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

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

which implies H

(Ω) × L

(Ω) membership of the third term. Regarding the re-

maining terms, the analysis is similar. It suﬃces to notice that w

− Dg(0) ∈

(Ω) = D(A

1/2

), Dg ∈ C(H

(Ω)) and



f(s)



∈ L

([D(A

∗1/2

)]



Thus, the conclusion follows from (2.6), (2.8), after noting that long time

behavior is controlled by the exponential decay of the semigroup. 

By using the formula above, along with the aforementioned properties of

analytic semigroups given in (2.6), (2.8) the following regularity for less regular

Cauchy data is easily deduced

Lemma 2.11. For

g ∈ L

1/2

(∂Ω)),g

∈ L

−1/2

(∂Ω)),f∈ L

([D(A)]



)

∈ H

(Ω),γw

= g(0),w

∈ H

−1

(Ω) (2.24)

we have W ∈ L

(Ω) × L

(Ω))

After these preliminary considerations, we shall proceed with the study of

regularity of solutions, assuming that the boundary data and initial data are more

regular. Some of the results stated below can be also obtained by energy methods

applied to appropriate homogenizations of the problem (see the next section).

However, in order to track precisely the regularity, particularly long time regularity,

and the eﬀects of compatibility condition, semigroup representations seem more

appropriate. Within this framework, one can take advantage of explicit singular

estimates and of exponential decays associated with the semigroup.

In order to proceed with our program, we shall exploit several diﬀerent rep-

resentations of the semigroup formula (these methods are known since [22]).

Lemma 2.12. The following representation holds for all Cauchy data as speciﬁed

in Theorem 1.1.

W (t)=



g(t)

(t)



+ e



−D

g(0)

− D

(0)



(2.25)

−



A(t−s)



(s)



ds +



A(t−s)



f(s)



ds.

Proof. The starting point is (2.23). By integration by parts and using semigroup

property valid for analytic semigroups Ae

,fort>0 we obtain the rep-

resentation stated above. Note that the present representation has no singularity

in the kernel, but it requires higher diﬀerentiability of the boundary data. 

Remark 2.13. Note that W =



g(t)

(t)



where U

satisﬁes damped wave

equation driven by the source f −D

with zero boundary conditions and initial

data given by



−D

g(0)

− D

(0)



Well-posedness for the Westervelt Equation 369

In order to obtain information on higher-order time derivatives of solutions,

diﬀerentiation, in the sense of distribution, of formula in (2.12) leads to the fol-

lowing representation of weak solutions.

Lemma 2.14. With reference to Cauchy data speciﬁed in Theorem 1.1, the following

representation holds.

(t)=



(t)

−f(t)



+ Ae



− D

g(0)

− D

(0)



−



A(t−s)



(s)



ds (2.26)



A(t−s)



(s)



ds + e



f(t)



The formula in Lemma 2.14 is initially derived in the topology of extended

spaces [28, 8]. However, taking advantage of analyticity of the semigroup Ae

∈

L(H),t > 0, each term in the formula represents a well-deﬁned element in H for

t>0. Semigroup formulas in 2.12 and Lemma 2.14, along with the properties (2.8)

of analytic semigroup e

, lead to the following – long time behavior – regularity.

Lemma 2.15. Let f =0,and

g ∈ L

∞

1/2

(Γ)) ,g

∈ L

∞

1/2

(Γ)),g

∈ L

∞

−1/2

(Γ))

∈ H

(Ω) ,γw

= g(0) ,w

∈ H

(Ω),γw

= g

(0)

we have

W ∈ L

∞

(D(A

1−

) ⊕ L

∞

(Ω) × H

(Ω)).

In particular w ∈ L

∞

(Ω)),w

∈ L

∞

(Ω))

Proof. Use directly formula in Lemma 2.12 along with the properties enjoyed by

the semigroup e

and listed in (2.6), (2.8). Particular emphasis should be paid to

exponential decay at inﬁnity and control of singularity at the origin.

Writing W (t)=I(t)+II(t)+III(t), we obtain the following estimates for

each term:

1−

III(t)|

≤



−ω(t−s)

(t − s)

1−

(s)|

−1/2

(∂Ω)

ds ≤ C|g

∞

−1/2

(∂Ω))

(2.27)

1/2

II(t)|

≤|A

1/2



− D

g(0)

−D

(0)



≤ C|A

1/2

−D

g(0))| + |A

1/2

− D

(0))|

≤ C[|w

(Ω)

| + |w

(Ω)

+ |g(0)|

1/2

(∂Ω)

+ |g

(0)|

1/2

(∂Ω)

≤ C

g,w

(2.28)

where we have used compatibility conditions on the boundary.

The identiﬁcation in (2.6) applied with θ =1/2 concludes the estimate for the

second term I(t) ∈ H

(Ω) ×H

(Ω) as long as g

(t) ∈ H

1/2

(∂Ω),g(t) ∈ H

1/2

(∂Ω).



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

Lemma 2.16. Let f =0and

g ∈ L

∞

3/2

)(∂Ω)) ,g

∈ L

∞

1/2

(∂Ω))),

∈ L

∞

−1/2

(∂Ω)) ∩L

1/2

(∂Ω)),g

ttt

∈ L

−3/2

(∂Ω))

∈ H

(Ω) ,γw

= g(0) ,w

∈ H

(Ω),γw

= g

(0). (2.29)

Then

W ∈ L

∞

(Ω) × H

(Ω))

∈ L

∞

(Ω)) ∩ L

(Ω)).

Proof. The proof follows by reading oﬀ from formula in Lemma (2.12) with a

supplement of the following result:

Proposition 2.17. Let the operator K be deﬁned as follows:

(Kf)(t) ≡



A(t−s)



f(s)



ds.

Let Z =(z,z

) ≡ e

+ Kf

Then, there exists constants C

> 0(independent on t>0) and ω

> 0 such

that

|Az(t)|

+ |A

1/2

(t)|

+ |z

(t)|



1/2

≤ C

−ω

[|Az(0)|

+ |A

1/2

(0)|

+ |f(0) − bAz

(0)|

]

+ C



−ω

(t−s)

[|A

−1/2

(s)|

+ |f|

]ds].

The constant ω

=min[ω, b

−1

Proof. The proof of this proposition follows from energy methods reported in

theorem 2.3. The important factor is that the constant C does not depend on

t. Here are the details. Noting that

(t)=e

(0) + (Kf

)(t)

the fourth formula in (2.8) implies

(t)|

≤ Ce

−2ωt

(0)|

+ C



−2ω(t−s)

−1/2

(s)|

ds. (2.30)

Hence

1/2

(t)|

+ |z

(t)|

≤ Ce

−2ωt

[|A

1/2

(0)|

+ |z

(0)|

]

+ C



−2ω(t−s)

−1/2

(s)|

ds. (2.31)

Well-posedness for the Westervelt Equation 371

On the other hand, applying the multiplier Az to the original equation satisﬁed

by z,gives

|Az(t)|

≤ C

−b

−1

|Az(0)|

+ C



−2b

−1

(t−s)

[|f(s)|

+ |z

(s)|

]ds. (2.32)

Combining the estimates in (2.31) and (2.32) leads, after some calculations, to the

ﬁnal conclusion stated in Proposition 2.17. 

To continue with the proof of Lemma 2.16 we write, as before, W (t)=I(t)+

II(t)+III(t), where W (t) is given by Lemma 2.12. Applying the proposition

with f = D

and Z(0) ≡



−D

g(0)

− D

(0)



gives the desired result for term

II + III. We note the use of compatibility conditions which allows to deduce that

Z(0) ∈D(A) ×D(A), as needed for application of the proposition.

The estimate for term I is straightforward, as it follows from a priori regu-

larity of the boundary data g(t) and regularity of harmonic extension D

. 

Finally, we shall need regularity of the third time derivative of parabolic

extension. This is given in the lemma below.

Lemma 2.18. Let f =0and (2.29) hold. Then:

ttt

∈ L

−1

(Ω)).

Proof. We use the formula in Lemma 2.14 and write

(t)=I(t)+II(t)+III(t) .

In order to read oﬀ regularity of w

ttt

we proceed with evaluation of W

. This leads

to calculation of

III(t)=III

which can be written as:

III

= Ae



(0)





A(t−s)



ttt

(s)



ds.

The assumptions imposed imply D

ttt

∈ L

([D(A

1/2

]



) hence



ttt

(t)



∈ L

([D(A

1/2

]



Moreover

1/2



(0)



∈ L

([D(A

1/2

]



This gives

III

∈ L

([D(A

1/2

)]



)=L

(Ω) × H

−1

(Ω)).

As for II

,wehave



− D

g(0)

− D

(0)



∈D(A) ×D(A)

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

since each term is in A (as in the argument given in previous lemma,) so Z(0) ∈

D(A). This gives

AZ(0) = A



w(0) − Dg(0)

− Dg

(0)



∈ H

hence

= Ae

AZ(0) ∈ L

[(D(A

∗1/2

)]



) ∈ L

(Ω)) × L

−1

(Ω)))

where we have used all the properties of fractional powers and characterization of

domains associated with strongly damped model.

The analysis for I





is straightforward. Calculations above tes-

tify that the second coordinate of W

belongs to L

−1

(Ω)), as desired. 

2.3.2. “Parabolic extension” of Cauchy data. The aim of this section is to intro-

duce “parabolic extension” into the interior of the cylinder Ω × (0,T)ofCauchy

data g,u

speciﬁed in Theorem 1.1. To this end we shall use harmonic extension

for the boundary data. This leads to the following problem. Consider

− c

Δw − bΔw

= f (t)in(0,T) × Ω (2.33)

w = g on ∂Ω,w(0) = w

(0) = w

, in Ω.

This will lead to an extension

¯g = w according to (2.33) with (2.34)

f =0,w

= u

We ﬁrst state a result using the harmonic extension (2.16).

Theorem 2.19. Let

1. f ∈ L

(Ω)) ∩ H

−1

(Ω))

2. g ∈ C(H

3/2

(∂Ω)), g

∈ C(H

1/2

(∂Ω)), g

∈ C(H

−1/2

(∂Ω)) ∩ L

1/2

(∂Ω)),

ttt

∈ L

−3/2

(∂Ω)).

3. g(0) = w

∂Ω

, w

∈ H

(Ω), g

(0) = w

∂Ω

, w

∈ H

(Ω), f(0) ∈ L

(Ω).

Then w ∈ C(H

(Ω)), w

∈ C(H

(Ω)), w

∈ C(L

(Ω)) ∩ L

(Ω)).

Proof. By introducing new variable u ≡ w−D

g equation (2.33) can be restated as

+ c

Au + bAu = f(t)+F (t) (2.35)

where F (t) ≡−D

and initial data becomes u(0) = w

− D

g(0),u

(0) =

− D

(0).

We shall apply the result of Theorem 2.3, see also [9].

For this we verify compatibility conditions:

− D

g(0) ∈ H

(Ω) ∩ H

(Ω) ,w

− D

(0) ∈ H

(Ω)

f(0) + F (0) −c

−bΔw

∈ L

(Ω)

Well-posedness for the Westervelt Equation 373

which are satisﬁed due to the imposed assumptions. Similarly, the conditions 2.

of Theorem 2.3 imposed on f and F are also satisﬁed by the assumptions. The

condition on the forcing term becomes

f, D

∈ L

(Ω)) ∩ H

−1

(Ω)) .

We shall use regularity of the Dirichlet map (2.17). The above translates into the

following regularity of g:

∈ L

−1/2

(∂Ω))),g

ttt

∈ L

−3/2

(∂Ω)).

With the above regularity, on the strength of Theorem 2.3 we obtain:

u ∈ C(H

(Ω) ∩ H

(Ω)),u

∈ C(H

(Ω)),u

∈ C(L

(Ω)) ∩ L

(Ω)). (2.36)

This implies the same regularity for the function w, provided that

g ∈ C(H

(Ω)),D

∈ C(H

(Ω))

∈ C(L

(Ω)) ∩ L

(Ω)).

Since D

∈L(H

(∂Ω) → H

s+1/2

(Ω)) for all real s we obtain the same

conclusion with g satisfying

g ∈ C(H

3/2

(∂Ω)),g

∈ C(H

1/2

(∂Ω)) (2.37)

∈ C(H

−1/2

(∂Ω)) ∩ L

1/2

(∂Ω))). 

3. Back to the nonlinear problem

3.1. The Westervelt equation with source term

Consider the Westervelt equation

(1 − 2ku)u

−c

Δu − bΔu

=2k(u

)

+ q, (3.1)

with zero Dirichlet boundary conditions and given initial conditions:

u =0on∂Ω,u(t =0)=u

(t =0)=u

. (3.2)

Here k = β

/(ρc

), and q plays the role of a given interior source.

We will show results on local and global well-posedness for this problem

before we carry out the proofs of Theorems 1.1, 1.2, 1.3.

3.1.1. Local well-posedness. The ﬁxed point operator T : W ⊆ X → W that

we will make use of for using Banach’s ﬁxed point theorem, will be deﬁned by

T (v) ≡ u with u solving the following linearization of the original equation

(1 − 2kv)u

− c

Δu − bΔu

=2kv

+ q, (3.3)

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

with boundary conditions v =0on∂Ω , and initial conditions v(t =0)=u

(t =0)=u

. Here,

W =



v ∈ C((0,T) ×Ω), ||v||

∞

((0,T )×Ω)

≤ m,

||Δv||

(Ω))

≤ ¯a, ||∇v

(Ω))

≤ ¯a, ||∇v

C(L

(Ω))

≤ ¯a,

v(0) = u

(0) = u



(3.4)

where m<

,¯a are to be chosen appropriately during the course of the proof.

The map T is well deﬁned, a fact that follows from linear analysis.

The main result of this section is the following:

Theorem 3.1. Let b>0, m<

, T arbitrary. We assume that

u,1

(0) ≤ ρ, q



−1

(Ω))

+ q

(Ω))

≤ ˜ρ

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

Then there exists a solution u ∈ W of (3.1), (3.2), which is unique in W and

satisﬁes Δu, u

, ∇u

∈ C(0,T; L

(Ω)) , ∇u

∈ L

(0,T; L

(Ω)) .

Proof. In order to show that the map T is a self-mapping and contraction on W

with suitable parameters m,¯a,wemakeuseofTheorem2.3with

α(t, x) ≡ 1 − 2kv(t, x), (3.5)

f(t, x) ≡ 2kv

(t, x)u

(t, x)+q(t, x). (3.6)

Note that Assumption 2.1 is obviously satisﬁed for A = −Δ with homo-

geneous Dirichlet boundary conditions, due to Poincar´e’s inequality, H

regu-

larity of solutions to the Laplace equation with L

right-hand side, and conti-

nuity of the embeddings H

(Ω) → L

(Ω), H

(Ω) → C(Ω). Moreover, we have

D(A

1/2

) ⊆ H

(Ω), D(A) ⊆ H

(Ω), with |A

1/2

w| = |∇w|, |Aw| = |Δw|. Addition-

ally we have, D(A) ⊂ W

(Ω) , with |∇w|

(Ω)

≤

|Δw|.

Step 1. We will ﬁrst show that for v ∈ W the functions α, f according to (3.5)

enjoy the regularity required in Theorem 2.3, i.e., α ∈ C

(H) and, according to

Assumption 2.2, 0 <α

≤ α(t, x) ≤ α

, so we deal with the non-degenerate

case. Moreover, according to the assumptions of Theorem 2.3 we have to show

f ∈ L

(H) ∩ H

([D(A

1/2

)]



). Note that 3. of the assumptions of Theorem 2.3

follows from E

u,1

(0) ≤ ρ. We recall that H = L

(Ω). For this purpose we can

make use of Proposition 4 in [16] and augment it in a straightforward way by the

source term:

Proposition 3.2. Let v ∈ W and α, f be deﬁned above by equation (3.5) and assume

that km ≤ 1/4.Then

•

=3/2 ≥ α(t, x) ≥ 1/2=α

•|α

C(H)

≤ 2C

k¯a

•



−1/2

dt ≤ 16C

¯a

(|u

C(H)

+ |A

1/2

(H)

)+2q



−1

(Ω))

•



|f|

ds ≤ 8C

¯a

)

+2q

(Ω))

Well-posedness for the Westervelt Equation 375

Note that on the strength of Assumption 2.1 and by deﬁnition of A we have

|u|

∞

((0,T )×Ω)

≤ C

|Au|

C(H)

; |Δu|

(H)

≤ T |Au|

C(H)

|∇u

(H)

≤|A

1/2

(H)

; |∇u

C(H)

≤

C(0,T )

. (3.7)

Step 2. To obtain the bounds m,¯a required for u ∈ W ,wewillnowmakeuseof

estimates (2.10), (2.11) from Theorem 2.3 together with Proposition 3.2.

Applying the estimate (2.10) with ˆ =  =

gives:

(t)+



1/2

(s)|

≤



(0) +

16C

¯a

(|u

C(H)

+ |Au

(H)

q



−1

(Ω))



× e

b/4

t(2C

k¯a)

(3.8)

which yields

max{|E

C(0,T )

1/2

(t)|

(H)

}

≤



(0) +

q



−1

(Ω))

+ φ(¯a)|E

C(0,T )



b/4

T (2C

k¯a)

, (3.9)

where

φ(¯a)=

16C

¯a

max





With ¯a suﬃciently small so that φ(¯a) ≤

, and ρ,˜ρ suﬃciently small so that

max





ρ +

˜ρ



b/4

T (2C

k¯a)

≤ ¯a

we get by (3.7) |u

)

≤ ¯a, |u

C(H

)

≤ ¯a as desired, and additionally

C(H)

≤

¯a

. (3.10)

Applying the estimate (2.11) gives:

b|Au(t)|

≤ b|Au(0)|

¯a

(Ω0)

q

(Ω))



1/2

≤ bρ +

˜ρ +

¯a

T ¯a

¯a

so that by possibly decreasing ρ,˜ρ,and¯a using (3.7) we can guarantee

|u|

∞

((0,T )×Ω)

≤ m, hence the ﬁnal conclusion T W ⊂ W follows.

Step 3. The proof of contractivity of T is exactly the same as without source term,

see [16]. 

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

3.1.2. Global well-posedness. We shall exploit the barrier’s method, typically used

in quasilinear hyperbolic problems [5, 32, 34, 23].

Theorem 3.3. For suﬃciently small initial data and source term 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

E(0) ≤ ρ (3.11)

and

q



(0,∞;H

−1

(Ω))

+ q

(0,∞;L

(Ω))

≤ ˜ρ, (3.12)

then

E(t) ≤ M for all t ∈ R

Proof. Let u be a local solution that exists for suﬃciently small data by Theorem

3.1. For this solution, we apply Proposition 2.6 with

α ≡ 1 − 2ku , f ≡ 2ku

+ q (3.13)

(see Proposition 6 in [16]) for which the following estimates can be obtained



|α

(t)|

dt ≤ 16C



1/2

(t)|

1/2

(t)|



|f(t)|

dt ≤ 8C



1/2

(t)|

dt +2



|q(t)|



−1/2

(t)|

dt ≤ 32C



1/2

(t)|

1/2

(t)|



−1/2

(t)|

dt.

Therewith, Proposition 2.6 yields

(T )+(α(T )u

(T ),u

(T )) +



(t)+|A

1/2

(t)|

)(1 − Φ(u, u

)(t))dt

≤ C

E(0) + 2





−1/2

(t)|

+ C

|q(t)|



dt (3.14)

where

b =min{1, 2/c

}

(t)=

1/2

(t)|

+ |Au(t)|

and

E(t)=E

(t)+

(α(t)u

(t),u

(t)) ,

and

Φ(u, u

)(t) ≤

C(E

(t)+E

(t)

) (3.15)

(see the proof of Theorem 3.2 in [16]).

Based on estimate (3.14), we now apply barrier’s method, i.e., we assume

that there exists some time when degeneration occurs in α or in (1 − Φ(u, u

)).

Well-posedness for the Westervelt Equation 377

Let T

be the ﬁrst such time instance, i.e., the ﬁrst time when either α(T

)=0or

Φ(u, u

)(T

)=1, which implies that either

k|Au(T

)|≥1 (3.16)

(see (3.13) and Assumption 2.1) or

C(E

)+E

)

) ≥ 1 (3.17)

(see (3.15)).

On the other hand, we can take ρ, ˜ρ in (3.11), (3.12) suﬃciently small so that

with

M = C

ρ +2max{C

}˜ρ



M<1and

M +

) < 1 , (3.18)

and apply (3.14), which holds up to t = T

and yields |Au(T

≤E

) ≤

Therewith

k|Au(T

)|≤2C



M<1

and

C(E

)+E

)

) ≤

M +

) < 1

a contradiction to (3.16), (3.17).

Note that since C

C are intrinsic constants independent of T ,for

any given M,wecanchoose

M such that

M ≤ M and the inequalities (3.18) are

satisﬁed. Then we choose ρ, ˜ρ such that C

ρ +2max{C

}˜ρ ≤

M.Withthese

ρ, ˜ρ (independent of T ) in (3.11), (3.12) we obtain the global energy estimate

E(t) ≤

M ≤ M for all t ∈ R



3.1.3. Decay rates.

Theorem 3.4. We assume that the initial data and source term are suﬃciently

small such that (3.11), (3.12) holds. Additionally, we assume that the source term

decays exponentially to zero

∀0 ≤ s<T :





(t)|

−1

(Ω)

+ |q(t)|

(Ω))



dτ ≤ C

−ω

. (3.19)

Then the energy decays exponentially fast to zero.

E(t) ≤ Ce

−ωt

E(0) (3.20)

where 0 <ω<min{

,ω

} with

b =min{1, 4/C

}min{1, 2/c

}

b,and

b, C

as in

(2.15).

Note that (3.19) follows from

∀t ≥ 0: |q

(t)|

−1

(Ω)

+ |q(t)|

(Ω))

≤ ω

−ω