v)) v 2.
− |∇ |
ZΩ
6
Invoking to the Young inequality, we infer that for all ε [0,1),
∈
(3.6) up−1(p(p 1)ϕ(v)χ(v) 2pϕ′(v)) u v
− − ∇ ·∇
ZΩ
(p(p 1)ϕ(v)χ(v) 2pϕ′(v))2
(1 ε)p(p 1) up−2ϕ(v) u 2+ up − − v 2.
≤ − − |∇ | 4(1 ε)p(p 1)ϕ(v) |∇ |
ZΩ ZΩ − −
Thus combination of (3.4), (3.5) and (3.6) yields that
(3.7)
d
upϕ(v)+εp(p 1) up−2ϕ(v) u 2 upF (v) v 2+ upϕ′(v)( v +u),
ϕ
dt − |∇ | ≤ |∇ | −
ZΩ ZΩ ZΩ ZΩ
where
ε p(p 1) pϕ′(s)2
F (s) := ϕ′′(s) pϕ′(s)χ(s)+ − χ(s)2ϕ(s)+ .
ϕ
− − 1 ε 4(1 ε) (1 ε)(p 1)ϕ(s)
− − − −
Noting that
ϕ′(s) = g′(s)ϕ(s) and ϕ′′(s) = g′′(s)ϕ(s)+g′(s)2ϕ(s),
we can rewrite the function F (s) as
ϕ
ε p(p 1) (εp+1 ε)g′(s)2
F (s) = g′′(s) pg′(s)χ(s)+ − χ(s)2 + − ϕ(s).
ϕ
− − 1 ε 4(1 ε) (1 ε)(p 1)
(cid:18) − − − − (cid:19)
Recalling by (3.2) that
r rk
g′(s) = − and g′′(s) = ,
(a+s)k (a+s)k+1
we see from (1.3) that
(3.8) F (s) H (s)ϕ(v),
ϕ ε
≤
where
(εp+1 ε)r2 εpKr kr p(p 1)K2
H (s) := − + + − .
ε
(1 ε)(p 1)(a+s)2k (1 ε)(a+s)2k − (a+s)k+1 4(1 ε)(a+s)2k
− − − −
Therefore we obtain from (3.7) together with (3.8) that
d
upϕ(v)+εp(p 1) up−2ϕ(v) u 2
dt − |∇ |
ZΩ ZΩ
( v+u)
upH (v)ϕ(v) v 2 r upϕ(v) − .
≤ ε |∇ | − (a+v)k
ZΩ ZΩ
Wefinallyconfirmfromtheboundedness of s (k 1)thatthereexistsc > 0satisfying
(a+s)k ≥
rv
upϕ(v) c upϕ(v),
(a+v)k ≤
ZΩ ZΩ
and thus we obtain (3.3).
7
4. Global existence and boundedness
In this section we will show the Lp-estimate for u with p > n by using Lemma 3.2.
2
4.1. Energy estimate in the case k > 1
We first derive the energy estimate in the case k > 1. In this subsection we assume that χ
satisfies (1.3) with some k > 1. Now we shall show the following inequality by modifying
the method in [11].
Lemma 4.1. Assume that (1.3) and (1.4) are satisfied with some k > 1, a 0 and
≥
K > 0. Then there exist p > n, r > 0 and ε (0,1) such that
2 ∈
H (s) 0 for all s η,
ε
≤ ≥
where H is defined in Lemma 3.2 and η > 0 is defined as (1.5), which implies that
ε
d ϕ(v)
(4.1) upϕ(v)+εp(p 1) up−2ϕ(v) u 2 c upϕ(v) r up+1 .
dt − |∇ | ≤ − (a+v)k
ZΩ ZΩ ZΩ ZΩ
Proof. We take p 1, r > 0 and ε [0,1) which will be fixed later. Due to the definition
≥ ∈
of H we write as
ε
H (s) = a (s)r2 +a (s)r+a (s),
ε 1 2 3
where
εp+1 ε
a (s) := − ,
1 (1 ε)(p 1)(a+s)2k
− −
εpK k
a (s) := ,
2 (1 ε)(a+s)2k − (a+s)k+1
−
p(p 1)K2
a (s) := − .
3 4(1 ε)(a+s)2k
−
Noting from the condition (1.4) that there exist p > n and ε (0,1) satisfying
2 ∈
(1 ε)k(a+η)k−1
K < − ,
εp+ p(εp+1 ε)
−
we have that p
εp+ p(εp+1 ε)
− K k(a+s)k−1 for all s η.
(cid:16) p1 ε (cid:17) ≤ ≥
−
This implies that the discriminant
D (s) = a (s)2 4a (s)a (s)
r 2 1 3
−
εpK k 2 p(εp+1 ε)K2
= −
(1 ε)(a+s)2k − (a+s)k+1 − (1 ε)2(a+s)4k
(cid:18) − (cid:19) −
1 εpK 2 p(εp+1 ε)K2
= k(a+s)k−1 −
(a+s)4k (1 ε) − − (1 ε)2
!
(cid:18) − (cid:19) −
8
is nonnegative for all s η. Finally, we show that there exists r > 0 such that
≥
(4.2) H (s) = a (s)r2 +a (s)r +a (s) 0
ε 1 2 3
≤
for all s η. Because D (s) is nonnegative, we can define
r
≥
a D (s)
2 r
r (s) := − ±
± 2a (s)
1p
(1 ε)(p 1) εpK
= − − +k(a+s)k−1 (a+s)2k D (s) .
r
2(εp+1 ε) −(1 ε) ±
− (cid:18) − (cid:19)
p
Then we see that H (s) 0 for each s η and all r (r (s),r (s)). Since the functions
ε − +
≤ ≥ ∈
p(εp+1 ε)K2 K p(εp+1 ε)
r (τ) := τ τ2 − τ −
± ±s − (1−ε)2 ≥ p 1−ε !
e
satisfy that
dr dr
+(τ) > 0 and −(τ) < 0
dτ dτ
e e
K√p(εp+1−ε)
for all τ , we obtain that
≥ 1−ε
K p(εp+1 ε) K p(εp+1 ε)
r (s) r − = r − r (s)
− ≤ − 1 ε + 1 ε ≤ +
p − ! p − !
e e
holds for all s η. Therefore if we put
≥
K p(εp+1 ε) (p 1)K2 p
r := r − = − > 0,
0 − 1 ε 2 εp+1 ε
p − ! r −
e
then r = r (r (s),r (s)) for all s η, which means that (4.2) holds for all s η. This
0 − +
∈ ≥ ≥
implies the end of the proof.
Now we are ready to show the Lp-estimate in the case k > 1.
Lemma 4.2. Assume that (1.3) and (1.4) are satisfied with some k > 1, a 0 and
≥
K > 0. Then there exist p > n and C > 0 such that
2
u( ,t) C for all t > 0.
Lp(Ω)
k · k ≤
Proof. The proof is similar as in [13]. From Lemmas 3.2 and 4.1 we obtain (4.1) with
some p > n, r > 0 and ε (0,1). We shall show the Lp-estimate for u by using (4.1).
2 ∈
From the positivity of u, v and ϕ we have that
ϕ(v)
(4.3) r up+1 0.
− (a+v)k ≤
ZΩ
9
We next deal with the term up−1ϕ(v) u 2. Noting the boundedness of ϕ:
Ω |∇ |
(4.4) R C ϕ(s) 1 (s η)
ϕ
≤ ≤ ≥
and combining the Gagliardo–Nirenberg inequality with the mass conservation property
(2.1), we deduce that there exists c > 0 such that
1
(4.5) upϕ(v) ≤ up = ku2pk2L2(Ω)
ZΩ ZΩ
2a
≤ c1 k∇up2kL2(Ω) +ku2pkLp2(Ω) kup2kL2(p21−(Ωa))
(cid:16) (cid:17)
= c1 k∇up2kL2(Ω) +ku0kLp21(Ω) 2aku0kpL(11(−Ωa))
(cid:16) (cid:17)
2n−n
with a = p 2 (0,1). From (4.1), (4.3) and combination of (4.5) and
2n+1−n ∈
p 2
Cϕ u2p 2 up−2ϕ(v) u 2
|∇ | ≤ |∇ |
ZΩ ZΩ
we see that there exist c ,c > 0 such that
2 3
1
d a
upϕ(v) c upϕ(v) c upϕ(v) +c ,
dt ≤ − 2 3
ZΩ ZΩ (cid:18)ZΩ (cid:19)
which implies that there exist p > n, r > 0 (determined in Lemma 4.1) and C > 0
2
satisfying
upϕ(v) C.
≤
ZΩ
Therefore we obtain from (4.4) that
up CC−1.
≤ ϕ
ZΩ
Thus we attain the goal of the proof.
4.2. Energy estimates in the case k = 1
In this section we assume that the sensitivity function χ satisfies (1.3) with k = 1. We
first show the following estimate for H (s).
0
Lemma 4.3. Assume that (1.3) and (1.4) are satisfied with k = 1 and with some a 0
≥
and K > 0. Then for all p (1, 1 ) there exists an interval I such that for all r I
∈ K2 p ∈ p
and s η,
≥
H (s) 0,
0
≤
where H is defined as in Lemma 3.2 with ε = 0, which means
0
d ϕ(v)
(4.6) upϕ(v) c upϕ(v) r up+1 .
dt ≤ − a+v
ZΩ ZΩ ZΩ
10