Nonlinear Drift Term: Convergence and Stability

Dans le document The DART-Europe E-theses Portal (Page 90-98)

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 weightθpxqx xrfor

r¥maxpr,¯ 2p 1q. (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 notationMpforMprUs

´ xpUpxqdx.

Theorem 4.3.1. Assume that f is a continuous function onr0, 8q which satisfies Assumption (4.9), thatγPp0,2sand that the fragmentation kernelκsatisfies Assumption(4.33). Then the nontrivial steady states of Equation (4.8)write

I8 Mp

µkpUpµ;q with I8PN, and for any solutionu,there exists I8PN Yt0usuch that

tlimÑ8

upt,q I8

MpµkpUpµ;q

H

0. (4.36)

Proof of Theorem 4.3.1. First step: u0PE.

Consider an initial distributionu0PE defined in (4.21), then there existW0¡0 andQ0¥0 such that u0pxqQ0UpW0µ;xq.

Letupt, xqbe the solution to Equation (4.8) and define W as the solution to

$

Then Corollary (4.2.3) ensures that

t, x¥0, upt, xqQ0UpWptqµ;xqeµ´0tpWpsq1qds we obtain a system of ODEs equivalent to Equation (4.8) inE

$

and proving convergence of uis equivalent to prove convergence of pW, Qq. To study System (4.38), we change the unknown toZ :WkpQ.ThenpW, Zqis solution to and we exhibit a Lyapunov functional for this equation. We define

GpWq:W1lnpWq

so, if we findαsuch that 1 α2pp1q?p,we will have d

dt kµ GpWq α2FpZq

µpW1q α?ppµfppZqq

2

¤0.

Forp1,there are two roots to the binomial

pp1qα22?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 bq2 pa αbq2¥ωpa2 b2q. (4.41) Proof. We prove the inequality

pa bq2 pa αbq2¥pα1qpa2 b2q (4.42) where

pxq 42

?

4 x2 x2 2 x2

?

4 x2 x22.

This function is positive except atx0 where Ωp0q0.Its maximum is Ωp2q2.

To prove (4.42), we definec: ab and we prove that Ωp1αqmin

c

pc 1q2 pc αq2 c2 1

. Define

Γαpcq: pc 1q2 pc αq2

c2 1 ,

compute

Γ1αpcq p1 αqpα21qcp1 αqc2

p1 c2q2 , and we find that the minimum of Γαis reached for

cα

1 2

α1

a

4 pα1q2 . As a consequence, for allaPRandb0,we have

pa bq2 pa αbq2¥Γαpcαqpa2 b2q and Γαpcαqp1αq.This is still true forb0 by passing to the limit.

Denoting α ?p1 1 and α ?p1

1, we define LpW, Zq : 2kµ GpWq pα2 α2

qFpZq. Then Lemma 4.3.2 ensures the existence ofω¡0 such that

d

dtLpWptq, Zptqq µpW1q α ?ppµfppZqq

2

µpW1q α

?ppµfppZqq

2

¤ω µ2pW1q2 α2ppµfppZqq2

:DpW, Zq, (4.43)

and DpW, Zq is positive outside of the steady states. Moreover Assumption (4.9) ensures that L and D are coercive in the sense that LpW, ZqÑ 8and DpW, ZqÑ 8when }pW, Zq}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. Iffp0q¡µ, LpW, ZqÑ 8ifW orZ tends to 0,so for anypW0¡0, Z0¡0qthe solutionpW, Zq converges to a critical point ofL, namelyp1,µkpI8Mp

q withI8

PN.If fp0q µ, then for any ¯W ¡0 we have that LpW, ZqÝpÝW,ZÝÝqÑpÝÝÝW ,0¯ÝÝÑq 8. So eitherpW, Zqconverges top1,µkpI8Mp

qwithI8 PN,or Z Ñ0.

To conclude to the convergence inH,we write

}upt,qZ8Upµ;q}

ˆ 8

0

pQptqUpWptqµ;xqZ8Upµ;xqq2px xrqdx

and we use dominated convergence. We know by Theorem 1 in [41] that under Assumption (4.33) and for ν 1, xαUpxq 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, Zq, 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 vphptq, xq:Wptqkupt, Wptqkxqeµpthptqq

with

W9 W k

f

ˆ xpu

µW

and

h9 W.

We recall that in this case h : R Ñ R is one to one. We choose the initial valuesWp0q 1 and hp0q0 to havevp0, xqup0, xqu0pxq.Thenvpt, xqis a solution to

B

Btvpt, xqµ B

Bx xvpt, xq

µvpt, xq Fγvpt, xq (4.44) and, thanks to the GRE, we conclude that

vpt, xqÝÝÝÑ

tÑ8 ̺0Upµ;xq where

̺0

ˆ 8

0

φpµ;xqv0pxqdx ˆ 8

0

φpµ;xqu0pxqdx and so

ˆ

xpupt, xqdx

tÑ8̺0µkp

ˆ

xpUpxqdx

Wptqkpeµphptqtqdx.

As a consequence it holds

W9 ptq

tÑ8

W

k fp WkpQ

µW

withQptq̺0eµphptqtq which satisfies the equation

Q9 µQpW 1q.

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

εptq:

´xpupt, xqdx

µkpMpQptqWkpptq1. (4.45)

Then we have

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

|εptq| Wkpeµpthptqq

The function x xx2pr is integrable under Assumption (4.35). Finally the dynamics of the solution u to Equation (4.8) is given by

#

tÑ8 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]

}upt,qQptqUpWptqµ;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 fp0q   µ and p ¥ 1. For this we integrate Equation (4.8) againstxp and find

d

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

which ensures the exponential convergence

}upt,q}H¤}u0}HepfpMpru0sqµqt.

For r ¥ p, we have Mpru0s ¤ C}u0}H, so for }u0} small enough, we have fpMpru0sq   µ and the exponential convergence occurs.

Whenfp0q¡µ,we have seen in the proof of Theorem 4.3.1 thatLpW, ZqÑ 8whenW orZ tends to zero. So the trivial steady state is unstable.

Stability of nontrivial steady states.

LetpW8, Z8

qbe a positive steady state to System (4.40). We want to prove that Z8Upµ;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

Upµ;q} ρ2 ñ t¡0, }upt,qZ8

Upµ;q} ρ1. We have already seen that

}upt,qZ8

So it is sufficient, to have the local stability ofpW8, Z8

q, to prove that VηpW8, 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 εη such that VηpW8, 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

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

UpxqC eβγxγ (4.52)

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

DC¡0, ρ¡0, pW, ZqPBppW8, Z8q, ρq, LpW, Zq¤CDpW, Zq. (4.55) Fix such aρ and fix η ¡ 0 such that VηpW8, Z8

q € BppW8, Z8, ρq. Consider}u0Z8

Upµ;q} small enough so that|εptq|remains smaller thanεη for all time. ThenVηpW8, Z8

qis stable for the dynamics of System (4.47). Now look at the termEpWptq, Zptq, εptqqforpW, Zqsolution to System (4.47) in this

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

pZ0Z8

This inequality ensures that the termspiqandpiiqdecrease to zero exponentially fast. It only remains to prove the same result forpiiiq,and for this we use the explicit formula (4.52). We obtain

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

|ψrpWq|¤CpW1q2

and it is the local exponential stability of the nontrivial steady states which satisfyf1pI8

q 0 in the case whenκ2.

Whenf1pI8q¡0,the steady state pW8, Z8qis 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, Zq for two different values of parameterp.The other coefficients areγ0.1, µ1 andfpxq2ex.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

Dans le document The DART-Europe E-theses Portal (Page 90-98)