Blow-up by the Concavity Method

whereΦ(s) =e^s−s−1, keeping in mind thatR

Ωv(t)dx= 0. Here and in what follows,L^Φ(Ω) denotes the Orlicz space which corresponds to the Young functionΦ(s), andk · k_LΦ(Ω)as its norm. Let's denote the Young function complementary to Φ by Ψ, so thatL^Ψ(Ω) denotes the Orlicz space, withk·k_LΨ(Ω)is its norm. It is also known thatΨ(s) = (s+1) log(s+1)−s.

Using Hölder's inequality for Orlicz spaces [2, 78], we see that F(u(t), v(t))≥ −|Ω|

e − Z

Ω

(u(t)−1)v(t)dx+ d2

2γk∇v(t)k²_L2(Ω)

≥ −|Ω|

e − kv(t)k_LΦ(Ω)ku(t)−1k_LΨ(Ω)+ d2

2γk∇v(t)k²_L2(Ω)

≥ −|Ω|

e − K˜

4εku(t)−1k_LΨ(Ω)+ d₂

2γ −ε

k∇v(t)k²_L2(Ω), whereε < _2γ^d2. Combining this with Lemma 6.3 of [26] and the fact that

Z

Ω

u(t)dx=|Ω| ∀t≥0, we get the result.

Remark 5.1. We remark that, except statement 6., all the statements of Theorem 5.14 and Lemma 5.15 are also true for blow-up solutions of (5.3) when Ω ⊂R² has a piecewise C² boundary.

Then there exists a time t^∗ after whichF blows up, and the inequality F F⁰⁰−(ν+ 1)(F⁰)² ≥0

holds.

Proof. Firstly, we observe that (5.50) means that F^−ν is concave with an initial positive derivative, and thus, the inequality

F^−ν(t)≤F^−ν(0) + (F^−ν)⁰(0)t ⇔F^−ν(t)≤F^−ν(0)−νF^−ν−1(0)F⁰(0)t, is true. From this, we get that

t≤t^∗= F^−ν(0)

νF^−ν−1(0)F⁰(0) = F(0)

νF⁰(0). (5.51)

So,t^∗ is an upper bound for the blow-up time of the functionF. Secondly, we note that

(F^−ν)⁰ = −νF^−ν−1F⁰

(F^−ν)⁰⁰ = −ν(−ν−1)F^−ν−2(F⁰)²−νF^−ν−1F⁰⁰

= −νF^−ν−2[F F⁰⁰−(ν+ 1)(F⁰)²].

From the conditions in (5.50), we obtain the inequality F F⁰⁰−(ν+ 1)(F⁰)²≥0, and the proof of the Lemma is complete.

We therefore have the following theorem regarding the system (3.1).

Theorem 5.17. Consider the problem (3.1). Assume thatv∈L^∞(Ω), v0∈W^1,∞(Ω), u0 ∈ L²(Ω), and set

F(t) =F = Z t

0

Z

Ω

u²(x, s)dx ds+ω, (5.52) for some constant ω >0. Then Lemma 5.16 holds true.

Proof. We start by dierentiatingF with respect tot to get F⁰ =

Z

Ω

u² dx= Z t

0

d dt

Z

Ω

u² dx

ds+ Z

Ω

u²₀ dx

= 2 Z t

0

Z

Ω

uu_tdx ds+ Z

Ω

u²₀ dx

= 2 Z t

0

Z

Ω

uutdx ds+C0,

(5.53)

whereC0=R

Ωu²₀ dx. Dierentiating in (5.53) with respect to tgives F⁰⁰= 2

Z

Ω

uutdx

= 2 Z t

0

d dt

Z

Ω

uutdx

ds+ 2 Z

Ω

uutdx 0

= 2 Z t

0

d dt

Z

Ω

uutdx

ds+C1,

(5.54)

where

C₁= 2 Z

Ω

uu_tdx 0

= 2 Z

Ω

u(d₁∆u− ∇ ·(uχ∇v))dx 0

= 2

χ Z

Ω

u₀∇u₀∇v₀ dx−d₁ Z

Ω

|∇u₀|²dx

= 2

χ Z

Ω

u0∇u₀∇v₀ dx−d1

Z

Ω

|∇u₀|²dx

.

(5.55)

Note that (5.54) can be rewritten as F⁰⁰=C

Z t 0

Z

Ω

u²_t dx ds+

2 Z t

0

d dt

Z

Ω

uutdx

ds−C Z t

0

Z

Ω

u²_t dx ds

+C1. Squaring both sides of (5.53) and using Young's inequality, we write that

(F⁰)² =

2 Z t

0

Z

Ω

uutdx ds+C0

2

= 4 Z t

0

Z

Ω

uu_tdx ds 2

+ 4C₀ Z _t

0

Z

Ω

uu_tdx ds+C₀²

≤ (4 +ε) Z t

0

Z

Ω

uu_tdx ds 2

+C_εC₀²,

for some constantC_ε. It then follows, by applying the Cauchy-Schwarz inequality, that (F⁰)² ≤ (4 +ε)

Z _t

0

Z

Ω

u²_t dx ds· Z _t

0

Z

Ω

u²dx ds+C_εC₀²

≤ (4 +ε)F Z t

0

Z

Ω

u²_t dx ds+CεC₀².

Thence, F F⁰⁰− C

4 +ε(F⁰)² ≥F

2 Z t

0

d dt

Z

Ω

uu_tdx

ds−C Z t

0

Z

Ω

u²_t dx ds+C₁

− C

4 +εC_εC₀². The expression between the square brackets above can be rewritten as follows;

2 Z t

0

d dt

Z

Ω

uu_tdx

ds−C Z t

0

Z

Ω

u_tu_tdx ds

= 2 Z t

0

d dt

Z

Ω

u(d1∆u− ∇ ·(uχ∇v))dx

ds−C Z t

0

Z

Ω

ut(d1∆u− ∇ ·(uχ∇v))dx ds

= 2d₁ Z t

0

(−k∇uk²₂)⁰ds+ 2 Z t

0

d dt

χ

Z

Ω

u∇u∇v dx

ds

−Cχ Z t

0

Z

Ω

u∇u_t∇v dx ds+Cd1

2 Z t

0

(k∇uk²₂)⁰ds

= 2d₁ Z t

0

(−k∇uk²₂)⁰ds+ 2χ Z t

0

Z

Ω

(u_t∇u∇v+u∇u_t∇v+u∇u∇v_t)dx ds

−Cχ Z _t

0

Z

Ω

u∇u_t∇v dx ds+Cd1

2 Z _t

0

(k∇uk²₂)⁰ds.

If we choose C= 2, then we get 2

Z t 0

d dt

Z

Ω

uu_tdx

ds−C Z t

0

Z

Ω

u²_t dx ds

=d1

h

k∇u₀k²_L2(Ω)− k∇uk²_L2(Ω)

i + 2χ

Z t 0

Z

Ω

(ut∇u∇v+u∇u∇v_t)dx ds.

SinceF ≥ω, we get F F⁰⁰− 2

4 +ε(F⁰)² ≥F

d1

h

k∇u₀k²_L2(Ω)− k∇uk²_L2(Ω)

i + 2χ

Z t 0

Z

Ω

(ut∇u∇v+u∇u∇v_t)dx ds +2k∇v₀k_L∞(Ω)ku₀k_L2(Ω)k∇u₀k_L2(Ω)−2d1k∇u₀k²_L2(Ω)−2Cεku₀k⁴_L2(Ω)

(4 +ε)ω )

≥F d₁n

k∇u₀k²_L2(Ω)− k∇uk²_L2(Ω)

o+ 2k∇v₀k_L∞(Ω)ku₀k_L2(Ω)k∇u₀k_L2(Ω)

−2d₁k∇u₀k²_L2(Ω)−2C_εku₀k⁴_L2(Ω) (4 +ε)ω

≥0,

and the proof of the Theorem is complete.

Attraction-Repulsion KS Equations in Scale of Banach Spaces

6.1 Introduction

In this chapter, we study the well-posedness and asymptotic global dynamics of the attraction- repulsion Keller-Segel system of equations admitting the following abstract formulation:











U_t+A_pU =P(u)U,

U(0) =U0 ∈E^α_q ×Er^β ×Er^β,0≤α−β <1, q, r≥1,

(6.1)

modelling aggregation of microglia in Alzheimer's disease, whereU = (u, v, w)^>have entries holding meanings as in (4.2). For notational convenience, we will set v =ψ₂, w= ψ₃. In (6.1), we have that

M_3×3(L^p(Ω,R³))3 A_p = diag[−∆,−∆ +λ2,−∆ +λ3]

:D(A_p)⊂L^p(Ω,R³)→L^p(Ω,R³) (6.2) with domain

D(A_p) :=

ϕ∈H^2,p(Ω,R³) :∂_~nϕ= 0 on Γ , (6.3) 144

considered for real valued vector functions dened on an open bounded subset Ω ⊂ R^N possessing a smooth boundaryΓ =∂Ω,~n denotes the unit outward pointing normal vector to Γ,

P(u)U =



−

3

X

j=2

Div(u(−1)^jχj∇ψ_j), a2u, a3u





>

, (6.4)

and the biophysical constants are as dened in (4.5), withd_i = 1, i= 1,2,3.

More precisely, the evolutionary equation (6.1) reads the following chemotaxis system of equations























ut = ∆u−P3

j=2Div(u(−1)^jχj∇ψ_j), vt = ∆v−λ2v2+a2u,

w_t = ∆w−λ₃w+a₃u, inΩ×I,˙ 0 = ∂_~nu=∂_~nψ_j on Γ×I˙,

u(0) = u₀, ψ_j(0) =ψ₀ inΩ,

(6.5)

whereI˙= (0, T),I = [0, T), and in simplication we have written

−

3

X

j=2

Div(u(−1)^jχj∇ψ_j) =−∇ ·(u(χ2∇v−χ3∇w)) =:P(u)ψ. (6.6) Recall that (6.6) can be viewed in the sense of distributions as the weak form

PΩ(u)ψ:=hP(u)ψ, ϕi_p⁰_,p=

3

X

j=2

Z

Ω

u(−1)^jχj∇ψ_j∇ϕ (6.7) in adequate function spaces.

It is clear to see that the system of equations (6.5) has L¹−spatial integrable solutions in taking the L^p− dual product with as test function ϕ= (1,1,1)^> in distributions sense.

More concretely, d dt

Z

Ω

u= 0⇒uΩ(t) = Z

Ω

u0(x), ∀t∈I,˙ d

dt Z

Ω

ψ=−λ Z

Ω

ψ+a Z

Ω

u⇒ψΩ(t) =e^−λtψ0+a|Ω|¯u0

λ (1−e^−λt), ∀t∈I,˙

whereu_Ω =R

Ωu=|Ω|u.Thus, if T =∞ we obtain M=

(

A∈R³:A=

|Ω|¯u₀,a₂|Ω|u¯₀

λ₂ ,a₃|Ω|u¯₀ λ₃

>)

, (6.8)

as the time limit set of L¹−spatially integrable solutions.

On the other hand, the stationary equations to (6.5), using La-Salle- Hale-Henry [30]

invariance principle, can be deduced associated to the system of equations (6.5) in following the works of [36] by means of the Lyapunov function

J(u, ψ) = Z

Ω

ulnu−κ Z

Ω

uψ+ κ ar

Z

Ω

(|∇ψ|^r+λ|ψ|^r), (6.9) where ψ = (ψ₂, ψ₃), λ = (λ₂, λ₃), a = (a₂, a₃), κ = sgnP3

j=2(−1)^jχ_j > 0 to which holding onto, if the system of equations is globally well-posedness in time, then it implies studying of the non-local elliptic problem







∆ψ−λψ+µe^κψ= 0 inΩ,

∂ψ

∂~n = 0 onΓ =∂Ω, (6.10)

where

µ=a

R

Ωu R

Ωe^{P3j=2(−1)jχjψ}

=a R

Ωu₀ R

Ωe^{P3j=2(−1)jχjψ} ,

using the implied conclusion in (6.8). Henceforth, in alternatives we can distinguish the following possible situations:

e^{P3j=2(−1)jχj}











1 if χ₂ χ₃,

≥ 1 ifχ2≥χ3, 1 ifχ2χ3

(6.11)

corresponding respectively to that:

Repulsion coecient dominating strongly the attraction coecient.

Attraction coecient dominating mildly the repulsion coecient. (6.12) Attraction coecient dominating signicantly the repulsion coecient.

We point-out here that, as technical basis for our analysis, we use abstract dynamical systems theory for evolutionary equations [30, 68, 66], where the approach dictates that, to solve the equations (6.5) and to understand their qualitative properties one has to seeks for solutions satisfying the integral equations

F(u, u₀)(t) :=e^∆tu₀+ Z t

0

e^∆(t−s)P(u)ψ(s)ds,

ψ(t, ψ0) :=e^(∆−λ)tψ0+a Z t

0

e^{(∆−λ)(t−s)}u(s)ds, ψ=ψj, j = 2,3,

(6.13)

and vice-versa. Note that if (6.13) is to be properly dened, then the non homogeneous terms of the equations (6.5) need to be such that they are mapped into the spaces of the initial data.

This chapter is organized as follows. In Section 6.2, we give some preliminaries on the function spaces, and E_q^α −Ep^β heat kernel estimates of the semigroup associated to the operator (6.2), which might not have been covered in Chapter 1.

Section 6.3, is devoted to the well-posedness of the system of equations (6.5) inL^σ( ˙I;L^p(Ω)) takingα, β = 0, i.e. u₀ ∈L^q(Ω), v₀, w₀ ∈L^r(Ω). In order, to gain control over the coupled

term in (6.6) of the cell density equation in (6.5) we introduce the Banach space Zq⁰ :=

n

∇z∈L^p(Ω);−Div(a(x)∇z)∈L^q0(Ω), a(x)∈L^Θ(Ω)⊃H^1,q(Ω) xed, 1

q = 1 Θ+1

p

⊂H^−1,q0(Ω),

(6.14)

endowed with the norm

ka(x)∇zk_Z

q0 =ka(x)∇zk_q+kDiv(a(x)∇z)k_q⁰ ∼=kDiv(a(x)∇z)k_q⁰ <∞.

Then we prove the following lemma.

Lemma 6.1. Assume in (6.6) that uχ∇ψ∈ Z_q⁰, q⁰ ≥ _N−2^2N . Then, PΩ(u)ψ∈ H^−1,q0(Ω) is well dened, and

kP_Ωk_L(H1,q(Ω),H^−1,q0(Ω)) = sup

k∇ϕk_q0≤1

|hP(u)ψ, ϕi_q,q⁰|

kuχ∇ψk_q ≤2(N eπ)⁻¹<1. (6.15) Moreover,

p≥q ≥ p

2 ⇐⇒ N ≥q≥ N

2, (6.16)

is a valid Sobolev spaces embedding relation, with q⁰ ≥p ifN <4 and p > q⁰ ≥q if N ≥4. Important to take note of is that, (6.16) imply studying the cells density equation up to the critical space H^1,N(Ω), and the reduction of the system of equations in the large time asymptotic dynamics to the non-local elliptic problem (6.10), to which the Moser-Trudinger inequality imply well-posedness only if

κ≤N ω

1 N−1

N−1, where ωN−1 = 2π^N2

Γ(N/2), (6.17)

denote the measure of the unit sphere in R^N, N ≥ 2. In the context of Lemma 6.1, we

obtain that the system of equations (6.5) admits a unique solution of at least class

X_q,r^p (I) := C(I;L^q(Ω))∩ L^∞_N

2

1 q−¹

p

(I;L^p(Ω))∩L^σ(I;L^p(Ω))×

×[C(I;L^r(Ω))∩C(I;H^1,p(Ω))∩C¹(I;L^p(Ω))]²

= V ×Z×Z. (6.18)

More precisely, we have the following theorem.

Theorem 6.2. Consider the system of equations (6.5) with u0 ∈ L^q(Ω),ψ0 ∈ L^r(Ω), and assume that Lemma 6.1 holds. Let u∈L^σ( ˙I;L^Θ(Ω)), forr, p≥Θ, be such that

1 σ + N

2Θ ≤min

1 +N 2r,1

2+ N 2p

. (6.19)

Then,

(i) ψ∈C(I;L^r(Ω))∩C( ˙I;H^1,p(Ω)). (ii) If (i) holds, then F(u, u₀)∈ L^∞_N

2Θ

( ˙I;L^p(Ω)) satises that the mapping L^q(Ω)×L^σ( ˙I;L^Θ(Ω))3(u₀, u)→ F(u, u₀)∈ L^∞N

2Θ

( ˙I;L^p(Ω)) (6.20) is linear and continuous. Furthermore, F(u, u₀) is locally Hölder continuous with values inL^p(Ω).

(iii) The system of equations (6.5) admits a unique solution in the class (6.18). That is U ∈Xq,r^p (I).

A priori uniform boundedness inΩ×I˙of the cells density solution is proven in Subsection 6.3.1, yielding, as a result, that the complete system solution is a global classical solution.

Independent to this conclusion, we obtain the following proposition.

Proposition 6.3. Consider the system of equations (6.5) in the context of Theorem 6.2.

Then,

lim sup

t%+∞

ku(t)k_p = 0, lim sup

t%+∞

k∇ψ(t)k_p= 0, (6.21) and the system of equations dene an extended or perturbed analytic semigroup in L^p− spaces. Moreover, in the global asymptotic dynamics ,it holds that

lim sup

t%∞

k(u(t), v(t), w(t))^>k_p =A^∗ ∈ M ∪ {0}, (6.22) where the limit setM, as dened in (6.8), corresponds to theL¹−spatial integrable solutions of the system equations in distributions sense.

In Section 6.4, we prove similar results to those of Section 6.3, but in a much more general function space setting, which includes one used in Chapter 4 of the scale of Hilbert spaces. More precisely, we give a treatment of the equations in Bessel potential spaces E_q^α, α∈R,1< q <∞. To this end, we rst establish the following nesting relation between the spaces;

E_p^α7−→E_q^α 7−→E_p^β 7−→E_q^β 7−→E^β_r, (6.23) and

E_q^α0 ←−^−→ E_p^α 7−→E_q^α7−→E_p^β0 7−→E_q^β0

←−−→ E_p^β. (6.24)

Then, prove the following counterpart to Lemma 6.1.

Lemma 6.4. Assume (6.23)-(6.24) hold, and let u ∈ E_q^α, ψ ∈ Ep^β, β ≥ ¹₂, 0 ≤ α−β <

1, p≥q. Then, if

1 2 +N

2p ≤α+β, and 1 + N

2p ≤2α+β, (6.25)

hold, then the product uχ∇ψ ∈ E_q^α, and the weak form P_Ω(u) ∈ E_q⁰0 ⊆ E_p^−β0 , for q⁰ ≥

2N

N−4α, p≥ _2α^N, are well dened, and kP_Ωk_L(Eα

q,E^−β

q0 ):= sup

kϕk_α,q0≤1

|hP(u)ψ, ϕi

E_q^α,E^−β_q0 | χku∇ψk_α,q ≤

2 N eπ

α+^β₂−¹₂

. (6.26)

In particular,PΩ∈ L_lip(E_q^α, E_q^β0) is satised.

To conclude, we give similar results to those in Theorem 6.2-Proposition 6.3 in the following theorem.

Theorem 6.5. Assume in the system of equations (6.1)-(6.4) that Lemma 6.4 holds. Then, (i) The system of equations admits a uniqueC¹−strong solution. Furthermore, there exists

a constant

ω = min





 1−

2 N eπ

α+β−¹₂ 3

X

j=2

χ_j+a_j q

,

1− 2

N eπ

α+^β₂−¹₂ 3

X

j=2

χ_j+a_j r







>0 (6.27) such that the coupled system dierential operator in (6.5) denes a perturbed analytic semigroup in Z^α+β =E^α_q ×Er^β ×Er^β spaces, and

lim sup

t%∞

k(u(t), v(t), w(t))^>k_α+β =A^∗∈ M ∪ {0} (6.28) where the limit set M is dened as in (6.8) corresponds to L¹− spatial integrable solutions of the system (6.5) of equations in distributions sense.

(ii) Assume that the rst condition in (6.25) is veried strictly. Then, the solution semi- group is a classical solution.

To make the proof of the results in above theorems accessible to the reader we give, in the next section, some preliminaries.