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

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378 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

Proof. Using (3.14) with ρ, ˜ρ in (3.11), (3.12) suﬃciently small so that



ρ +2˜ρ ≤ 1/2

we get 1/2 ≤ α(t, x) ≤ 3/2 , hence

E(t) ∼E

(t)+|u

, E(t) ≤ 2max{C

}˜ρ.

Moreover, with

1/2

≥

, (3.21)

and using the fact that in the proof of (3.14), t = 0 can obviously be replaced by

t = s, we get (see the proof of Theorem 3.3 in [16])

E(T )+



E(τ) dτ ≤ C

E(s)+2C

max{C

−ω

(3.22)

for any s<T.

We consider

E(τ)=E(τ)+λe

−ω

with λ =2C

max{C

}/(C

−

and assume w.l.o.g. that C

≥ 1andω

so that λ>0. (If ω

≤

< 1, we can replace C

in (3.22) by

> max{

, 1}≥C

Therewith, we get



E(τ) dτ ≤ C

E(s) so that we can apply a standard

semigroup argument (see, [31] or, e.g., Theorem 8.1 in [20]) to obtain

E(t) ≤

E(0)e

−

and therewith (3.20). 

3.2. The Westervelt equation with nonhomogeneous Dirichlet boundary data

We return to the Westervelt equation with zero source term

(1 − 2ku)u

−c

Δu − bΔu

=2k(u

)

, (3.23)

but nonhomogeneous Dirichlet boundary conditions

u = g on (0,T) × ∂Ω (3.24)

and initial conditions

u(t =0) = u

(t =0) = u

. (3.25)

For this purpose we transform to a problem with homogeneous boundary and

initial conditions for u

in the decomposition u = u

+¯g with ¯g being selected as

the extension of boundary data given by (2.4).

Substituting u into (3.23) gives

(1 − 2ku)u

− c

Δu

−bΔu

=2ku

+ q (3.26)

u =0, on (0,T) × ∂Ω

(0) = 0 ,u

(0) = 0, in Ω.

where the “forcing” q is given by

q ≡ 2ku¯g

+2ku

¯g

Well-posedness for the Westervelt Equation 379

Thus, we are looking for solution u = u

+¯g such that u

satisﬁes (3.26)

with zero Cauchy (boundary and initial) data. The above leads to a ﬁxed point

formulation

: W ⊂ X → W, with W given by (3.4)

and T

(v) ≡ u,whereu = u

+¯g with u

given by

(1 − 2kv)u

− c

Δu

− bΔu

=2kv

+ q

(3.27)

=0on(0,T) ×∂Ω

(0) = 0 ,u

(0) = 0 .

and

≡ 2kv¯g

+2kv

¯g

. (3.28)

Since now v ∈ W does not satisfy homogeneous boundary conditions, we will have

to use the following estimates following from Assumption 2.1 and the decomposi-

tion w =(w −D

γw)+D

γw,wherew −D

γw satisﬁes homogeneous Dirichlet

boundary conditions.

(Ω) ⊂ L

(Ω) , with |w|≤

(|∇w| + |γ ˜w|

1/2

(∂Ω)

) , (3.29)

and |∇w|≤

(|Δw| + |γ ˜w|

3/2

(∂Ω)

) , (3.30)

(Ω) ⊂ L

(Ω) , with |w|

(Ω)

≤

(|∇w| + |γ ˜w|

1/2

(∂Ω)

) , (3.31)

and |∇w|

(Ω)

≤

(|Δw| + |γ ˜w|

3/2

(∂Ω)

) , (3.32)

(Ω) ⊂ C(Ω) , with |w|

∞

(Ω)

≤

(|Δw| + |γ ˜w|

3/2

(∂Ω)

) . (3.33)

3.2.1. Local well-posedness: Proof of Theorem 1.1. Intheﬁxedpointproofoflocal

existence we will fall back to the case of the nonlinear problem with the source

and homogeneous boundary data.

Toseethis,wenoticethatu

= T v where T v is deﬁned by (3.3) with q

replaced by q

and zero Cauchy data. This leads to the following description of

the action of the map T

v = T v +¯g

where ¯g is given by (2.34) and q

is given by

≡ 2kv¯g

+2kv

¯g

. (3.34)

To formulate a smallness assumption on the boundary data, we will estimate

q



−1

(Ω))

+ q



(Ω))

according to (3.34) by some quantity C(¯g,m, ¯a)=

(¯g, m,¯a)}. Then we will use the extension theorems from Section 2.3 to estimate

(¯g, m,¯a) from above in terms of appropriate norms of g.

For this, we use the following estimate:

Proposition 3.5.

−1/2

[v¯g

ttt

]|(t) ≤ (b|∇¯g

(t)| + c

|∇¯g

(t)|)(|v(t)|

∞

(Ω)

+ C

|∇v(t)|

(Ω)

380 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

Proof. Since we do not have an estimate of ¯g

ttt

, we use the respective PDEs to

express two time derivatives via space derivatives, i.e., for any φ ∈ L

(Ω) with

|φ| =1,weevaluate

−1/2

(v¯g

ttt

)(t),φ)=(¯g

ttt

(t),v(t)A

−1/2

φ)=(c

Δ¯g

+ bΔ¯g

,vA

−1/2

φ).

Since A

−1/2

φ ∈ H

(Ω), we obtain

(Δ¯g

,vA

−1/2

φ)=(∇¯g

,v∇A

−1/2

φ + ∇vA

−1/2

φ)

≤|∇¯g

(t)|(|v(t)|

∞

(Ω)

|φ| + |∇v(t)|

(Ω)

−1/2

φ|

(Ω)

) .

A similar argument applies to the term Δ¯g

. 

The following lemma provides the estimate of ||∇v||

C(L

(Ω)

using the infor-

mation ||Δv||

(Ω))

≤ ¯a, ||∇v

(Ω))

≤ ¯a, ||∇v

C(L

(Ω))

≤ ¯a, see (3.4).

Lemma 3.6. For any φ ∈ L

(Ω)) ∩ H

(Ω)) with φ(0) ∈ L

(Ω) we have

φ ∈ C(L

(Ω)) with

||φ||

C(L

(Ω))

≤



φ

1/2

(Ω))

φ

3/2

(Ω))

φ



(Ω))

+ |φ(0)|

(Ω)



1/3

where C

is the norm of the continuous embedding H

1/4

(0,T) → L

(0,T)

Proof.

|φ(t)|

(Ω)



|φ(s, x)|

dx ds + |φ(0)|

(Ω)



|φ(s, x)|

sign(φ(s, x))φ

(s, x) dx ds + |φ(0)|

(Ω)

≤ 3



|φ(s)|

(Ω)

|φ

(s)|

(Ω)

ds + |φ(0)|

(Ω)

≤ 3φ

(Ω))

φ



(Ω))

+ |φ(0)|

(Ω)

≤ 3C

φ

1/4

(Ω))

φ



(Ω))

+ |φ(0)|

(Ω)

≤ 3C

φ

1/2

(Ω))

φ

3/2

(Ω))

φ



(Ω))

+ |φ(0)|

(Ω)

where we have used standard interpolation in the last inequality:

||φ||

1/4

(Ω))

≤ C||φ||

1/2

(Ω))

||φ||

3/2

(Ω))



Therewith we can give the following estimates of the ¯g terms arising from

homogenization:

Proposition 3.7. For q

given by (3.34) with v ∈ W (see (3.4)) the estimate

q



−1

(Ω))

+ q



(Ω))

≤ C

∗

(¯g, m,¯a)

Well-posedness for the Westervelt Equation 381

holds with

∗

(¯g, m,¯a)=C

(¯g, m,¯a) (3.35)

=2k(m + θ(T,m,¯a, g, u

))(b∇¯g



(Ω))

+ c

∇¯g



(Ω))

)

+4k

(¯a + g



C(H

1/2

(∂Ω))

)¯g



3/2

(Ω))

+2k

(¯a + g



1/2

(∂Ω))

)¯g



∞

3/2

(Ω))

+2km¯g



(Ω))

+2k

(¯a + g



C(H

1/2

(∂Ω))

)¯g



(Ω))

where

θ(T,m,¯a, g, u

)

≡|∇u

(Ω)

+3C



(¯a + g

3/2

(∂Ω))

)

+ T ¯a



1/4

3/2

(¯a + g

3/2

(∂Ω))

)

3/2

√

T ¯a.

(3.36)

Proof. We use Proposition 3.5 and Lemma 3.6, which together with ψ

(0,T )

≤

1/2

ψ

C(0,T )

yields

||∇v||

C(L

(Ω))

≤ θ(T,m,¯a, g, u

)

and the estimate |ab|

−1

(Ω)

≤|ab|

6/5

(Ω))

≤|a|

(Ω)

|b|

3/2

(Ω)

that follows from

duality and continuity of the embedding H

(Ω) → L

(Ω) to obtain

q



−1

(Ω))

=2kv¯g

ttt

+2v

¯g

+ v

¯g



−1

(Ω))

≤ 2k(m + θ(T,m,¯a, g, u

))(b∇¯g



(Ω))

+ c

∇¯g



(Ω))

)

+4k

(¯a + g



C(H

1/2

(∂Ω))

)¯g



3/2

(Ω))

+2k

(¯a + g



1/2

(∂Ω))

)¯g



∞

3/2

(Ω))

where we have used Proposition 3.5 and Lemma 3.6, as well as

q



(Ω))

=2kv¯g

+ v

¯g



(Ω))

≤ 2km¯g



(Ω))

+2k

(¯a + g



C(H

1/2

(∂Ω))

)¯g



(Ω)).



It only remains to combine Proposition 3.7 with the estimates according to

Section 2.3, which under condition (1.4) gives

q



−1

(Ω))

+ q



(Ω))

(3.37)

≤



C(T,m,¯a, u

C(g

3/2

(∂Ω))

+ g



C(H

1/2

(∂Ω))

+ g



1/2

(∂Ω))

)



with X according to (1.5), and a constant

C(T,m,¯a, u

) independent of g and

small for small m,¯a, |∇u

(Ω)

,aswellasaconstant

Since when evaluating T

v,weadd¯g to u

, we additionally need smallness

of g

(cf., (2.37)).

Therewith, along the lines of the proof of Theorem 3.1 (note that the proof

of contractivity of T is exactly the same as in case of homogeneous Dirichlet

boundary conditions, see [16]) we arrive at the following result:

382 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

Theorem 3.8. Let b>0, m<

, T arbitrary. We assume that E

u,0

(0) < ∞,

u,1

(0) ≤ ρ, g

≤ ˜ρ

with ¯a, ρ, ˜ρ suﬃciently small (but possibly depending on T ),

Then there exists a solution u ∈ W of (3.23), (3.24), (3.25), which is unique

in W and satisﬁes Δu, u

, ∇u

∈ C(0,T; L

(Ω)) , ∇u

∈ L

(0,T; L

(Ω)) .

3.2.2. Global well-posedness: Proof of Theorem 1.2. By reducing the situation of

nonhomogeneous Dirichlet data to the situation of nonzero source term we obtain

a global well-posedness result.

Theorem 3.9. For suﬃciently small initial data and boundary data the solutions

are global in time. Moreover, the size of initial data does not depend on time. This

is to say: For any given M>0 there exist ρ>0 and ˜ρ>0 such that if (1.4),

E(0) ≤ ρ, (3.38)

l=0



g

∞

3/2−l

)(∂Ω))

l=0



3−l

g

−3/2+2l

(∂Ω))

≤ ˜ρ (3.39)

then E(t) ≤ M, t ∈ R

Proof. We use the extension (2.34) and the fact that u = u

+¯g,whereu

satisﬁes

(1 − 2k(u

+¯g))u

− c

Δu

−bΔu

=2k(u

)

+ q

(3.40)

=0on∂Ω

(0) = 0 ,u

(0) = 0 .

and

≡ 2ku

¯g

+4ku

¯g

+2k¯g¯g

+2k(¯g

)

. (3.41)

Similarly to Proposition 3.7 we obtain, using

=2ku

¯g

ttt

+6ku

¯g

+4ku

¯g

+6k¯g

¯g

+2k¯g¯g

ttt

the estimate

(t)|

−1

(Ω)

+ |q

(t)|

(Ω)

≤ 64



2k(b|∇¯g

(t)| + c

|∇¯g

(t)|)

+ C

|Ω|

1/6

)

|Δu

(t)|

+6kC

|∇u

(t)|

|¯g

(t)|

3/2

(Ω)

+4kC

|∇u

(t)|

|¯g

(t)|

3/2

(Ω)

+6k

(|∇¯g

(t)| + |g

(t)|

1/2

(∂Ω)

)

|¯g

(t)|

3/2

(Ω)

+2k(b|∇¯g

(t)| + c

|∇¯g

(t)|)

(

|Ω|

1/6

)

(|Δ¯g(t)| + |g(t)|

3/2

(∂Ω)

)



Well-posedness for the Westervelt Equation 383

+16



2kC

|Δu

(t)|

|¯g

(t)|

+4kC

3/2

1/2

3/2

1/2

|∇u

(t)|

(|∇¯g

(t)| + |g

(t)|

1/2

(∂Ω)

)

+2k

(|Δ¯g(t)| + |g(t)|

3/2

(∂Ω)

)

|¯g

(t)|

+2k

(|∇¯g

(t)| + |g

(t)|

1/2

(∂Ω)

)



≤ 256kC

|∇u

(t)|

|¯g

(t)|

3/2

(Ω)

+ Ck



|Δu

(t)|

+ |∇u

(t)|

+ |Δ¯g(t)|

+ |∇¯g

(t)|

+ |¯g

(t)|

+|g(t)|

3/2

(∂Ω)

+ |g

(t)|

1/2

(∂Ω)

+|∇¯g

(t)|



|Δu

(t)|

+(|Δ¯g(t)| + |g(t)|

3/2

(∂Ω)

)



(3.42)

≤ C



˜ρ|A

1/2

(t)|

+ E

(t)

+ |Δ¯g(t)|

+ |∇¯g

(t)|

+ |¯g

(t)|

(Ω)

+|g(t)|

3/2

(∂Ω)

+ |g

(t)|

1/2

(∂Ω)



(3.43)

for some generic constant C, where we have used Lemmas 2.16, 2.18.

Therewith, similarly to (3.14) and again using Lemmas 2.16, 2.18 as well as

φ

(0,∞)

≤φ

∞

(0,∞)

φ

(0,∞)

we get

(T )+(α(T )u

(T ),u

(T )) (3.44)

b −C ˜ρ)



(t)+|A

1/2

(t)|

)(1 − Φ(u

)(t))dt ≤ C

E(0) + C ˜ρ

with E

as in the proof of Theorem (3.3) with u replaced by u

and Φ deﬁned by

Φ(u

)(t)=C

16C

1/2

+ C

1/2

+ C

16C

1/2

+ CE

(t)

see (3.43) and the equation before (50) in [16], hence satisfying (3.15).

Therewith, the rest of the proof can be carried out analogously to the one of

Theorem (3.3) to yield global existence of u

and therewith of u = u

+¯g. 

3.2.3. Decay rates: Proof of Theorem 1.3.

Theorem 3.10. We assume that the initial and boundary data are suﬃciently small

such that (3.38),and(3.39) holds. Additionally, we assume that the boundary data

decays exponentially to zero in the sense speciﬁed in Theorem 1.3, i.e.:

|g(t)|

3/2

(∂Ω)

+ |g

(t)|

1/2

(∂Ω)

+ |g

(t)|

1/2

(∂Ω)

+ |g

ttt

(t)|

−3/2

(∂Ω)

≤ C

−ω

(3.45)

Then the energy decays exponentially fast to zero.

E(t)+E

u,0

(t) ≤ Ce

−˜ωt

384 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

where 0 < ˜ω<min{

,ω,

,ω

} with

b =min{1, 4/C

}(min{1, 2/c

}

b −Ck˜ρ),

as in (2.15),andω as in (2.7).

Proof. Again we use the extension (2.34).

In order to proceed along the lines of the proof of Theorem 3.4, we consider

(3.40) with (3.41) and estimate (3.42), which similarly to (3.14), (3.44) yields

(T )+(α(T )u

(T ),u

(T ))

b −C ˜ρ)



(t)+|A

1/2

(t)|

)(1 − Φ(u

)(t))dt

≤ C

E(s)+C





|Δ¯g(t)|

+ |∇¯g

(t)|

+ |¯g

(t)|

+ |g(t)|

3/2

(∂Ω)

+ |g

(t)|

1/2

(∂Ω)

+ |∇¯g

(t)|



|Δu

(t)|

+(|Δ¯g(t)| + |g(t)|

3/2

(∂Ω)

)



dt (3.46)

with E

as in the proof of Theorem (3.3) with u replaced by u

and Φ satisfying

(3.15).

Since by Theorem 3.9 and Lemma 2.16, the term



|Δu

(t)|

+(|Δ¯g(t)| +

|g(t)|

3/2

(∂Ω)

)

is in L

∞

, it therefore remains to show that (3.45) implies expo-

nential decay





|Δ¯g(t)|

+ |∇¯g

(t)|

+ |¯g

(t)|

+ |∇¯g

(t)|



dt ≤ Ce

−˜ωt

. (3.47)

From Lemma 2.12 with w =¯g, w

= u

, w

= u

we get



¯g(t) − D

g(t)

¯g

(t) − D

(t)



= e



−D

g(0)

− D

(0)



+ K[−D

](t)

which by Proposition 2.17 with t = 0 replaced by t = s yields

|Az(t)|

+ |A

1/2

(t)|

+ |z

(t)|



1/2

(τ)|

dτ

≤ C

−ω

(t−s)

[|Az(s)|

+ |A

1/2

(s)|

+ |D

(s)+bAz

(s)|

]

+ C



−ω

(t−τ )

[|A

−1/2

ttt

(τ)|

+ |D

(τ)|

]dτ] (3.48)

for z(t)=¯g(t) −D

g(t), where ω

=min[ω, b

−1

]. To this end, note that in Propo-

sition 2.17 we have Z(t)=e

A(t−s)

Z(s)+



A(t−τ )



f(τ)



dτ.

With s = 0 in (3.48), using (3.45) and (2.17) this implies

z,1

(t)=|Δz(t)|

+ |∇z

(t)|

+ |z

(t)|

≤

−˜ωt

(3.49)

which yields





|Δz(t)|

+ |∇z

(t)|

+ |z

(t)|



dt ≤

−2˜ωs

. (3.50)

Well-posedness for the Westervelt Equation 385

Now we use (3.48) with t = T and (3.45) to obtain



1/2

(τ)|

dτ

≤ C

−ω

(T −s)

[|Az(s)|

+ |A

1/2

(s)|

+ |D

(s)+bAz

(s)|

−˜ωs

In here, we can use the identity (2.19) that implies bAz

(s)=−¯g

(s) −c

Az(s)

to get



1/2

(τ)|

dτ

≤ C

−˜ω(T −s)

[|Az(s)|

+ |A

1/2

(s)|

+ |z

(s)+c

Az(s)|

−˜ωs

Inserting (3.49) with t = s gives



1/2

(τ)|

dτ ≤

−˜ωs

which together with

(3.50) and



⎛

⎝

|ΔD

g(t)

B CD E

+ |∇D

(t)|

+ |D

(t)|

+ |∇D

(t)|

⎞

⎠

dt ≤ Ce

−˜ωs

yields (3.47). So we obtain

(t) ≤ Ce

−˜ωt

This together with (3.49) yields

u,1

(t) ≤ 2E

(t)+2E

¯g,1

(t) ≤ Ce

−˜ωt

and therewith the assertion regarding the higher energy E

u,1

(t). Regarding the

lower energy E

u,0

(t), we evoke (3.29), (3.30), to obtain:

u,0

(t) ≤

(|∇u

(t)| + |g

(t)

1/2

(∂Ω)

)

(|Δu

(t)| + |g(t)

3/2

(∂Ω)

)

≤ 2max{

}(E

u,1

(t)+|g

(t)

1/2

(∂Ω)

+ |g(t)

3/2

(∂Ω)

). 

Acknowledgment

1. Support by the NSF under Grant DMS 0606682, the German Science Foun-

dation under Grant KA 1778/5-1, the Elite Network of Bavaria within the In-

ternational Doctorate Program Identiﬁcation, Optimization and Control with

Applications in Modern Technologies, and the Cluster of Excellence SimTech,

University of Stuttgart, is gratefully acknowledged.

2. The second-named author performed part of the work while visiting the De-

partment of Mathematics, University of Warsaw within the European Union

Project “Modern University”. Warm hospitality and partial support is grate-

fully acknowledged.

386 B. Kaltenbacher, I. Lasiecka and S. Veljovi´c

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Barbara Kaltenbacher and Slobodan Veljovi´c

University of Graz, Austria

e-mail: barbara.kaltenbacher@uni-graz.at

slobodan.veljovic@uni-graz.at

Irena Lasiecka

Department of Mathematics

University of Virginia

Charlottesville, VA 22901, USA

and

Systems Research Institute

Polish Academy of Sciences, Warsaw

e-mail: il2v@virginia.edu