Nonlinear Drift Term: Convergence and Stability

Consider the nonlinear growth-fragmentation equation (4.8) where the transport term depends on the p-moment of the solution itself. This dependency may represent the influence of the total population of individuals on the growth process of each individual. We study the long-time asymptotic behavior of the solutions in the positive coneH with the weightθ^px^qx x^rfor

r^¥max^pr,¯ 2p 1^q. (4.35)

We prove that there is always convergence to a steady state, provided that the functionf is less thanµat the infinity. This result is precised in Theorem 4.3.1 which immediately leads to Theorem 4.1.1 and the stability of the steady states is given in Theorem 4.1.2. We use the notationMpforMp^rUs

´ x^pUpx^qdx.

Theorem 4.3.1. Assume that f is a continuous function on^r0, ^8q which satisfies Assumption (4.9), thatγ^Pp0,2^sand that the fragmentation kernelκsatisfies Assumption(4.33). Then the nontrivial steady states of Equation (4.8)write

I⁸ Mp

µ^kpU^pµ;^q with I^8PN, and for any solutionu,there exists I^8PN ^Yt0^usuch that

tlim^Ñ8

u^pt,^q I8

M_pµ^kpUpµ;^q

H

0. (4.36)

Proof of Theorem 4.3.1. First step: u0^PE.

Consider an initial distributionu0^PE defined in (4.21), then there existW0^¡0 andQ0^¥0 such that u0^px^qQ0U^pW0µ;x^q.

Letu^pt, x^qbe the solution to Equation (4.8) and define W as the solution to

$

Then Corollary (4.2.3) ensures that

t, x^¥0, u^pt, x^qQ0U^pW^pt^qµ;x^qe^{µ´0tpWpsq1qds} we obtain a system of ODEs equivalent to Equation (4.8) inE

$

and proving convergence of uis equivalent to prove convergence of ^pW, Q^q. To study System (4.38), we change the unknown toZ :W^kpQ.Then^pW, Z^qis solution to and we exhibit a Lyapunov functional for this equation. We define

G^pW^q:W1ln^pW^q

so, if we findαsuch that 1 α^2pp1^q2α^?p,we will have d

dt kµ G^pW^q α²F^pZ^q

µ^pW1^q α^?p^pµfp^pZ^qq

2

¤0.

Forp1,there are two roots to the binomial

pp1^qα²2^?pα 10 which are

α

?p1 p1

1

?p1.

For these two values, the dissipation is nonpositive but we want to have negativity outside of the steady states in order to conclude to convergence. We use a combination of this two values and the following lemma.

Lemma 4.3.2. For any α1,there existsω^¡0 such that

a, b, ^pa b^{q2 p}a αb^q2¥ω^pa² b^2q. (4.41) Proof. We prove the inequality

pa b^{q2 p}a αb^q2¥Ω^pα1^qpa² b^2q (4.42) where

Ω^px^q 42

?

4 x² x² 2 ^x₂

?

4 x^{2 x}₂².

This function is positive except atx0 where Ω^p0^q0.Its maximum is Ω^p2^q2.

To prove (4.42), we definec: ^a_b and we prove that Ω^p1α^qmin

c

pc 1^{q2 p}c α^q2 c² 1

. Define

Γα^pc^q: ^pc 1^{q2 p}c α^q2

c² 1 ,

compute

Γ¹_α^pc^{q p}1 α^qpα²1^qc^p1 α^qc²

p1 c^2q2 , and we find that the minimum of Γαis reached for

cα

1 2

α1

a

4 ^pα1^q2 . As a consequence, for alla^PRandb0,we have

pa b^{q2 p}a αb^q2¥Γα^pcα^qpa² b^2q and Γα^pcα^qΩ^p1α^q.This is still true forb0 by passing to the limit.

Denoting α ^?_p¹ ₁ and α ^?_p¹

1, we define L^pW, Z^q : 2kµ G^pW^{q p}α² α²

qF^pZ^q. Then Lemma 4.3.2 ensures the existence ofω^¡0 such that

d

dtL^pW^pt^q, Z^pt^qq µ^pW1^q α ^?p^pµf_p^pZ^qq

2

µ^pW1^q α

?p^pµf_p^pZ^qq

2

¤ω µ^2pW1^q2 α²p^pµf_p^pZ^qq2

:D^pW, Z^q, (4.43)

and D^pW, Z^q is positive outside of the steady states. Moreover Assumption (4.9) ensures that L and D are coercive in the sense that L^pW, Z^{qÑ 8}and D^pW, Z^{qÑ 8}when ^}pW, Z^{q}Ñ 8}.So Lis a Lyapunov functional for System (4.40) and we can conclude to the convergence of the solution to a steady state. Iff^p0^q¡µ, L^pW, Z^{qÑ 8}ifW orZ tends to 0,so for any^pW0^¡0, Z0^¡0^qthe solution^pW, Z^q converges to a critical point ofL, namely^p1,_µkp^I8Mp

q withI8

PN.If f^p0^q µ, then for any ¯W ^¡0 we have that L^pW, Z^{qÝpÝW,ZÝÝqÑpÝÝÝW ,0¯ÝÝÑq 8}. So either^pW, Z^qconverges to^p1,_µkp^I8Mp

qwithI^{8 P}N,or Z ^Ñ0.

To conclude to the convergence inH,we write

}u^pt,^qZ⁸U^pµ;^q}

ˆ ⁸

0

pQ^pt^qU^pW^pt^qµ;x^qZ⁸U^pµ;x^qq2px x^rqdx

and we use dominated convergence. We know by Theorem 1 in [41] that under Assumption (4.33) and for ν 1, x^αU^px^q is bounded in R for all α ^¥0, so it ensures that the integrand is dominated by an integrable function. Then the ponctual convergence is given by the convergence of ^pW, Z^q, so convergence (4.36) occurs.

Second step: general initial distributionu0.

Departing fromua solution to Equation (4.8) not necessarily inE,we define as in Section 4.2.3 v^ph^pt^q, x^q:W^pt^qku^pt, W^pt^qkx^qe^µpthptqq

with

W9 W k

f

ˆ x^pu

µW

and

h9 W.

We recall that in this case h : R Ñ R is one to one. We choose the initial valuesW^p0^q 1 and h^p0^q0 to havev^p0, x^qu^p0, x^qu0^px^q.Thenv^pt, x^qis a solution to

B

Btv^pt, x^qµ ^B

Bx xv^pt, x^q

µv^pt, x^q F_γv^pt, x^q (4.44) and, thanks to the GRE, we conclude that

v^pt, x^qÝÝÝÑ

t^Ñ8 ̺0U^pµ;x^q where

̺0

ˆ ⁸

0

φ^pµ;x^qv0^px^qdx ˆ ⁸

0

φ^pµ;x^qu0^px^qdx and so

ˆ

x^pu^pt, x^qdx

t^Ñ8̺0µ^kp

ˆ

x^pU^px^qdx

W^pt^qkpe^µphptqtqdx.

As a consequence it holds

W9 ^pt^q

t^Ñ8

W

k f_p W^kpQ

µW

withQ^pt^q̺0e^µphptqtq which satisfies the equation

Q9 µQ^pW 1^q.

The interpretation of this is that System (4.37) represents asymptotically the dynamics of the solutions to Equation (4.8). More rigorously, define

ε^pt^q:

´x^pu^pt, x^qdx

µ^kpMpQ^pt^qW^kppt^q1. (4.45)

Then we have

and, using the Cauchy-Schwarz inequality and the exponential decay theorem of [15],

|ε^pt^q| W^kpe^µpthptqq

The function _{x x}^x2pr is integrable under Assumption (4.35). Finally the dynamics of the solution u to Equation (4.8) is given by

#

t^Ñ8 0.Now we look what becomes the Lyapunov functional of the first step for this system and we obtain

d

Thanks to Assumption (4.9), we know thatfpis bounded, soW which is solution to W9 W to the spectral gap theorem of [15]

}u^pt,^qQ^pt^qUpW^pt^qµ;^q}H Q

Proof of Theorem 4.1.2. Stability of the trivial steady state.

We start with the stability of the zero steady state when f^p0^q   µ and p ^¥ 1. For this we integrate Equation (4.8) againstx^p and find

d

thanks to the mass conservation assumption (4.3). Thus´

which ensures the exponential convergence

}u^pt,^q}H^¤}u0^}He^{pfpMpru0sqµqt}.

For r ^¥ p, we have Mp^ru0^{s ¤} C^}u0^}H, so for ^}u0^} small enough, we have f^pMp^ru0^sq   µ and the exponential convergence occurs.

Whenf^p0^q¡µ,we have seen in the proof of Theorem 4.3.1 thatL^pW, Z^{qÑ 8}whenW orZ tends to zero. So the trivial steady state is unstable.

Stability of nontrivial steady states.

Let^pW8, Z8

qbe a positive steady state to System (4.40). We want to prove that Z8U^pµ;^qis locally asymptotically stable. Since Theorem 4.3.1 ensures the convergence of any solution to Equation (4.8) toward a steady state, it only remains to prove the local stability ofZ8

Upµ;^q, namely

ρ1^¡0, ^Dρ2^¡0, ^}u0Z8

U^pµ;^q} ρ2 ^ñ t^¡0, ^}u^pt,^qZ8

U^pµ;^q} ρ1. We have already seen that

}u^pt,^qZ8

So it is sufficient, to have the local stability of^pW8, Z8

q, to prove that V_η^pW8, Z8

qis stable. This is true for System (4.40) since in this caseL is a Lyapunov functional. Then, by continuity of f, there exists ε_η such that V_ηpW8, Z8 enough and the local asymptotical stability of the nontrivial steady states is proved.

Now assume thatκ 2 and prove the local exponential stability of Z8

U^pµ;^q. In this case we have (see examples in [41]) the explicit formula

Upx^qC e^βγxγ (4.52)

so the following local “entropy - entropy dissipation” inequality holds

DC^¡0, ρ^¡0, ^pW, Z^qPB^ppW⁸, Z^8q, ρ^q, L^pW, Z^q¤CD^pW, Z^q. (4.55) Fix such aρ and fix η ^¡ 0 such that V_ηpW8, Z8

q € B^ppW8, Z8, ρ^q. Consider^}u0Z8

Upµ;^q} small enough so that^|ε^pt^q|remains smaller thanε_η for all time. ThenV_ηpW8, Z8

qis stable for the dynamics of System (4.47). Now look at the termE^pW^pt^q, Z^pt^q, ε^pt^qqfor^pW, Z^qsolution to System (4.47) in this

Then, thanks to the equivalences (4.53) and (4.54) and becauseh^pt^qt when t ^Ñ8, there areb ^¡0 Using the Cauchy-Schwarz inequality as in (4.50), we find that

pZ0Z8

This inequality ensures that the terms^pi^qand^pii^qdecrease to zero exponentially fast. It only remains to prove the same result for^piii^q,and for this we use the explicit formula (4.52). We obtain

ˆ ⁸ Thanks to a Taylor expansion, we find that, locally,

|ψ_r^pW^q|¤C^pW1^q2

and it is the local exponential stability of the nontrivial steady states which satisfyf^1pI8

q 0 in the case whenκ2.

Whenf^1pI^8q¡0,the steady state ^pW⁸, Z^8qis a saddle point ofLso it is unstable.

We remark that the structure of the reduced system (4.40) is different for p   1 and p ^¡ 1. The nontrivial steady states are focuses in the case when p   1 and nodes forp ^¥ 1 (see Figure 4.1 for a numerical illustration in the case of Corollary 4.1.3).

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0

0.5 1 1.5 2 2.5

W

Z

p<1

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

W

Z

p>1

Figure 4.1: Solutions to System (4.40) are plotted in the phase plan ^pW, Z^q for two different values of parameterp.The other coefficients areγ0.1, µ1 andf^px^q2e^x.We can see that the steady state is a focus forp 1 (left) and a node forp^¡1 (right).

4.4 Nonlinear Drift and Death Terms: Stable Persistent

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