Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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708 G. Str¨ohmer

and

= {z ∈ C | there is a U ∈ D (L) \{0} with LU = zU}

for the set of eigenvalues of L.

We prove that spectral stability implies dynamic stability, i.e., that the ab-

sence of eigenvalues of the formal linearization with non-negative real part implies

the non-linear stability of the equilibrium. In addition, for constant equilibria and

for the functions F and G given in (1.7) we reproduce the result of Chen, Friedman

and Hu that Conditions 1 and 2 imply stability. We work in Sobolev spaces W

with p>3. To be precise, we prove

Theorem 1.1. Assume that

∩{z ∈ C | Re (z) ≥ 0} = Ø. (1.10)

Then there is a number η>0 such that if ϕ

∈ W

and v

∈ W

and ϕ



v



≤ η, then there exists a unique solution (v, ϕ) of equations (1.9) for

t ∈ [0, ∞) such that v ∈ C



[0, ∞) ,W

(Ω)



∩ C

([0, ∞) ,L

(Ω)), ϕ ∈ C



[0, ∞) ,W

(Ω)



∩ C



[0, ∞) ,W

(Ω)



∂ϕ

∂n

=0and ϕ (0) = ϕ

, v (0) = v

.Also

ϕ (t)

+ v (t)

→ 0 as t →∞.

Condition (1.10) is implied by Conditions 1 and 2 if F and G are either indepen-

dent of x or of the form given in equations (1.7).

The system (1.2) is similar to the equations for viscous compressible ﬂuids in

that it combines a hyperbolic equation with a parabolic one. Therefore it seems

reasonable to transform the equations into Lagrange coordinates produced by the

vector ﬁeld ∇φ. Doing this, we encounter the problem that if we control k spatial

derivatives of φ, we only control k − 1 derivatives of the Lagrange coordinates

and therefore k − 2 of those of the transform of the normal n, leading to a priori

estimates of only k − 1 derivatives of φ.Thuswehavealossofregularity,which

does not occur in the original coordinates. To avoid this we will transform the

dependent variables in our problem.

2. Notation

Let Ω ⊂ R

be a domain. By L

(Ω)wedenoteforp ∈ (1, ∞)thesetofall

measurable real (or complex, the distinction will usually not cause any diﬃculty)

functions for which the pth power of their absolute value is integrable, by W

(Ω)

for integer k>0 and again p ∈ (1, ∞) the subset of L

(Ω) of all functions having

distributional derivatives up to order k belonging to L

(Ω) also. If the set is the

domain Ω mentioned above, it is usually omitted. If B is any Banach space, we

denote by C

(Ω,B)thesetofallk-times continuously diﬀerentiable functions

from Ω to B,byC



Ω,B



the subset of functions for which all these derivatives

have a continuous extension to Ω.

Stability for a Parabolic-hyperbolic System 709

3. Local existence

We will ﬁrst show that for given initial values a solution exists for a time interval

of a length going to inﬁnity as the size of the initial values goes to zero. A similar

result in H¨older spaces was already proved in [3]. There is not much genuinely new

in this section, the point is just to state precisely what we need here and give an

outline of a proof. We conﬁne ourselves to initial values such that ϕ

∈ W

∂ϕ

∂n

and v

∈ W

The operator Δ with the boundary conditions

∂ϕ

∂n

= 0 generates an analytic

semigroup on the space L

(Ω) with the domain of deﬁnition

D (Δ) =



ϕ ∈ L

(Ω) | ϕ ∈ W

(Ω) ,

∂ϕ

∂n



We will prove the existence of a solution by a ﬁxed point argument in the space



(v, ϕ) | v ∈ C



[0,T] ,W

(Ω)



∩C

([0,T] ,L

(Ω)) ,

ϕ ∈ C



[0,T] ,W

(Ω)



∩ C



[0,T] ,W

(Ω)



with the norm

(v, ϕ)

=max

0≤t≤T



v (t)

+ v

(t)

+ ϕ (t)

+ ϕ

(t)



The result is the following.

Theorem 3.1. There is a positive number C such that if

η = ϕ



+ v



and

∂ϕ

∂n

=0, then a unique solution (v, ϕ) of the system (1.9) exists on any

interval [0,T] with T ≤−C − C

−1

log (η) if this number is positive. In addition

then

(v, ϕ)

≤ Ce

η ≤ 1

and if T ≥ 1,thenalso

ϕ (T )

≤ C

(ϕ, v)

(

[T −1,T ],W

)

+ (ϕ

)

([T −1,T ]×Ω)

Proof. To reformulate our problem as a ﬁxed-point equation let us assume (v, ϕ) ∈

and (v, ϕ)

≤ 1. Then we deﬁne (v, ϕ) as the solution of the system of

equations

+ ∇v∇ϕ = −∇u

∗

∇ϕ +



G (v, ϕ, x) −(v + u

∗

)Δϕ = g,

− Δϕ =



F (v, ϕ, x)

(3.1)

with the initial values ϕ (0) = ϕ

,v(0) = v

. We can diﬀerentiate the second

equation with respect to time and obtain for f = ϕ

that

− Δf =



(v, ϕ, x) v



(v, ϕ, x) ϕ

710 G. Str¨ohmer

which implies by standard estimates for semigroups – a fairly abstract and general

version is contained in Theorem 5.2.1. in [1] – that

f (t)

≤ C



ϕ



+ v





+ Ct

1/2

(v, ϕ)

and therefore

ϕ (t)

+ ϕ

(t)

≤ C



ϕ



+ v





+ Ct

1/2

(v, ϕ)

by standard elliptic estimates. (See, among many others, [2].) Introducing charac-

teristic coordinates by

= ∇ϕ (T

(x) ,t) ,T

(x)=x,

we obtain

T



−1

≤ C (T ) (v, ϕ)

. (3.2)

Therefore also

g ◦ T



≤ C (T ) (v, ϕ)

(for the deﬁnition of g see equations (3.1)) and

v

(t)

+ v (t)

≤ C v



+ TC(T ) (v, ϕ)

with an increasing function C (T ). We therefore have

(v, ϕ)

≤ C



ϕ



+ v





+ T

1/2

C (T ) (v, ϕ)

where C (T ) now denotes a diﬀerent increasing function. If we make T = T

small that T

1/2

C (T

) ≤ 1/2 and then restrict the initial values by the inequality

ϕ



+ v



≤

= η

the mapping taking (v, ϕ)to(v, ϕ) maps the ball (v, ϕ)

≤ 1toitself.Itisalso

easy to see that at least suﬃciently high powers of the mapping are contractions

in lower-order norms, and thus it has a unique ﬁxed point. Also with C

=2C

we have

ϕ (t)

+ v (t)

≤ C



ϕ



+ v





for t ∈ [0,T

]. If the initial value was small enough we can repeat this process, and

we then obtain the existence of a solution on the interval [0,T

k]ifηC

k−1

≤ η

and the inequality

ϕ (t)

+ v (t)

≤ C



ϕ



+ v





for 0 ≤ t ≤ T

k. From this we can easily derive our claims with the exception of

the last inequality. To obtain that inequality, observe that the right-hand side of

the equation for ϕ belongs to C

1/2

([T − 1,T] ,L

) and this norm can be estimated

by the right-hand side above. Using Theorem 2.5.3, Section III in [1], we then have

that ϕ

1/2

(

[T −1/2,T ],W

)

is bounded. Then we can diﬀerentiate the equation with

respect to t and use the usual parabolic estimates to get ϕ



2,1

([T −1/3,T ]×Ω)

and

Stability for a Parabolic-hyperbolic System 711

therefore ϕ

(t)

2−2/p

for t ∈ [T − 1/4,T] is bounded. (See, e.g., Theorem 7.20

in [9].) Using standard elliptic estimates, we can then obtain our claims. 

4. Long term existence and stability

Throughout the following calculations we assume that (v, ϕ)

≤ 1. In this sec-

tion we will prove a priori estimates for small solutions of the equations (1.2) by

considering them as solutions of the inhomogeneous version of the linearized equa-

tions (1.8). They are not particularly hard to solve, but in switching to Lagrange

coordinates, as was already indicated, if we do not look at the non-linear problem

from the right angle, we have a loss of regularity. In order to avoid this problem,

we transform the equation, using the second equation in (1.9) to remove u

∗

Δϕ

from the ﬁrst. Then we have

+ ∇v∇ϕ = −∇u

∗

∇ϕ +



G (v,ϕ, x) − u

∗



−



F (v, ϕ, x)



− vΔϕ,

−Δϕ =



F (v, ϕ,x) .

(4.1)

This leads to the equations

(v + u

∗

ϕ)

+ ∇(v + u

∗

ϕ) ∇ϕ

= ∇(u

∗

ϕ) ∇ϕ −∇u

∗

∇ϕ +



G (v, ϕ,x)+u

∗



F (v, ϕ, x) − vΔϕ,

− Δϕ =



F (v, ϕ, x) ,

suggesting the deﬁnition w = v + u

∗

ϕ. With these new variables the equilibrium

in question is still located at w =0,ϕ=0, and for (w, ϕ)wehavetheequations

+ ∇w∇ϕ

= −∇u

∗

∇ϕ + u

∗

|∇ϕ|

+ ϕ∇u

∗

∇ϕ



G (w −u

∗

ϕ, ϕ, x)+u

∗



F (w − u

∗

ϕ, ϕ, x) − (w − u

∗

ϕ)Δϕ,

− Δϕ =



F (w − u

∗

ϕ, ϕ, x) .

(4.2)

Letting

G (P, w, ϕ, x)=u

∗

|P |

+ ϕ∇u

∗

P +



G (w −u

∗

ϕ, ϕ, x)+u

∗



F (w − u

∗

ϕ, ϕ, x)

and

F (w, ϕ, x)=



F (w − u

∗

ϕ, ϕ, x)

we obtain

+ ∇w∇ϕ = −∇u

∗

∇ϕ + G (∇ϕ, w, ϕ, x) − (w − u

∗

ϕ)Δϕ,

+ |∇ϕ|

−Δϕ = F (w, ϕ, x)+|∇ϕ|

(4.3)

Let us deﬁne

A (x)=G

(0, 0, 0,x) ,B(x)=G

(0, 0, 0,x) ,

C (x)=F

(0, 0,x) ,D(x)=F

(0, 0,x)

712 G. Str¨ohmer

and

= u

∗

|∇ϕ|

+ ϕ∇u

∗

∇ϕ − (w − u

∗

ϕ)Δϕ + G (0,w,ϕ,x) − A (x) w − B (x) ϕ,

= F (w, ϕ, x) − C (x) w − D (x) ϕ + |∇ϕ|

Then

R



≤ C



w

+ ϕ

+ Δϕ



R



≤ C



ϕ

+ w



and

+ ∇w∇ϕ = −∇u

∗

∇ϕ + A (x) w + B (x) ϕ + R

+ ∇ϕ∇ϕ −Δϕ = C (x) w + D (x) ϕ + R

(4.4)

Now we introduce Lagrange coordinates by



(y)=∇ϕ (T

(y) ,t) ,T

(y)=y.

Then let w (y, t)=w (T

(y) ,t)and

∇

(f ◦T

)=(∇f ) ◦ T

(f ◦T

)=Δf ◦ T

We refrain from stating the exact form of these diﬀerential operators, although

they will be needed later in the proof, but we leave these calculations to the

reader. Then, in these new coordinates, equation (1.9) is equivalent to

w

= −∇

u

∗

∇

ϕ +



A w +



B ϕ +



ϕ

−L

ϕ =



C (x) w +



D (x) ϕ +



(4.5)

This is why we consider the evolution equation

w

= −∇

u

∗

∇

ϕ +



A w +



B ϕ,

ϕ

−L

ϕ =



D ϕ +



C w

(4.6)

with the boundary values

n ·∇

ϕ =0.

We will ﬁnd out that this is now accessible to arguments considering this as a

perturbation of the equation

w

= −∇u

∗

∇ϕ + A w + B ϕ,

ϕ

− Δ ϕ = D ϕ + C w

(4.7)

with boundary condition

∂ ϕ

∂n

=0.Wenowomitthein the notation. We want to

show we can use the theory of analytic semigroups by studying the operator

−∇u

∗

∇ϕ + Aw + Bϕ

Δϕ + Dϕ + Cw

Then A : D (A) → B with

B =



U =

| w ∈ W

,ϕ∈ L



Stability for a Parabolic-hyperbolic System 713

and

D (A)=

.

∈ B | ϕ ∈ W

∂ϕ

∂n



Then equation (4.7) becomes U

= AU and we have the following result.

Lemma 4.1. Assume that no z with Re (z) ≥ 0 is an eigenvalue of A. Then this

half-plane belongs to the resolvent set of A and there exists a number C such that

(zI − A)

−1

L(B)

≤ C (1 + |z|)

−1

for z with Re (z) ≥ 0. There also is an ε>0 such that

exp (tA)

L(B)

≤ Ce

−εt

Proof. First we prove the resolvent estimate. If

zU − AU = F =

then

(z − A) w = −∇u

∗

∇ϕ + Bϕ + F

zϕ − Δϕ = Dϕ + Cw + F

(4.8)

We have

A =

∂G

∂w

(0, 0, 0,x)=

∂



∂v

(0, 0,x)+u

∗

∂



∂v

(0, 0,x)

∂G

∂u

∗

,φ

∗

,x)+u

∗

∂F

∂u

∗

,φ

∗

,x)=a + cu

∗

< 0,

thus

A = a + cu

∗

(4.9)

and max

x∈Ω

A (x) < 0. From the ﬁrst equation in (4.8) for Re (z) ≥ 0

w =(z −A)

−1

(−∇u

∗

∇ϕ + Bϕ + F

) (4.10)

and

zϕ − Δϕ = Dϕ + C (−∇u

∗

∇ϕ + Bϕ + F

)(z − A)

−1

+ F

From standard elliptic estimates we then obtain

ϕ

+ |z|ϕ

≤ C



ϕ

+ F 



From equation (4.10) then

w

(1 + |z|) ≤ C



F 

+ ϕ



This proves

(zI − A)

−1

L(B)

≤ C |z|

−1

714 G. Str¨ohmer

for large z. By standard compactness arguments one also has that if the only

solution of AU = zU is zero for all z with Re (z) ≥ 0, then the resolvent set

contains all such z and

(zI − A)

−1

L(B)

≤ C (1 + |z|)

−1

for such z. The existence of solutions for very large z is obvious in view of the

above considerations, and then it carries over to smaller z owingtotheestimates

we proved. The remainder is a consequence of, e.g., Theorem 2.1, Part II in [5]. 

Lemma 4.2. For U =

ϕv+ u

∗

the equation AU = zU is equivalent to

zv = −∇u

∗

∇ϕ + av + bϕ − u

∗

Δϕ

and

zϕ =Δϕ + dϕ + cv.

Proof. Using equations (4.8) and (4.9) and remembering the deﬁnition of A, B, C,

and D in equations (4.4), this is an easy calculation. 

Now we want to obtain the information we need about the long-term behavior

by considering the problem as a perturbation of this linearization. To do this we

also need to consider the equation

= −∇u

∗

∇ϕ + Aw + Bϕ + f

− Δϕ = Dϕ + Cw + f

∂ϕ

∂n

= g on ∂Ω.

(4.11)

It is easy to show the following result.

Lemma 4.3. Assume f

belongs to L



(0,T) ,W



, f

to L

((0,T) ×Ω) and g to

1−1/p,1/2−1/2p

((0,T) × ∂Ω) as well as g (0) = 0. Then there is exactly one pair

of functions ϕ ∈ W

2,1

,w ∈ C



[0,T] ,W



with w

∈ L

((0,T) × Ω) fulﬁlling

equation (4.11) and ϕ (0) = 0, w (0) = 0 as well as the estimate

ϕ

2,1

+ w

(

[0,T ],W

)

+ w



((0,T )×Ω)

≤ C (T )



f



((0,T )×Ω)

+ g

1−1/p,1/2−1/2p

((0,T )×∂Ω)

+ f



(

(0,T ),W

)



Proof. For suﬃciently short time intervals we can treat this system as a pertur-

bation of the single equations w

= f

and ϕ

− Δϕ = f

∂ϕ

∂n

= g.Thenwecan

continue step by step, also using the information about the operator A we already

derived. There we can use Theorem 5.4 in [8]. A proof can also be done in a similar

way to that of Theorem 13 in [10]. 

Now the solution ϕ, v constructed in Theorem 3.1 has the property that

(v, ϕ)

≤ C (T )



v (0)

+ ϕ (0)



We also have

T

−id



−1

−id

≤ C (T )



v (0)

+ ϕ (0)



Stability for a Parabolic-hyperbolic System 715

Thus



(t)



(t)

≤ C (T )



v (0)

+ ϕ (0)



it is also easy to see that the remaining error terms one has to incorporate into f

and f

to make w and ϕ solutions of (4.11) can be estimated in the same way. If

we write w and ϕ in the form w = w

+ w

and ϕ = ϕ

+ ϕ

where (w

,ϕ

)solve

equation (4.11) with f

= f

=0and(w

,ϕ

)(0) = (w

,ϕ

), while (w

,ϕ

)solve

equation (4.11) with the given f

and f

and (w

,ϕ

) (0) = 0, then combining

Lemmas 4.1, 4.2 and 4.3 we have

ϕ

2,1

((T −1,T )×Ω)

+ w

(

[T −1,T ],W

)

+ w



((T −1,,T )×Ω)

≤ C (T )



v



+ ϕ





+ Ce

−εT



ϕ



+ v





Then we can use the last inequality in Theorem 3.1, which one can easily see is

still true if we replace v by w, to obtain

ϕ(T )

+v(T )

≤C (T )



ϕ



+v





+Ce

−εT



ϕ



+v





For T so large that Ce

−εT

< 1/2 and initial values small enough to assure the

existence of a solution on [0,T]wethenhave

ϕ (T )

+ v (T )

≤



ϕ



+ v





+ C



ϕ



+ v





For suﬃciently small initial values then

ϕ (T )

+ v (T )

≤



ϕ



+ v





This allows us to continue the solution indeﬁnitely and implies that it converges

to zero as claimed. This concludes the proof of the ﬁrst part of Theorem 1.1.

5. Excluding eigenvalues

For Re (z) ≥ 0wehavetoconcludev =0,ϕ=0from

zϕ − Δϕ

−∇u

∗

∇ϕ + av + bϕ −u

∗

Δϕ

cv + dϕ

and

∂ϕ

∂n

= 0, where we have suppressed the variable x throughout. Solving the

lower equation for Δϕ we obtain

Δϕ =(z −d) ϕ − cv.

We can use this to replace Δϕ in the upper equation, which results in

zv = −∇u

∗

∇ϕ + av + bϕ −u

∗

((z − d) ϕ − cv) .

We assumed −a −u

∗

c>0, thus −a −u

∗

c + z =0, and we can solve for v to obtain

v =

b + u

∗

d −u

∗

z − a − u

∗

ϕ −

∇u

∗

∇ϕ

z − a − u

∗

716 G. Str¨ohmer

Replacing v in the second equation gives us

Δϕ =

ad − bc −z (a + d)+z

z − a − u

∗

ϕ +

c∇u

∗

z − a − u

∗

∇ϕ.

To duplicate the conditions of Chen, Friedman and Hu, we assume F and G do

not explicitly depend on x and neither does u

∗

, and D = ad−bc > 0,τ = a+d<0

and Γ = a + u

∗

c<0. If ϕ = 0 it has to be an eigenfunction of −ΔwithNeumann

boundary conditions for an eigenvalue λ

≥ 0, thus

D −τz + z

z − Γ

= −λ

and

−λ

Γ+D +(λ

− τ) z + z

=0.

This is an equation which cannot have any roots with a non-negative real part, as

−λ

Γ+D>0andλ

−τ>0.

For variable equilibria, considering the case with a variable δ we have

Γ=−δ, D = μδ, τ = −

and therefore

Δϕ =

μδ + z

δ + z

ϕ −

∇δ

δ + z

∇ϕ

and

div ((δ + z) ∇ϕ)=(δ + z)Δϕ + ∇δ∇ϕ =



μδ + z

δ + z



ϕ.

If we multiply the equation by ϕ and integrate over Ω, we obtain after an integra-

tion by parts that

−



(δ + z) |∇ϕ|

dx =





μδ + z

δ + z



|ϕ|

dx.

Therefore



μδ |ϕ|

+ δ |∇ϕ|

dx + z



δ |ϕ|

+ |∇ϕ|

dx + z



|ϕ|

dx =0.

If ϕ =0, we have a quadratic equation with only positive coeﬃcients which again

cannot have a root in the half-plane described by Re (z) ≥ 0.

References

[1] Amann, H.: Linear and quasilinear parabolic problems, Volume I: Abstract linear

theory, Birkh¨auser, Basel, 1995.

[2] Browder, F.E.: Estimates and existence theorems for elliptic boundary value prob-

lems. Proc. Nat. Acad. Sci. U.S.A. 45 (1959) 365–372.

[3] Chen, X.; Friedman, A.; Hu, B.: A Parabolic-Hyperbolic Quasilinear System, Comm.

in PDE 33, 969–987.

Stability for a Parabolic-hyperbolic System 717

[4] Dalibard, A.L.; Perthame, B.: Existence of solutions of the hyperbolic Keller-Segel

model, Trans. Amer. Math. Soc. 361 (2009) 2319–2335.

[5] Friedman, A.: Partial diﬀerential equations, Holt, Rinehart, Winston, New York,

1969.

[6] Horstmann, D.: From 1970 until present: The Keller-Segel model in chemotaxis and

its consequences, Jahresber. Deutsch. Math.-Verein. 105 (2003) 103–165.

[7] Horstmann, D.: From 1970 until present: the Keller-Segel model in chemotaxis and

its consequences. II. Jahresber. Deutsch. Math.-Verein. 106 (2004) 51–69.

[8] Ladyzhenskaya, O.A.: Boundary value problems in mathematical physics III, Proc.

Steklov Inst. Math. 83 (1965).

[9] Lieberman, G.: Second order parabolic diﬀerential equations, World Scientiﬁc, River

Edge, N.J., 1996.

[10] Str¨ohmer, G.: Asymptotic estimates for a perturbation of the linearization of an

equation for compressible ﬂuid ﬂow, Studia math. 185 (2008) 99–125.

Gerhard Str¨ohmer

Department of Mathematics

University of Iowa

Iowa City

IA 52242-1419, USA

e-mail: gerhard-strohmer@uiowa.edu