Analysis on Keller-Segel Models in Chemotaxis Li CHEN Universit at - - PowerPoint PPT Presentation

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Analysis on Keller-Segel Models in Chemotaxis

Li CHEN

Universit¨ at Mannheim

03.2019, Potsdam

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

1

Keller-Segel model with linear diffusion

2

Keller-Segel equation with nonlinear diffusion in multi-D

3

Keller-Segel model with nonlinear nonlocal reaction

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Keller-Segel model with linear diffusion

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

The parabolic-elliptic Keller-Segel model with linear diffusion is ρt = ∆ρ − div(ρ∇c), −∆c = ρ.

• case ∇K = C x

|x|n

• .

Typical quantities of the system Conservation of mass m0(t) =

• ρ(x, t)dx =
• ρ0(x)dx = m0

Entropy dissipation relation for F(ρ) =

• (ρ ln ρ − ρc

2 )dx d dt F(ρ) +

• ρ|∇ ln ρ − ∇c|2dx = 0. ⇒ F(ρ) ≤ F(ρ0)

Key feature of the system: Global existence vs. finite time blow up. Since 1990’s, J¨ ager and Luckhaus, Biler, Herrero, Horstmann, Medina, Nagai, Stevens, Velazquez, Winkler......

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

The critical mass 8π in 2-D

ρt = ∆ρ − div(ρ∇c), c = − 1 2π

• log

1 |x − y|ρ(y)dy Entropy: F(ρ) =

• ρ ln ρdx − 1

8π ρ(x, t)ρ(y, t) log 1 |x − y|2 dxdy. Logarithmic Hardy-Littlewood-Sobolev inequality

• ρ log ρdx − 1

m0 ρ(x)ρ(y) log 1 |x − y|2 dxdy + C(m0) ≥ 0, where m0 =

• ρ(x)dx, C(m0) := m0(1 + log π − log m0).

A direct application of this inequality gives F(ρ(·, t)) ≥ (1 − m0 8π )

• ρ log ρdx − m0

8π C(m0), F(ρ(·, t)) ≥ ( 1 m0 − 1 8π ) ρ(x, t)ρ(y, t) log 1 |x − y|2 dxdy − C(m0). m0 < 8π: global existence, Blanchet, Dolbeault, Perthame, 2006. Another proof: Carrillo, Chen, Liu, and Wang, based on Delort’s theory of 2-D incompressible Euler equation, 2012.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

blow up in finite time

2-D, m0 > 8π, Dolbeault, Perthame, 2004. Idea: Second moment, m2(t) := |x|2 2 ρdx, m′

2(t) = 2m0(1 − m0

8π ) < 0. multi-D, m2 < Cm

n n−2

, Perthame, 2005. multi-D, m

n−2 n

n

(0) <

1 (n−1)2n|Sn−1|m 2− 2

n

, Chen, Siedentop, 2017. Idea: n-th moment mn(t) :=

• |x|nρdx,

m′

n(t) ≤ 2n(n − 1)m

n−2 n

n

m

2 n

0 − n21−n

|Sn−1|m2

0.

Remark: The global existence only holds for “small” initial data. It is an interesting question whether the reverse inequality would imply global

• existence. If not, what additional condition can help?

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Main idea with the n-th moment The following Keller-Segel system is

considered (with an additional point source) ∂tρ − ∇ · (∇ρ − ρ∇c) = 0, −∆c = ρ − Zδ, x ∈ Rn, t ≥ 0, ρ(x, 0) = ρ0(x) ≥ 0. Use |x|n as test function m′

n(t)

= 2n(n − 1)

• |x|n−2ρ(x)dx +

nZ |Sn−1|

• ρ(x)dx

− n 2|Sn−1| (|x|n−1 x |x| − |y|n−1 y |y|) x − y |x − y|n

• =:V

ρ(x)ρ(y)dxdy We can show that V ≥ 22−n, therefore m′

n(t) ≤ 2n(n − 1)mn−2 − n21−n

|Sn−1|m2

0 +

nZ |Sn−1|m0.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Further interpolation shows that m′

n(t) ≤ 2n(n − 1)m

n−2 n

n

m

2 n

0 − n21−n

|Sn−1|m2

0 +

nZ |Sn−1|m0. In particular, we have a shrinking n-th moment, if the initial moments fulfill m

n−2 n

n

(0) < 1 (n − 1)2n|Sn−1|m

2− 2

n

− Z 2(n − 1)|Sn−1|m

1− 2

n

. In case of no external point source (Z = 0) and n = 2, we have 1 < m0 8π .

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Keller-Segel equation with nonlinear diffusion in multi-D

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

To balance the aggregation, porus media type nonlinear diffusion is considered since 2005, ρt = ∆ρm − div(ρ∇c) m ≥ 1 = div

• ρ∇
• m

m − 1ρm−1 − c

• , x ∈ Rn, t ≥ 0.

m∗ = 2 − 2

n, Idea: scaling invariance of the total mass. (if ρ(x, t)

is a solution, then λnρ(λx, t) is also a solution). Many results for m = m∗ since 2005: Bedrossian, Bertozzi, Blanchet, Carrillo, Cie´ slak, Horstmann, Ishida, Kowalczyk, Laurencot, Luckhaus, Rodr´ iguez, Sugiyama, Szyma´ nska ,Winkler, Yao, Yokota ... However, the results are not as “beautiful” as the 2-D case. mc =

2n n+2, Idea: stationary solutions and conformal invariant of

the entropy, Chen, Liu, Wang, 2012.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Stationary solutions in 2-D, ln ρ − c = 0, −∆c = ec in R2, has a family of solutions Cλ,x0(x), and the stationary solution for ρ is Uλ,x0(x) = eCλ,x0(x) = 8

• λ

λ2 + |x − x0|2 2 , with Uλ,x0L1 = 8π. Stationary solutions in Multi-D,

m m−1ρm−1 − c = 0,

−∆c = (m − 1 m )

1 m−1 c 1 m−1 in Rn.

Gidas, Spruck (1981) 1 ≤

1 m−1 < n+2 n−2, then c ≡ 0. m∗ belongs to this case. 1 m−1 = n+2 n−2, or m = 2n n+2, then c = Cλ,x0(x), and the stationary solution is

Uλ,x0(x) = 2

n+2 4 n n+2 2

• λ

λ2 + |x − x0|2 n+2

2

with Uλ,x0m

Lm = K(n).

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Results with exponent mc = 2n/(n + 2). Radially symmetric solutions, Chen, Liu, Wang (2012)

ρ0(|x|) < Uλ0(|x|), global existence, ρ(|x|, t)

t→∞

− → 0 in L1

loc(Rn).

ρ0(|x|) > Uλ0(|x|), blow-up, ∃t∗ ≤ +∞, i.e., ∃r(t)

t→t∗

− → 0 s.t.

• B(0,r(t))

ρ(|x|, t)dx ≥ C > 0.

For general initial data, Chen, Liu, Wang (2012)

ρ0Lmc < Cs < UλLmc , global existence, for t ≫ 1, it holds ρ(·, t)Lmc ≤ Ct−

1 mc (β−1) ,

β = 2m2

c − 3mc + 2

mc(mc − 1) > 1. m2(0) < ∞, F(ρ0) < F(Uλ) and ρ0Lmc > UλLmc , blow up in the sense that ∃T ∗ < ∞ s.t. lim

t→T ∗ ρ(·, t)Lmc = +∞.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Results with exponent mc < m < m∗. Chen, Wang, (2014)

For ρ0 ∈ L1

+(Rn) ∩ L

2n n+2 (Rn), F(ρ0) < F∗, the following holds,

ρ0

L

2n n+2 (Rn) < (s∗) n−2 2n(m−1) , global existence

ρ0

L

2n n+2 (Rn) > (s∗) n−2 2n(m−1) , m2(0) < ∞, finite time blow up.

where F∗ = K1(n, m)mα1(m,n) > 0, s∗ = K2(n, m)mα2(m,n) . Remarks If F(ρ0) < F∗, L

2n n+2 norm of the initial data can not be (s∗) n−2 2n(m−1) .

Thus the classification of the initial data is complete. F(ρ0) < F∗ gives a relation between the mass and the free energy, F(ρ0)M

m(n+2)−2n 2n−2−mn

< 2 − 2

n − m

(m − 1)(1 − 2

n)

2n2α(n) C(n)

• n(m−1)

2n−2−mn .

This tells that in this exponent region, total mass is not the appropriate quantity to classify global existence and blow up.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Summary n = 2, m = 1, L1 norm, and the second moment are the “right” quantity, Dolbeault, Perthame; n > 2, m = 1, Ln/2 norm gives the possibility to show existence, Perthame; while the n-th moment is the “right” way to describe blow up, Chen, Siedentop; n > 2, m = m∗, L1 norm and the second moment are the “right” quantity, Blanchet, Carrillo, Lauren¸ cot; n > 2, m = mc, Lmc norm and the second moment are the “right” quantity, Chen, Liu, Wang.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Keller-Segel model with nonlinear nonlocal reaction

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

As a warm up, we start from a linear diffusion equation with nonlinear nonlocal reaction. ut − ∆u = uα

• 1 − σ
• Rn uβ(x, t)dx
• ,

x ∈ Rn, t > 0, u(x, 0) = u0(x) ≥ 0, x ∈ Rn, where u is the density, α, β ≥ 1, σ > 0. Main result (Cauchy problem) u0 ≥ 0 and u0 ∈ Lβ(Rn) ∩ L∞(Rn), n ≥ 1. If α satisfies 1 ≤ α < 1 + (1 − 2/p)β, where p is the exponent from the Sobolev embedding theorem, i.e. p =

2n n−2, n ≥ 3, 2 < p < ∞, n = 2 and p = ∞, n = 1. then the

Cauchy problem has a bounded nonnegative solution with the following estimate β ≤ k ≤ ∞, u(t)Lk(Rn) ≤ C

• u0L∞(Rn), u0Lβ(Rn)
• .

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

REMARK For σ = 0, Fujita exponent.

α < 1 + 2/n: NO global solution for any nonnegative initial data. α > 1 + 2/n, global solution for small initial data, NO global solutions for large initial data.

For σ > 0, β = 1,

α < 1 + 2/n: global solution for any nonnegative initial data. α > 1 + 2/n, global solution for small initial data (not done, but easy), it is NOT known for large initial data.

Summary In case β = 1, for 1 < α < 1 + 2/n, σ = 0 gives the finite time blow up of the solution, while σ > 0 gives always global existence

• f the solution. In other words, switching on the nonlocal effect

before the blow up time will prevent the solution’s blow-up behavior.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Keller-Segel system with reaction (local or nonlocal) ut = ∆u − χ∇ · (uσ∇v) + f (u), x ∈ Ω, t > 0, with Neumann BC.

For σ = 1, f (u) ≤ a − bu2, a, b > 0, [Tello, Winkler] 2007. If b > n−2

n χ, there exists a global bounded solution.

For σ > 1, f (u) = µu(1 − uα), Galakhov, Salieva, Tello, 2016. If α > σ

• r α = σ, µ >

nα−2 nα+2(σ−1)χ, the solution is global and unique.

For σ = 1, f (u) = u(a0 − a1u − a2

• Ω udx/|Ω|), Tello, Negreanu, 2013.

a0 : ”Malthusian parameter” induces an exponential growth for low density (a2/|Ω|)

• Ω udx : the impact of the total mass As population

grows, the competitive effect of the local linear term becomes more

• influential. the reaction term behaves like f (u) ∼ u(a0 − a1u)

If a1 > 2χ + |a2|, then

• u −

a0 a1+a2

• L∞(Ω) → 0, t → ∞.

......

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Keller-Segel system with nonlinear nonlocal Fisher-KPP reaction ut = ∆u − ∇ · (uσ∇v) + uα −

• uβdx
• uα.

Motivation: When σ + 1 > α, global solution? The influence of nonlocal term is unclear Investigate the influence of nonlocal nonlinear exponents in the dampening term Growth term uσ+1, uα; Dampening term −uα uβdx Goal: under which conditions α, β can prevent blow-up Method: Local existence and uniqueness and a priori estimates Tool: Banach fixed point theorem and modified Moser iteration.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Global existence and uniqueness, Bian, Chen, and Latos, 2018 Theorem for global existence Let n ≥ 3, assume α > 1, β > 1, σ ≥ 1. If either σ + 1 ≤ α < 1 + 2β/n

• r

α < σ + 1 and σ + 1 − α β n + 2 2n < 1 n − α − 1 2β , then for any initial data u0 ∈ L1 ∩ L∞, the model possesses a unique global solution which is uniformly bounded.

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

A priori estimates . Step 1: basic energy estimates. d dt

• ukdx + 4(k − 1)

k ∇uk/2 2dx + k

• uk+α−1dx
• uβdx

=k

• uk+α−1dx + k(k − 1)

k + σ − 1

• uk+σdx,

⋆ Firstly using ∇uk/2 2dx, by Gagliardo-Nirenberg inequality for max n(α − 1) 2 , nσ 2 , β

• < k′ < min
• k + α − 1, k + σ
• we have

k

• uk+α−1dx ≤ k − 1

k ∇uk/2 2dx + C(k)ubα

Lk′,

k(k − 1) k + σ − 1

• uk+σdx ≤ k − 1

k ∇uk/2 2dx + C(k)ubσ

Lk′,

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

key point: ⋆ Secondly take advantage of

• uk+α−1dx
• uβdx

ubα

Lk′ ≤

• uk+α−1

Lk+α−1uβ Lβ

• bαθ

k+α−1 u

bα(1−θ−

θβ k+α−1)

Lβ

and ubσ

Lk′ ≤

• uk+α−1

Lk+α−1uβ Lβ

• bσθ

k+α−1 u

bσ(1−θ−

θβ k+α−1)

Lβ

, letting k′ = k + α − 1 + β 2 , then ubα

Lk′ ≤

• uk+α−1

Lk+α−1uβ Lβ

• bαθ

k+α−1

and ubσ

Lk′ ≤

• uk+α−1

Lk+α−1uβ Lβ

• bσθ

k+α−1 Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

bαθ k + α − 1 < 1 ⇔ 1 ≤ α < 1 + 2β/n. and bσθ k + α − 1 < 1 ⇔ σ + 1 − α β n + 2 2n < 1 n − α − 1 2β . By Young’s inequality C(k)ubα

Lk′ ≤ k

4uk+α−1

Lk+α−1uβ Lβ + C1(k),

C(k)ubσ

Lk′ ≤ k

4uk+α−1

Lk+α−1uβ Lβ + C2(k).

Then for β − (α − 1) < k < ∞ d dt

• ukdx + k

2

• uβdx
• uk+α−1dx + 2(k − 1)

k

• ∇uk/2

2

L2 ≤ C(k)

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Step 2: Uniformly boundedness of Lk norm for k ≥ β + α − 1. For k = β + α − 1, uβ+α−1

Lβ+α−1 ≤ β + α − 1

2 uβ+2(α−1)

Lβ+2(α−1) uβ Lβ +

1 2(β + α − 1). u(·, t)Lβ+α−1 ≤ C(u0Lβ+α−1). In addition, by G-N inequality and Young’s inequality

• ukdx ≤2(k − 1)

k

• ∇uk/2

2

L2 + C(n, k)uk L

β+k+α−1 2

≤2(k − 1) k

• ∇uk/2

2

L2 + k

2

• uβdx
• uk+α−1dx + C(n, k).

holds true for β + α − 1 < k < ∞. Step 3: uniformly boundedness. Moser type iteration uL∞ ≤ C(u0L∞, u0Lβ+α−1).

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

Heuristic scaling analysis: ut = ∆u − ∇ · (uσ∇v) + uα 1 −

• uβdx
• We take the growth factor, σ + 1 = α.

If u(x, t) is a solution, then uλ(x, t) = λn/βu(λx, λ2t) is also a solution if α − 1 = 2β/n, where the scaling preserves Lβ norm in space uλ(x, t)Lβ = u(x, t)Lβ. diffusion: ∆uλ = λn/β+2∆u aggregation term: λnα/β∇ · (uσ∇v) Growth: uα

λ

• uβ

λdx = λαn/βuα

1 −

• uβdx
• Balancing the indices : α − 1 = 2β/n

Li CHEN Keller-Segel Models in Chemotaxis

Keller-Segel model with linear diffusion Keller-Segel equation with nonlinear diffusion in multi-D Keller-Segel model with nonlinear nonlocal reaction

THANK YOU!

Li CHEN Keller-Segel Models in Chemotaxis