Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping

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Ann. I. H. Poincaré – AN 31 (2014) 1079–1100

www.elsevier.com/locate/anihpc

Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping

M.M. Cavalcanti

, V.N. Domingos Cavalcanti

, V. Komornik

, J.H. Rodrigues

aDepartamento de Matemática, Universidade Estadual de Maringá, 87020-900, Maringá, PR, Brazil bDépartement de Mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Received 5 December 2012; received in revised form 17 July 2013; accepted 12 August 2013 Available online 13 September 2013

Abstract

We consider the KdV–Burgers equationut+uxxx−uxx+λu+uux=0 and its linearized versionut+uxxx−uxx+λu=0 on the whole real line. We investigate their well-posedness their exponential stability whenλis an indefinite damping.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:primary 35Q53; secondary 93D15

Keywords:KdV–Burgers equation; Well-posedness; Stabilization by feedback; Decay rate

1. Introduction

The goal of this work is to prove the exponential stability of the Cauchy problem u_t+u_xxx−u_xx+λu+αuu_x=0 inR×(0,∞),

u(0)=u0 inR, (1.1)

whereλ∈L^∞(R)is a function which is allowed to change sign andαis a constant which assumesα=0 orα=1.

These assumptions made onαimply that we are considering both the linear and the nonlinear problem.

In this work we were inspired by the equation

u_t+u_xxx−u_xx+uu_x=0 inR×(0,∞). (1.2)

This equation gained some popularity when the necessity to attach dissipation to nonlinearity and dispersion arises in modelling unidirectional propagation of planar waves.

* Corresponding author. Tels.: +33 3 68 85 02 07 (office), +33 9 53 69 57 12 (home).

E-mail addresses:mmcavalcanti@uem.br(M.M. Cavalcanti),vndcavalcanti@uem.br(V.N. Domingos Cavalcanti), vilmos.komornik@math.unistra.fr(V. Komornik),jh.rodrigues@ymail.br(J.H. Rodrigues).

1 Research of Marcelo M. Cavalcanti partially supported by the CNPq Grant 300631/2003-0.

2 Research of Valéria N. Domingos Cavalcanti partially supported by the CNPq Grant 304895/2003-2.

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.08.003

Hereu=u(x, t )is a real-valued function of two real variablesx andt, which in applications corresponds to the distance in the direction of propagation and to passed time, respectively. The dependent variable may represent a dis- placement of the underlying medium or a velocity, for example. Eq.(1.2)is referred to as Korteweg–de Vries–Burgers equation (KdVB equation) because it represents the union of the Korteweg–de Vries equation

ut+uxxx+uux=0 (1.3)

and the Burgers equation

u_t−u_xx+uu_x=0. (1.4)

Solutions of(1.2)should approach zero ast goes to infinity. A natural question which arises is about the rate at whichu(t )approaches zero, where · is some norm for real-valued functions of a real variable. In[1]the authors prove that the solution of(1.2)corresponding to the initial datau₀∈L¹(R)∩H²(R)satisfies the inequality

u(·, t )

L²(R)Ct⁻¹⁴

for allt >0 with some positive constantC. Moreover, this estimate is optimal because

Ru₀(x) dx=0.

The study of decay of the energy associated with dispersive nonlinear equations is very interesting and a consider- able number of researchers have determined significant advances to the development of this subject. In[4]the authors gave a new contribution with respect to the decay of the energy related to mild solutions for the damped Korteweg–de Vries (KdV) type equation given by

ut+bux+uxxx+uux+a(x)u=0, (1.5)

whereu=u(x, t )is a real-valued function,bis a real constant anda=a(x)is a non-negative function. In this paper, they consider the initial value problem (IVP)

u_t+bu_x+u_xxx+uu_x+a(x)u=0 inR×(0,∞),

u(x,0)=u0(x) forx∈R, (1.6)

and the initial boundary value problem (IBVP)

⎧⎨

⎩

u_t+bu_x+u_xxx+uu_x+a(x)u=0 inR×(0,∞),

u(0, t )=0 fort0,

u(x,0)=u₀(x) forx0,

(1.7) whereR₊=(0,∞).

In both cases, the non-negative functiona(x)is responsible for the dissipative effect.

Eq.(1.5)is a generalization of the well-known KdV equation,

u_t+bu_x+u_xxx+uu_x=0. (1.8)

In the case of the initial value problem related to(1.8), the valueu(·, t )²_L2(R) can be interpreted as the energy. It is obvious that for a smooth and decaying at infinity solutionu(x, t )to(1.8), the energy is a constant of motion, that is

R

u²(x, t) dx=

R

u²₀(x) dx, (1.9)

so there is no decay of the energy as t→ ∞. On the other hand, if we considera(x)α₀>0,∀x∈R, then the solution to Eq.(1.6)satisfies

R

u²(x, t) dxe^−2α0t

R

u²₀(x) dx. (1.10)

The main goal of this article was to establish the decay of the energy in the cases whena(x)≡0 but it is not assumed the existence of a positive constantα₀such thata(x)α₀. It was the main novelty of this paper since in the previous results in the related literature, the functiona(x)was considered strictly positive at∞.

A first result on exponential decay of the energy for the IBVP(1.7)considering a localized damping was established in[16]. In this papera(x)α0>0 in(0, δ)∪(R,∞)for some 0< δ < R. In[19]it was pointed out that the interval (0, δ)can be dropped and it was proved the exponential decay of the energy in weighted spaces with exponential and power weights. The decay of spatial derivatives of solutions was also derived.

We remark that in[2]for the non-homogeneous IBVP(1.7), considering a constant dampinga(x)=α₀>0 and small but not decaying boundary data, the exponential decay at theH^k-level fork=1,3 or 4 is obtained (without restrictions on the size of initial data).

Taking into account the KdV equation posed on a finite interval with localized damping, the exponential decay of the energy was established in[17]and[18]. Considering KdV type equations with more general nonlinearities, these results were extended in[15]and[22]while periodic problems were studied in[10,11,14,24]. A good review regarding these topics is given in[23].

Note that for small initial and boundary data, exponential decay of the energy is obtained for KdV equation posed on a finite interval, without any damping. In addition, the internal dissipation is responsible for the validity of such result even for small anti-damping (see[6,7,12,13,3]). In[8]nontrivial stationary smooth solution to the KdV equation posed on a finite interval with zero boundary data is constructed.

In[9]the solutions to problem(1.8)with initial boundary conditions as in(1.7)and small initial data are considered and, the pointwise decay ast→ ∞is established.

In spite of having many works dealing with the KdV equation in the existing literature, the same cannot be asserted to the KdV–Burgers equation. This lack of results becomes more evident when we are interested in the asymptotic behaviour of its solutions. In this context, we can cite the article[5], where the author used global attractors the- ory in order to study the asymptotic behaviour at the H²(R)-level of the semigroup associated to the generalized KdV–Burgers equation

ut+ δuxx+g(u)

x−νuxx+γ u=f (x), t >0, x∈R,

whereδ, ν >0 andγ0 are constants,f ∈H²(R), andgis a Lipschitz function of classC²(R).

The solutions to the considered problems, regarded in the distributional sense, are called weak solutions. Weak solu- tions on a time interval(0, T )are called mild solutions ifu(·, t )∈C([0, T];H¹(I ))anduu_x+au∈L²(0, T;L²(I )), where eitherI =RorI=R+. The main reason to select such a class of solutions from the set of weak solutions is that they can be regarded as solutions to the corresponding linear problems

ut+bux+uxxx=f (x, t ), (1.11)

wheref ∈L²(0, T;L²(I )).

This article is organized as follows. In Sections2 and3 we investigate the well-posedness and stability of the corresponding linear problem. Sections4,5 and6are dedicated to the nonlinear problem. Under the effect of an indefinite damping mechanism, global well-posedness and exponential stability results are established.

2. Well-posedness for the linear problem

In this section we consider the following problem:

ut+uxxx−uxx=0 inR×(0,∞),

u(x,0)=u₀(x) forx∈R. (2.1)

The next result ensures that this problem is well posed inL²(R).

Proposition 2.1.The operatorA:= −∂_x³+∂_x²defined onD(A):=H³(R)generates a semigroup in the Hilbert space H:=L²(R).

Proof. According to the Lumer–Phillips theorem it is sufficient to check thatAis dissipative and thatI−Ais onto.

The dissipativity follows by a direct computation: ifu₀∈H³(R)is real-valued, thenuis also real-valued and

(Au, u)_H= ∞

−∞

(−u_xxx+u_xx)u dx

= ∞

−∞

uxxux−u²_xdx

=1 2

∞

−∞

d

dxu²_xdx− ∞

−∞

u²_xdx

= − ∞

−∞

u²_xdx

0.

SinceAv=A(v)for allv∈H³(R), it follows that (Au, u)H= −

∞

−∞

|ux|²dx0 (2.2)

for allu0∈H³(R).

It remains to show that for everyf ∈L²(R)there existsu∈H³(R)satisfying the equality u−u_xxx+u_xx=f.

Taking the Fourier transform it is equivalent to ˆ

u(ξ ) 1−(iξ )³+(iξ )²

= ˆf (ξ ) or to

ˆ

u(ξ )= f (ξ )ˆ

1−(iξ )³+(iξ )². (2.3)

In the last step we have used the fact that the denominator h(ξ ):=1−(iξ )³+(iξ )²

never vanishes. Since, moreover,h(ξ )is a continuous function satisfying|h(ξ )| → ∞as|ξ| → ∞, 1/ his bounded, and therefore the last equation has a unique solutionuˆ∈L²(R).

Finally, since the function 1+ |ξ| + |ξ|²+ |ξ|³

|1−(iξ )³+(iξ )²|

tends to 1 as|ξ| → ∞and hence it is bounded by some constantMonR, we conclude that (iξ )^ju(ξ )ˆ Mf (ξ )ˆ , j=0,1,2,3.

Sincefˆ∈L²(R), this implies the regularity propertyu∈H³(R). 2

3. Exponential decay rates for the linear problem

The proof of the dissipativity in the preceding section shows that the strong solutions of (2.1) (for which u₀∈H³(R)) satisfy the relation

d dt

∞

−∞

u(x, t )²dx= −2 ∞

−∞

ux(x, t)²dx0. (3.1)

This is, however, not sufficient for the exponential stability because the nonzero constant functions solve(2.1).

In order to ensure the exponential stability of the KdV–Burgers equation in H :=L²(R), we thus need some additional damping mechanism. Inspecting the proof ofProposition 2.1, it is natural to consider the following modified problem:

ut+uxxx−uxx+λu=0 inR×(0,∞),

u(x,0)=u₀(x) forx∈R, (3.2)

whereλis some given non-negative function.

Proposition 3.1.Ifλ∈L^∞(R), then the operatorA_λdefined by the formulaA_λu:= −u_xxx+u_xx−λuonD(A_λ):=

H³(R)generates a semigroup in the Hilbert spaceH:=L²(R).

Proof. It suffices to observe thatA_λis a bounded perturbation of the operatorAofProposition 2.1and therefore it is also the infinitesimal generator of a semigroup. 2

Next we prove the following

Proposition 3.2.Ifλ∈L^∞(R)has a positive lower boundλ, then the problem(3.2)is exponentially stable and its solutions satisfy the decay estimates

u(t )

L²(R)e^−λtu₀L²(R) for allt0. (3.3)

Proof. Ifu₀∈H³(R), then repeating the proof of(2.2)and(3.1)withAreplaced byA_λwe obtain that d

dt ∞

−∞

|u|²dx= −2 ∞

−∞

|ux|²dx−2 ∞

−∞

λ|u|²dx

−2λ ∞

−∞

|u|²dx (3.4)

in(0,∞); this yields(3.3).

The estimate (3.3)remains valid for mild solutions, too. Indeed, for any given u0∈L²(R) we may choose a sequence(u0,n)⊂H³(R), converging tou0inL²(R). Then the corresponding strong solutionsunsatisfy the estimates

un(t)

L²(R)e^−λ0tu0,nL²(R)

for eachnandt0. Lettingn→ ∞this yields(3.3)becauseun(t)→u(t )inL²(R)for each fixedt0. 2 The preceding proposition ensures the exponential stability only for dampings which are effective on the whole real line. Our next result allows us to weaken this assumption. Set

cp:=

1− 1

2p 2

p _2p−¹₁

for 1p <∞.

Proposition 3.3.Letλ∈L^∞(R). If there exist a positive numberλ₀and a functionλ₁∈L^p(R)for some1p <∞ such that

λλ₀+λ₁ almost everywhere and

λ₁L^p(R)<

λ₀ c_p

1−_2p¹

,

then the problem(3.2)is exponentially stable and its solutions satisfy the decay estimates u(t )

L²(R)e^−λtu₀L²(R) for allt0 with

λ:=λ0−cpλ1¹_L^+p₍^2p_R¹⁻₎¹ >0.

In the sequel we often write · pinstead of · L^p(R).

Proof of Proposition 3.3. For λ₁=0 the proposition reduces to the preceding one. Henceforth we assume that λ₁L^p(R)>0. In the sequel all integrations take place onR. Hence we omit the integration limits±∞and we write · pinstead of · L^p(R)for brevity.

It suffices to establish the following estimate:

d

dtu²2−2λu²2. (3.5)

Taking the real parts and applying a density argument it suffices to consider real-valued smooth solutions. We recall from the preceding proof the following identity:

d

dt u²dx

= −2 u²_xdx+

λu²dx

. (3.6)

Using elementary estimates, Hölder and interpolation inequalities we have

−

u²_xdx−

λu²dx= −

u²_xdx−

(λ−λ₁)u²dx−

λ₁u²dx −u_x²₂−λ₀u²₂+ λ₁pu²_2q

−u_x²₂−λ₀u²₂+ λ₁pu₂^q2u_∞^p2, whereqsatisfy _p¹+_q¹=1.

Next we observe that

v²_∞2v2v_x2 (3.7)

for allv∈H¹(R). Indeed, ifv∈C_c^∞(R)andy∈R, then we have the following inequality:

v(y)²=

y

−∞

2vvxdx 2

∞

−∞

|v| · |v_x|dx2v2v_x2,

proving our estimate for smooth functions. The general case follows by density.

Using(3.7)and applying the Young inequality we deduce from the preceding inequality for any fixedε >0 that

−

u²_xdx−

λu²dx−ux²2−λ0u²2+2^1pλ1pu

2p−1 p

2 ux₂^1p

= −u_x²2−λ₀u²2+ 1

ελ₁pu₂^2p−1p ε2^p1u_x2^p1 −u_x²₂−λ₀u²₂+(¹_ελ₁pu₂^2p−1p )^2p2p−1

2p 2p−1

+(ε2^1pu_x2^p1)^2p

2p .

Choosingεsuch that 4ε^2p=2p, the termsu_x²₂eliminate each other and we obtain that

− u²_xdx+

λu²dx

u²₂

−λ₀+2p−1 2p

2 p

¹

2p−1

λ₁p^2p2p−1

= −λu²₂. Combining this with(3.6)we obtain(3.5). 2

Remarks 3.4.

• We observe that the functionλinProposition 3.3may have negative values.

• The proposition and its proof remains valid in the limiting casep= ∞under the conditionλ₁L^∞(R)< λ₀with λ:=λ₀− λ₁L^∞(R)>0 This is equivalent toProposition 3.2above.

4. Well-posedness for the nonlinear problem Since the global well-posedness of the problem

ut+uxxx−uxx+λu+uux=0 inR×(0,∞),

u(0)=u₀ inR, (4.1)

whenu₀∈L²(R), may be proved in a standard way, we only give a brief sketch.

First we consider the corresponding linear inhomogeneous initial value problem u_t+u_xxx−u_xx+λu=f inR×(0, T ),

u(0)=u₀ inR, (4.2)

for some given 0< T <∞. Setting

A:= −∂_x³+∂_x²−λI, D(A)=H³(R), it can be written in the form

u_t=Au+f, u(0)=u₀.

According to Section2,Agenerates a strongly continuous semigroup{S(t)}t0of contractions inL²(R). Hence for any givenu₀∈L²(R),T >0 andf ∈L¹(0, T;L²(R)),(4.2)has a unique mild solutionu∈C([0, T];L²(R)), given by the formula

u(t )=S(t)u0+ t 0

S(t−s)f (s) ds, t∈ [0, T], and depending continuously on the data:

uC([0,T];L²(R)):= sup

t∈[0,T]uL²(R)u₀L²(R)+ fL¹(0,T;L²(R)).

In fact, the solution of(4.2)has an additional space regularity. Let us introduce the Banach space B=BT :=C [0, T];L²(R)

∩L² 0, T;H¹(R)

with the norm

u_B= uC([0,T];L²(R))+ ∂_xuL²(0,T;L²(R)). We have the following

Proposition 4.1.Ifu0∈L²(R)andf ∈L¹(0, T;L²(R)), then the solutionuof (4.2)belongs toB, and u_Bc_T u₀L²(R)+ fL¹(0,T;L²(R))

withc_T =2e^Tλ∞.

Furthermore, the following energy identity holds for allt∈ [0, T]:

1 2u(t )²

L²(R)+ t 0

∂_xu(s)²

L²(R)ds+ t 0

R

λ(x)u(x, s)²dx ds

=1

2u0²_L2(R)+ t 0

R

f (x, s)u(x, s) dx ds.

Now we turn to the nonlinear problem. Let u₀∈L²(R). Motivated by the preceding considerations, by amild solutionof(4.1)we mean a functionu∈BT,T >0, satisfying

u(t )=S(t)u₀− t 0

S(t−s)u(s)∂_xu(s) ds, t∈ [0, T].

By aglobal mild solutionof(4.1)we mean a functionu: [0,∞)→H¹(R)whose restriction to every bounded interval [0, T]is a mild solution of(4.1).

We have the following

Theorem 4.2.For any givenu₀∈L²(R)the problem(4.1)has a unique global mild solution.

Furthermore, the following energy identity holds for allt0:

1 2u(t )²

L²(R)+ t 0

∂xu(s)²

L²(R)ds+ t 0

R

λ(x)u(x, s)²dx ds=1

2u0²_L2(R). (4.3)

For the proof we need a lemma (see[20, Proposition 4.1]):

Lemma 4.3.

(a) Ifu∈L²(0, T;H¹(R)), thenMu:=u∂xu∈L¹(0, T;L²(R)).

(b) If, moreover,u, v∈B, then

Mu−MvL¹(0,T;L²(R))√

2T^1/4 u_B+ v_B

u−v_B.

Following the ideas contained in[21], using this lemma we may establish the local well posedness using a fixed point argument:

Lemma 4.4.Givenu0∈L²(R), the problem(4.1)has a unique mild solution for every sufficiently smallT >0.

Furthermore, the solution satisfies the identity(4.3)for allt∈ [0, T].

Now the proof ofTheorem 4.2may be completed as follows. The uniqueness follows from the local uniqueness in Lemma 4.4. For the global existence we only have to show that the solution cannot blow up at any finite timeT.

It follows fromTheorem 4.2that, for each fixedT >0, the solution map

A:L²(R)→B, u0→u, (4.4)

whereu=Au₀is the corresponding solution of the problem(4.1), is well defined.

Next we investigate the well-posedness of the nonlinear problem for more regular data. We define for each any fixeds∈ [0,3]andT >0 the space

Bs,T:=C [0, T];H^s(R)

∩L² 0, T;H^s+1(R) endowed with norm

u_Bs,T := sup

t∈[0,T]

u(t )

H^s(R)+ uL²(0,T;H^s+1(R)).

In fact, using a method first introduced by Tartar[25]and adapted by Bona and Scott[26], we prove the following result

Theorem 4.5.LetT >0 andλ∈H¹(R). For everyu₀∈H^s(R),0s3, the nonlinear problem(4.1)admits a unique solutionu, which belongs to the classBs,T. Also, there exists a continuous functionC:R⁺×(0,∞)→R⁺, nondecreasing in its first variable, such that

u_Bs,T C u02, T

u0H^s(R).

To prove the above theorem we need first to present the cited method and we also need to prove some auxiliary results.

LetB0andB1be two Banach spaces such thatB1⊂B0with the inclusion map continuous. Letf ∈B0and, for t0, define

K(f, t )= inf

g∈B1

f−gB0+tgB1

.

For 0< θ <1 and 1p∞, define [B₀, B₁]θ,p=B_θ,p=

f ∈B₀: fθ,p:=

∞ 0

K(f, t )^pt^−θp−1dt ¹

p

<∞

with the usual modification for the casep= ∞. ThenB_θ,pis a Banach space with norm · θ,p. Given two pairs (θ₁, p₁)and(θ₂, p₂)as above, then(θ₁, p₁)≺(θ₂, p₂)means

θ₁< θ₂, or

θ₁=θ₂ andp₁> p₂.

If(θ1, p1)≺(θ2, p2)thenBθ₂,p₂⊂Bθ₁,p₁with the inclusion map continuous.

The interpolation result is the following.

Theorem 4.6.(See Bona and Scott[26, Theorem 4.3].) LetB₀^j andB₁^j be Banach spaces such thatB₁^j⊂B₀^j with continuous inclusion mappings,j =1,2. Let αandq lie in the ranges0< α <1and 1q∞. SupposeAis a mapping such that

(i) A:B_α,q¹ →B₀²and forf, g∈B_α,q¹ , Af −Ag_B²

0 C₀ fB¹_α,q+ gB_α,q¹

f−g_B¹

0

and

(ii) A:B₁¹→B₁²and forh∈B₁¹, Ah_B²

1C₁ hB_α,q¹

h_B¹

1,

whereC_j:R⁺→R⁺are continuous nondecreasing functions,j=0,1.

Then if(θ, p)(α, q),AmapsB_θ,p¹ intoB_θ,q² and forf ∈B_θ,p¹ we have Af_B²

θ,pC fB_α,q¹

f_B¹

θ,p, wherer >0,C(r)=4C0(4r)^1−θC₁(3r)^θ.

Using classical theory of linear semigroup and a contraction map theorem argument as presented in[26]we estab- lish the following existence theorem for the problem(4.1).

Theorem 4.7.LetT >0andλ∈H¹(R). For anyu₀∈H³(R)there exists a unique solutionuof (4.1)in the class B3,T withu_t ∈B0,T. Also, there exists a continuous function β₃:R⁺×(0,∞)→R⁺, nondecreasing in its first variable, such that

u_B3,T β3 u02, T

u0H³(R).

The proof of the previous result also requires an adaptation ofLemma 4.3as follows.

Lemma 4.8.LetT >0. For anyu, v∈B3,T such thatut, vt∈B0,T we have 1

2(uv)_x∈W^1,1 0, T;L²(R)

and 1

2(uv)_xx∈L² 0, T;L²(R) .

In addition, the following estimates hold 1

2(uv)_x

W^1,1(0,T;L²(R))

√

2T¹⁴ u_B3,Tv_B3,T+ u_B3,Tv_tB0,T + u_tB0,Tv_B3,T

and

1 2(uv)xx

L²(0,T;L²(R))

2³²T¹²u_B3,Tv_B3,T.

Proof of Theorem 4.7. LetT >0 be fixed,λ∈H¹(R)andu₀∈H³(R). For 0< θT andR >0 define Sθ,R:=

u∈B3,θsuch thatut∈B0,θandu_B3,θ + ut_B0,θR .

It follows from Lemma 4.8 that ¹₂(v²)_x∈W^1,1(0, θ;L²(R)) and ¹₂(v²)_xx ∈L²(0, θ;L²(R)). Therefore, using classical linear semigroup results, it follows that the linear inhomogeneous problem

⎧⎨

⎩

u_t+u_xxx−u_xx+λu=1 2 v²

x inR×(0, θ ),

u(0)=u₀ inR,

has a unique solutionuin the classB3,θsuch thatut is the solution of

⎧⎨

⎩

w_t+w_xxx−w_xx+λw=1 2 v²

xt inR×(0, θ ),

w(0)=w₀ inR,

withw₀:= −u_0xxx +u_0xx−λu₀+ ¹₂(v(0)²)_x∈L²(R)andu_t ∈B0,θ. Also, there exists a positive constant c_3,θ verifying

u_B3,θ+ u_tB0,θ c_3,θ

u₀H³(R)+ 1

2 v²

x

W^1,1(0,θ;L²(R))

+ 1

2 v²

xx

L²(0,θ;L²(R))

.

Therefore, the applicationΓ defined by v∈S_θ,R→Γ v=(u, u_t)∈B3,θ×B0,θ,

which associates to eachv∈S_θ,Rthe corresponding solutionsuandu_t, respectively. Also, usingLemma 4.8and the fact thatc_3,θc_3,T for everyθT, we prove

Γ v_B3,θ×B0,θc_3,Tu₀H³(R)+c_3,T2³² θ¹⁴+θ¹² R². DefiningR=2c3,Tu0H³(R)and choosing 0< θT such that

c_3,T2⁵² θ¹⁴ +θ¹² R1

2

we conclude thatΓ v_B3,θ×B0,θ R. For this choice ofRandθ, ifu, v∈Sθ,Rwe have Γ u−Γ v_B3,θ×B0,θc_3,θ

1

2 (u−v)(u+v)

x

W^1,1(0,θ;L²(R))

+ 1

2 (u−v)(u+v)

xx

L²(0,θ;L²(R))

c_3,T2³² θ¹⁴ +θ¹² u_B3,θ×B0,θ+ v_B3,θ×B0,θ

u− v_B3,θ×B0,θ c_3,T2⁵² θ¹⁴ +θ¹²

Ru− v_B3,θ×B0,θ

1

2u− v_B3,θ×B0,θ,

whereu=(u, u_t), which proves thatΓ is a contraction. So,Γ admits a unique fixed point inS_θ,R. In particular, we have

sup

t∈[0,θ]

u(t )

H³(R)+ sup

t∈[0,θ]

u_t(t)

L²(R)2c3,Tu0H³(R).

So, using standards arguments, we may extendθtoT. Finally, the proof is completed definingβ₃(s)=2c3,T for every s∈R⁺. 2

To proveTheorem 4.5we need a last result.

Proposition 4.9.The solution map(4.4)is locally Lipschitz continuous, i.e., there exists a continuous functionβ0: R⁺×(0,∞)→R⁺, nondecreasing in its first variable, such that for anyu0, v0∈L²(R)we have

Au₀−Av_0Bβ₀ u₀2+ v₀2, T

u₀−v₀2.

Proof. Let 0θT to be determined later. Foru0, v0∈L²(R)letAu0=uandAv0=v be the corresponding solutions of(4.1)on[0, T]. We denote

Bθ:=C [0, θ];L²(R)

∩L² 0, θ;H¹(R) .

The restriction ofuandvto[0, θ]belong toBθ. Also,u−vis the solution of the linear inhomogeneous problem (4.2)withf =vv_x−uu_x and initial datau0−v0. So, combiningProposition 4.1withLemma 4.3and applying Gronwall’s inequality to(4.3)we get

u−v_Bθcθu0−v02+c²_θ√

2θ¹⁴ u02+ v02

u−v_Bθ, wherec_θ=2e^θλ∞.

Since 0< θT, we havec_θc_T and choosingθverifying c³_T√

2θ¹⁴ u₀2+ v₀2

1 2, we obtain

u−v_Bθ 2cTu₀−v₀2. Suppose that 2θ < T and set

B2θ:=C [θ,2θ];L²(R)

∩L² θ,2θ;H¹(R) .

The restrictions ofuandvon[θ,2θ]are the solutions of(4.1)on[θ,2θ]with respect to the initial datau(θ )andv(θ ), respectively. Also, proceeding as before we have

u−v_B2θcTu(θ )−v(θ )

2+c²_T√

2θ¹⁴ u(θ )

2+v(θ )

2

u−v_B2θ. Sinceu(θ )−v(θ )2u−v_Bθ, we have

u(θ )−v(θ )

22cTu0−v02,

while applying Gronwall’s inequality to(4.3)provides u(θ )

2+v(θ )

2c_θ u02+ v02

.

Gathering these inequalities and observing thatc_θc_T we conclude u−v_B2θ2c²_Tu₀−v₀2+c_T³√

2θ¹⁴ u₀2+ v₀2

u−v_B2θ

and for the choice ofθwe finally have u−v_B2θ4c²_Tu₀−v₀2.

Proceeding similarly, using the number of steps needed to cover the interval [0, T], we conclude the desired re- sult. 2

Sketch of the proof of Theorem 4.5. LetT >0 andλ∈H¹(R). We define B₀¹=L²(R), B₀²=BT, B₁¹=H³(R), B₁²=BT ,3

and letAbe the solution map for the nonlinear problem(4.1). Choosingp=2 we have, for 0< s <3 andθ=^s₃, B_θ,p² =Bs,T, B_θ,p¹ =H^s(R).

Theorem 4.2together withProposition 4.9provide assumption (i) in the interpolation theorem, whileTheorem 4.7 provides assumption (ii). So, if we applyTheorem 4.6, we conclude the proof. 2

As a consequence, we have the following result which shows that every mild solution of(4.1)is a regular solution when not considering the origin.

Corollary 4.10. Under the assumptions of Theorem4.5, for anyu0∈L²(R), the corresponding solutionuof (4.1) belongs to

B3,[ε,T]=C [ε, T], H³(R)

∩L² ε, T;H⁴(R) for everyT >0and0< ε < T.

Proof. Let T >0 and 0< ε < T. So, for u₀∈L²(R), it follows fromTheorem 4.5 that the problem(4.1)has a unique solutionuin the classB0,T. So, we haveu(t )∈H¹(R)for almost everyt∈ [0, T]. Lett₀∈(0, ε)such that u(t₀)∈H¹(R). ApplyingTheorem 4.5withu₀=u(t₀), we conclude that the restriction ofuto[t₀, T]is the solution of(4.1), with respect to the initial datau(t₀), in the classB1,[t0,T]. So, we have thatu(t )∈H²(R)for almost every t∈ [t0, T]. Lett1∈(t0, ε)such thatu(t1)∈H²(R). It follows fromTheorem 4.5that the restriction ofuto[t1, T] is the solution of(4.1), with respect to the initial datau(t1), in the class B2,[t₁,T]. Finally, sinceu(t )∈H³(R)for almost everyt∈ [t₁, T], it follows fromTheorem 4.5that fort₂∈(t₁, ε)such thatu(t₂)∈H³(R), the restriction ofu to[t₂, T]is the solution of(4.1), with respect to the initial datau(t₂), in the classB3,[t₂,T]and the result follows. 2 5. Exponential decay for the nonlinear problem

The result established in this section is similar toProposition 3.3. Hence we only sketch the proof.

We recall the notation c_p:=

1− 1

2p 2

p ¹

2p−1

for 1p <∞.

Theorem 5.1.Letλ∈L^∞(R). If there exist a positive numberλ₀and a functionλ₁∈L^p(R)for some1p <∞ such that

λλ₀+λ₁ almost everywhere and

λ1L^p(R)<

λ₀ c_p

1−2p¹

,

then the problem(4.1)is exponentially stable and its solutions satisfy the decay estimates u(t )

L²(R)e^−λtu0L²(R) for allt0 with

λ:=λ₀−c_pλ₁¹_L⁺p(^2pR¹⁻)¹ >0.

Remark 5.2.The result and its proof are also valid forp= ∞withλ:=λ₀− λ₁L^∞(R), under the assumption that this valueλis positive.

Proof. We omit the variables as well as the integration limits of the spatial variable. We are going to prove the estimate for smooth solutions, the general case follows by density. It suffices to establish the inequality

d dt

|u|²dx−2λu²2. (5.1)

From the energy identity we have d

dt

|u|²dx= −2

|∂_xu|²dx−2

λ|u|²dx (5.2)

−2

|∂_xu|²dx−2λ0

|u|²dx+2

λ₁|u|²dx.

Using Hölder’s inequality and interpolation we obtain the following estimate:

λ₁|u|²dx2^1pλ₁pu₂^2p−12 ∂_xu₂^2p2 c_pλ₁

2p 2p−1

p u²₂+ ∂_xu²₂. (5.3)

Inequalities(5.3)and(5.2)imply(5.1)as desired. 2