UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação

UNIVERSIDADE DE SÃO PAULO

Instituto de Ciências Matemáticas e de Computação

ISSN 0103-2577

_______________________________

STRONGLY DAMPED WAVE EQUATION AND ITS YOSIDA APPROXIMATIONS

M.C. BORTOLAN A. N. CARVALHO

N^o 432

_______________________________

NOTAS DO ICMC

SÉRIE MATEMÁTICA

São Carlos – SP

Jan./2017

APPROXIMATIONS

M.C. BORTOLAN¹AND A.N. CARVALHO²

Abstract. In this work we study the continuity for the family of global attractors of the equations u_tt−∆u−∆u_t−∆u_tt =f(u) at = 0 when Ω is a bounded smooth domain ofRⁿ, withn>3, and the nonlinearityf satisfies a subcritical growth condition. Also, we obtain an uniform bound for the fractal dimension of these global attractors.

1. Introduction

We study the continuity of global attractors of the following semilinear evolution equation of second order in time

(1.1)















utt−∆u−∆ut−∆utt =f(u), t >0, (u(0), u_t(0)) = (u₀, v₀),

u|_∂Ω = 0,

and we give an uniform bound for the fractal dimension of these global attractors.

We know that, for = 0, this equation is the usual strongly damped wave equation, and its asymptotic dynamics - related to global atrtactors - has already been vastly explored; see for instance [6, 7, 9, 12, 15, 22, 23, 26, 27, 28]. However, for each > 0 fixed, we have a special form of the improved Boussinesq equation (see [4, 19, 20, 25]) with damping −∆u_t, which, among other things, is used to describe ion-sound waves in plasma (see [20, 21]).

For each >0 fixed, this equation has been studied in[8], in terms of existence and uniqueness of solutions, existence of global attractors and asymptotic bootstrapping; in this case, the linear part of the equation (after a change of variables) is a bounded operator. Here, since we want to study the continuity of attractors at = 0, we will use the properties of the limiting problem with = 0 (local and global well posedness, regularity and existence of global attractors) as reported in [6, 7].

1Partially supported by FAPESP 2012/23724-1. ²Partially supported by CNPq 305230/2011-5.

1

Throughout this paper, we will assume that f :R→Ris a continuously differentiable function, respecting a growth condition with subcritical exponent; that is, there exist constants c >0 and ρ < ⁿ⁺²_n−2 such that for alls1, s2 ∈R

(1.2) |f(s1)−f(s2)|6c|s₁−s2|(1 +|s₁|^ρ−1+|s₂|^ρ−1),

and also, if λ1 denotes the first eigenvalue of −∆ with Dirichlet boundary conditions in Ω, we assume the following dissipation condition

(1.3) lim sup

|s|→∞

f(s) s < λ₁.

To begin our study, we will write further A for−∆ with the Dirichlet boundary conditions. Our problem then takes the form

(1.4)







u_tt+Au+Au_t+Au_tt =f(u), t >0 (u(0), v(0)) = (u0, v0).

and it is well-known thatA:H₀¹(Ω)∩H²(Ω)⊂L²(Ω)−→L²(Ω) is a closed, densely define operator which has the following properties:

(O1) A is self-adjoint with compact resolvent;

(O2) A is an operator of positive type;

(O3) σ(A) = σ_p(A) ={λ_n}_n∈N,λ₁ >0, λ_i ≤λ_i+1, for all i>1 (repeated to take into account the multiplicity),λ_n^n→∞→ ∞and ifv_n∈L²(Ω) are unitary eigenvectors associated withλ_n then{v_n}_n∈N constitutes an orthonormal basis forL²(Ω).

Remark 1.1. We included in Appendix A the proof of the main results of functional analysis we will use, in order to make explicit the uniformity of the constants obtained for ∈[0,1].

The key point in our analysis is the observation that the differential equation in(1.4), for >0, can be obtained from its limit, for = 0, with a suitable exchange of the unbounded operator A by itsYosida approximation Λ (see definition below). The techniques developed here to deal with these singular perturbation problem may be of aid to deal with other natural singular perturbation problems that appear in the literature in this form (see for example the Navier-Stokes-Voight problem in [14]).

Definition 1.2. Let Abe a closed, densely defined operator such thatR⁺⊂ρ(−A). Then, for each ∈[0,1]we define the operator Λ :D(Λ)⊂X→X, given by

D(Λ) ={x∈X: (I+A)⁻¹x∈D(A)}, and for x∈D(Λ) we set

Λx=A(I +A)⁻¹x.

The operators Λ are called Yosida approximationsof A.

In fact the differential equation in (1.4) can be rewritten asutt+ Λu+ Λut= (I +A)⁻¹f(u) with Λu0

−→→0 Au0 for all u0 ∈ D(A) and (I +A)⁻¹u0

−→→0 u0 for all u0 ∈ X. We exploit this feature and a suitable change of variables to fix (independently of ) the phase space to carry on our analysis.

Now, if X .

=L²(Ω), we will consider the double sided fractional power scales

• {X^α, α∈R}, generated by (X, A);

• {X^α, α∈R}_∈[0,1], generated by (X,Λ) (seeDefinition 1.2);

• {X˜^α, α∈R}_∈[0,1] generated by (X, I+A);

whereA, Λ andI+Ahave domainsX¹,X¹and ˜X¹, respectively, and are positive type operators.

Now we consider the following isometric isomorphism Φ :X¹² ×X˜

1

2 −→X×X given in its matrix form by

Φ =





A¹² 0 0 (I+A)¹²



, for each ∈[0,1].

If we apply the change of variables [^w_z] = Φ[u^ut], problem(1.4)can be rewritten as

(1.5)















(I+A)¹²z_t+A¹²w+A(I+A)⁻¹²z=f(A⁻¹²w) wt=A¹²(I+A)⁻¹²z,

(w(0), z(0)) = (A¹²u₀,(I+A)¹²v₀)

or

(1.6)















zt+A¹²(I+A)⁻¹²w+A(I+A)⁻¹z= (I+A)⁻¹²f(A⁻¹²w) w_t=A¹²(I+A)⁻¹²z,

(w(0), z(0)) = (A¹²u0,(I+A)¹²v0).

The later is a first order ODE that can writen in X×X as (1.7)





 d

dt[^w_z] +A[^w_z] =F([^w_z]) in [0,∞) (w(0), z(0)) = (w₀, z₀),

where (w₀, z₀) = Φ(u₀, v₀), in variables (t, w, z), where A:D(A)⊂X×X →X×X is a linear operator given by

D(A) = ("

w z

#

∈X×X

1

2 : w+ Λ

1

2z∈X

1

2

) , and

A

"

w z

#

=





−Λ

1

2z Λ

1

2(w+ Λ

1

2z)



. Of course, if [^w_z]∈X

1

2 ×X¹ we have that A

"

w z

#

=





−Λ

1

2z Λ

1

2w+ Λz



=





0 −Λ

1

2

Λ

1

2 Λ





"

w z

# ,

with X

1

2 ×X¹ being a dense subset ofD(A) and a locally Lipschitz map

(1.8) F([^w_z]) =h

0 f^e(w)

i , where f^e(w) = (I+A)⁻¹²f(A⁻¹²w).

Remark 1.3. It is important to notice that for each >0, D(A) =X×X and A∈ L(X×X).

The characterization above becomes important when dealing with the case = 0, since A₀ is an unbounded operator. The primary concern of our work is to deal with the uniformity in ∈ [0,1]

of the class of problems (1.4), hence placing the problems under the same framework is crucial.

We divide our work from now on in six sections and an appendix. InSection 2we deal with the linear problem associated with equation (1.7). More specifically, we prove that −A generates an analytic semigroup {e^−A : t>0}, and we obtain convergence in the uniform norm of operators of the associated semigroups when→0⁺ as follows:

Theorem 1.4. For any α∈[0,¹₂) and γ ∈[0,1]there exists a constant C_γ >0 such that ke^−At−e^−A0tk_L(X×X)6C_γ^αγt^−γe^−ω1t,

for all t > 0. In particular, e^{−At L(X×X)}−→ e^−A0t as → 0⁺, with uniform convergence for any interval [T,∞), T >0.

InSection 3we prove local and global well posedness results for equation(1.1)and we deal with all the cases at once. For each > 0, these results are contained in Theorems 1.1 and 1.2 of [8]

as for the case = 0 these results are contained in the results of Section 3 of [6]. To this end, a fine analysis of the fractional powers of the operators −A is required (such analysis is done in Subsection 2.2). The main results of this section can be summarized in the results below:

Theorem 1.5. For any initial data [^u_v0⁰] lying in a bounded subset B of X¹² ×X˜

1

2 there exists a number κ = κ(B, ) and a unique solution [0, κ) 3 t 7→ [^uv] (t, u0, v0) ∈ X¹² ×X˜

1

2 of (1.4) which depends continuously on its variables (t, u0, v0) ∈ [0, κ)×X¹² ×X˜

1

2 and such that, for any s∈h_{(ρ−1)(n−2)}

4 ,1

i

andγ ∈ 0,1−^s₂ ,

[^uv] (·, u₀, z0)∈C

(0, τ),(X¹² ×X˜

1

2)^γ

∩C¹

(0, τ),(X¹² ×X˜

1

2)^γ−

, and either κ=∞ or k[^uv] (t, u0, v0)k

X¹²×X˜

1 2

→ ∞ as t→κ⁻. Moreover, the solution satisfies in X¹² ×X˜

1

2 the variation of constants formula [^u_v] (t, w₀, z₀) =e^−A˜t[^u_v0⁰] +

Z t 0

e^{−A˜(t−s)}G([^u_v] (s, u₀, v₀))ds, t∈[0, κ), where

G([^u_v]) = Φ⁻¹_0,FΦ_0,([^u_v]).

Theorem 1.6. Problem (1.1) defines a C⁰-semigroup {S(t) : t > 0} on X¹² ×X˜

1

2 for each ∈[0,1], which has bounded orbits of bounded sets, defined by

S(t) [^uv⁰0] = Φ⁻¹ T(t)Φ[^uv⁰0], or equivalently

S(t) [^u_v0⁰] =e^−A˜t[^u_v0⁰] + Z t

0

e^{−A˜(t−s)}G(S(s) [^u_v0⁰])ds, for allt>0.

In Section 4 we prove the existence of global attractors for the semigroups {S(t) : t > 0}

generated by equations (1.1), which is given by

Theorem 1.7. The semigroup {S(t) : t > 0} has a global attractor A˜ in X¹² ×X˜

1

2, for each ∈[0,1].

In[8]the authors prove the existence of global attractors for each >0 and also provide bounds (dependent) for the global attractors (see Theorem 1.3 of this reference). In [9] they prove the same for the case= 0 (see the results of Subsection 4.2 in this reference); however, simply joining the results would not lead to a uniform bound for∈[0,1]. We also prove the following

Theorem 1.8. If s∈h

0,1−^{(ρ−1)(n−2)}₄

, then ∪_∈[0,1]A˜ is bounded in X^s+12 ×X^s2.

InSection 5 we are able to prove the upper semicontinuity of the global attractors{A˜}_∈[0,1] at = 0:

Theorem 1.9. The family {A˜}_∈[0,1] is upper semicontinuous in= 0, in X¹² ×X.

This result was also proven in[24], using a different technique, dealing with energy estimates of solutions (see Lemma 5.12 in this reference). Under (natural) additional assumptions we can also prove the lower semicontinuity

Theorem 1.10. Assume that f is a C² function on R with f, f⁰ and f⁰⁰ bounded in R. Also, assume that the set E of equilibrium points of(1.7)is finite and that each point of E is a hyperbolic point for (1.7) with = 0. Then the family of global attractors {A˜}_∈[0,1] is lower semicontinuous at = 0.

Lastly, inSection 6using some further uniform estimates for the semigroup generated by equation (1.7)we obtain an uniform estimate for the fractal dimension c( ˜A) of the global attractor ˜A. Theorem 1.11. There exists a number τ₀ >0 such that for any ∈[0,1]

c( ˜A)6τ0.

In [24, Lemma 5.10], the authors prove an estimate for the fractal dimension of the global attractors using exponential attractors, but the bound depends on∈[0,1].

Remark 1.12. We note that, most of our results are proved using techniques from functional anal- ysis, resorting to energy estimates when is absolutely necessary. We were able to obtain some fine estimates using a bootstrapping argument in the subcritical case. This equation has been considered

in [11], where they proved the upper semicontinuity of the global attractors of (1.1)as well as to ob- tain bounds for the fractal dimension of the attractors, but is not uniform in∈[0,1]. Here we also prove the lower semicontinuity of the global attractors, besides recovering the upper semicontinuity and obtaining uniform (w.r.t. ) bounds for the dimension using a different technique.

2. The linear problem and the uniform convergence of the linear semigroups In this section we study the linear problem associated with equations(1.7)inX×X, given by





 d

dt[^w_z] +A[^w_z] = 0, t >0 h_w(0)

z(0)

i

= [^wz0⁰]∈X×X

more precisely, we will prove that the family of operators{A}_∈[0,1]is uniformly sectorial; that is, we can find φ∈(0,^π₂),M >1 and a real numberω such that the sector

S_ω,φ={λ∈C:φ6|arg(λ−ω)|6π, λ6=ω}

is in the resolvent set of A for all ∈[0,1] and k(λ− A)⁻¹k_L(X×X)6 M

|λ−ω|, for all λ∈Sω,φ,

and moreover we will prove that we can take ω < 0, which will give us an uniform exponential decay for the generated analytic semigroups {e^−At: t>0}_∈[0,1].

2.1. Uniform sectoriality. In this subsection, our goal is to prove the uniform sectoriality of {A}_∈[0,1]in order to obtain a convergence of the generated linear semigroups{e^−At: t>0}_∈[0,1]

as→0⁺.

First we begin obtaining an uniform decay in time for the generated semigroups, and to this purpose we define the notations of the inner products we will use throughout our work.

Definition 2.1. In X we denote the usual inner product (·,·) and in X×X we use the inner product h·,·i given by

h[^w_z1¹],[^w_z2²]i .

= (w₁, w₂) + (z₁, z₂).

With this notation set, we are able define for each pair (, β)∈[0,1]×[0,1], a map from (X×X)² intoC by

h[^w_z1¹],[^wz2²]i_,β =h[^w_z1¹],[^wz2²]i+β

2(w1,Λ⁻

1

2z2) +β

2(z1,Λ⁻

1

2w2).

In what follows we will need a result of basic functional analysis, that we state below.

Proposition 2.2. The family of operators{Λ^−β }(,β)∈[0,1]×[0,1]is uniformly bounded. In particular, there exists a constant µ >0 such that

(Λ^βx, x)>µkxk²_X, for all x∈X¹. Proof: See Appendix A.

With this result we can prove a uniform equivalence between h·,·i_,β and h·,·i.

Proposition 2.3. For all (, β)∈[0,1]×[0,1], if we define k[^w_z]k²_,β .

=h[^w_z],[^w_z]i_,β, we have

1−βµ 2

k[^w_z]k²_X×X 6k[^w_z]k²_,β 6

1 +βµ 2

k[^w_z]k²_X×X. Proof: We have, since Λ⁻

1

2 is self-adjoint, that h[^w_z],[^w_z]i_,β=h[^w_z],[^w_z]i+βRe(w,Λ⁻

1

2z) =k[^w_z]k²_X×X +βRe(w,Λ⁻

1

2z), but

|Re(w,Λ⁻

1

2z)|6|(w,Λ⁻

1

2z)|6kwk_XkΛ⁻

1

2zk_X 6kΛ⁻

1

2k_L(X)kwk_Xkzk_X 6 kΛ⁻

1

2k_L(X)

2 k[^w_z]k²_X×X, By Proposition 2.2,kΛ⁻

1

2k_L(X) 6µand hence

|Re(w,Λ⁻

1

2z)|6 µ

2 k[^w_z]k²_X×X, which concludes the proof.

Corollary 2.4. There exists β0 ∈ (0,1] such that h·,·i_,β is an inner product in X ×X for all (, β)∈[0,1]×[0, β0].

Proof: Almost all the properties of an inner product are easily verified; and for the coercivity it suffices to choose β₀ ∈(0,1] such that 1−^β0₂^µ >0 in the previous proposition.

So far we are able to construct uniform equivalent norms inX×X and the next step is to prove that there exists a positive constant δ >0 such thatA−δI is acretive, for all ∈[0,1].

Proposition 2.5. There exist β₁ ∈(0, β₀] and a constant δ >0 such that Reh(A−δI) [^w_z],[^w_z]i_,β

1 >0, for all ∈[0,1] and [^w_z]∈D(A).

Proof: We have for [^w_z]∈X

1

2 ×X¹ that hA[^w_z],[^w_z]i_β =

−Λ

1 2z Λ

1 2w+Λz

,[^w_z]

,β

=−(Λ

1

2z, w) + (Λ

1

2w, z) + (Λz, z)−β

2(z, z) +β

2(w, w) +β 2(Λ

1

2w, z), which implies, since Λ

1

2 is self-adjoint, that RehA[^w_z],[^w_z]i=kΛ

1

2zk²_X −β

2kzk_X+ β

2kwk²_X +β 2Re(Λ

1

2z, w).

But |Re(Λ

1

2z, w)|6 ¹₂(kΛ

1

2zk²_X +kwk²_X) and hence RehA[^w_z],[^w_z]i>(1−β

4)kΛ

1

2zk²_X −β

2kzk²_X+β 4kwk²_X

>

(1− β

4)µ²− β 2

kzk²_X+β 4kwk²_X.

Now we choose β1∈(0, β0] such that (1−^β₂¹)µ²−^β₂¹ >0 and thus, by Proposition 2.3, we have RehA[^w_z],[^w_z]i_,β

1 >δh[^w_z],[^w_z]i_,β

1, where δ= (1 + ^β1₂^µ1)⁻¹min{(1−^β₂¹)µ²₁−^β₂¹,^β₄}>0, and therefore

Reh(A−δI) [^w_z],[^w_z]i_,β

1 >0.

From this we conclude that each operatorδI− A generates a strongly continuous semigroup of contractions in X×X with the normk · k_,β1, which in turn, usingProposition 2.3, lead us to the following result:

Theorem 2.6. There exist constants M >1 andδ >0, such that

ke^−Atk_L(X×X)6M e^−δt, for allt>0 and ∈[0,1].

Proof: Since

e^(δI−A)t[^w_z]

,β1 6k[^w_z]k_,β

1, Proposition 2.3 imples that there existsM >1 such that

e^(δI−A)t[^w_z]

X×X 6Mk[^w_z]k_X×X, and concludes the proof.

Corollary 2.7. Given ϕ∈(^π₂, π), there exists a constant Mϕ >1 such that k(λ− A)⁻¹k_L(X×X)6 M_ϕ

|λ−δ|, for allλ∈S_δ,ϕ and∈[0,1].

Proof: From the Inverse Laplace Transform, we know that (λ− A)⁻¹ =−

Z ∞ 0

e^λte^−Atdt, for all λ∈Csuch that Reλ < δand therefore

k(λ− A)⁻¹k_L(X×X) 6 M δ−Reλ, for all ∈[0,1]. Now, givenϕ∈(π/2, π), we have that

k(λ− A)⁻¹k_L(X×X)6 M

|cosϕ|

1

|λ−δ|, for all λ∈Sδ,ϕ and ∈[0,1].

So far we have proven that each−Agenerates a strongly-continuous semigroup inX×X (which for >0 is trivial, sinceAis bounded inX×X) and furthermore we proved an uniform exponential decay for the generated semigroups for ∈ [0,1]. But we would like to prove the convergence of e^−At toe^−A0t inL(X×X) as →0⁺, and to this purpose, we will need to work a little more.

For ∈[0,1] define D(B) =D(A), D(P) =X×X and B .

=A+_{0 0}

0I

,P .

=

h _{I 0}

Λ⁻

1 2 I

i , so that

P⁻¹ =h _{I 0}

−Λ⁻

1 2 I

i . Also, if we setD(D) =

[^w_z]∈X×X:P⁻¹[^w_z]∈D(B) , we can define D .

=PBP⁻¹.

Remark 2.8. It is simple to see that D(D) =X×X¹ and hence D=h

I−Λ

1 2

0 Λ

i . Finally, defineD( ˜D) =D(D) =X×X¹ and

D˜ .

=_{I 0}

0 Λ

.

For what follows we will need the definition and one result concerning the numerical range of an operator, which are given below.

Definition 2.9. If B : D(B) ⊂ Z → Z is a closed densely defined operator in a complex Hilbert space Z with inner product h·,·i, then the numerical range W(B) of B is the set

W(B) ={hBz, zi: z∈D(B), kzk_Z = 1}.

Theorem 2.10. Let B:D(B)⊂Z →Z be a closed densely defined operator in a complex Hilbert space Z, W(B) its numerical range and Σ an open connected set in C\W(B). If Σ∩ρ(B) 6= ∅ then Σ⊂ρ(B) and

k(λ−B)⁻¹k_L(Z)6 1

d(λ, W(B)), for allλ∈Σ, where d(λ, W(B)) is the distance betweenλ and W(B).

Proof: See Theorem 21.11 of [3].

With this result at hand, we can prove our first lemma.

Lemma 2.11. The operators D˜ : D( ˜D) ⊂ X×X −→ X×X constitute a family of uniformly sectorial operators.

Proof: Using again Proposition 2.2, there existsµ >0 such that for all z∈X¹ we have (Λz, z)>µ(z, z).

Thus, for [^w_z]∈D( ˜D) we obtain DD˜[^w_z],[^w_z]

E

= (w, w) + (Λz, z)>(w, w) +µ(z, z)>µ˜h[^w_z],[^w_z]i,

where ˜µ= min{1, µ} >0 and therefore the numerical imageW( ˜D) is contained in [˜µ,∞), for all ∈ [0,1]. Defining Σ .

= C\[˜µ,∞) we have that 0 ∈ Σ∩ρ( ˜D) for all ∈ [0,1] and hence, by Theorem 2.10, Σ⊂ρ( ˜D), for all ∈[0,1], and

k(λ−D˜)⁻¹k_L(X×X)6 1

d(λ, W( ˜D)) 6 1

d(λ,[˜µ,∞)), for allλ∈Σ.

Now givenφ∈(0, π/2), ifλ∈S_µ,φ˜ we have that

d(λ,[˜µ,∞))>|λ−µ|˜ sinφ, and hence

k(λ−D˜)⁻¹k_L(X×X)6 1

sinφ|λ−µ|˜ , for all λ∈S_µ,φ˜ and ∈[0,1].

To continue, we will need well know results in functional analysis, concerning interpolation of fractional powers of an operator, which we will state below.

Definition 2.12. Let C >1. A closed densely defined linear operatorB :D(B) ⊂Z → Z is said an operator of positive type with constantC if [0,∞)∈ρ(−B) and

(1 +s)k(s+B)⁻¹k_L(Z)6C, for alls∈[0,∞).

The set of all operators of positive type in Z with constantC will be denoted by PC(Z).

Proposition 2.13. Let A be an operator of positive type with constant C in X, then the Yosida approximations Λ of A are positive type operators with constant 1 +C, for all ∈[0,1].

Proof: See Appendix A.

Theorem 2.14. Assume that B∈PC(Z) and06α61, then there exists a constantK >0 such that

kB^αzk_Z 6KkBzk^α_Zkzk^1−α_Z , for allz∈D(B);

moreover, the constant K depends only on the constant C and not on the particular operator B.

Proof: See Theorem 1.4.4 of[17].

Corollary 2.15. There exists a constant K >0, independent of ∈[0,1], such that if 06α61 we have

kΛ^αxk_X 6KkΛxk^α_Xkxk^1−α_X , for allx∈X¹.

Lemma 2.16. The operators D : D(D) ⊂ X×X −→ X×X constitute a family of uniformly sectorial operators.

Proof: We have that ˜D− D= h

0 Λ

12

0 0

i

, and hence for all [^w_z]∈X×X¹ k( ˜D− D) [^w_z]k_X×X =kΛ

1

2zk_X 6KkΛzk

1 2

Xkzk

1 2

X

6 Kη

2 kΛzk_X + K 2ηkzk_X,

for allη >0, whereK >0 is the constant given in Theorem 2.14, which is independent of ∈[0,1]

and therefore

k( ˜D− D) [^w_z]k_X×X 6 Kη

2 kD˜[^w_z]k_X×X + K

2ηk[^w_z]k_X×X.

By Theorem 1.3.2 of[17]andLemma 2.11we have that the family{D}_∈[0,1]is uniformly sectorial.

Lemma 2.17. The operators B : D(B) ⊂ X×X −→ X×X constitute a family of uniformly sectorial operators.

Proof: We have for all λ∈Cthat

(λ− D) =P(λ− B)P⁻¹,

hence ρ(B) = ρ(D), and since the operators P,P⁻¹ are uniformly bounded in X ×X (see Proposition 2.2), Lemma 2.16implies that{B}_∈[0,1] is uniformly sectorial.

Theorem 2.18. The operators A:D(A)⊂X×X −→X×X constitute a family of uniformly sectorial operators.

Proof: Since

_{0 0}

0I

[^w_z]

X×X 6ηkB[^w_z]k_X×X +k[^w_z]k_X×X,

for allη >0,Lemma 2.17and Theorem 1.3.2 of[17]imply that {A}_∈[0,1] is uniformly sectorial.

So far, with our efforts, Theorem 2.18 implies the existence of constants M > 1, ω ∈ R and φ∈(0, π/2) such that

k(λ− A)⁻¹k_L(X×X)6 M

|λ−ω|, for all λ∈S_ω,φ and ∈[0,1],

but ω ∈R can be a negative real number (and using the results reported in[17], we can see that the numberω∈Robtained is, in fact, negative), which does not guarantee an uniform exponential decay for the generated semigroups. But these results together withCorollary 2.7give us conditions to obtain the desired uniform sectoriality of {A}_∈[0,1] with a uniform exponential decay:

Theorem 2.19. There exist constantsM >1, ω >0 andϕ∈(0, π/2)such that ρ(A)⊃S_ω,ϕ and k(λ− A)⁻¹k_L(X×X)6 M

|λ−ω|, for all λ∈S_ω,ϕ and ∈[0,1].

Proof: This follows from Corollary 2.7and Theorem 2.18.

Corollary 2.20. −A is the infinitesimal generator of an analytic semigroup {e^−At: t>0} for each ∈[0,1]and

e^−At= 1 2πi

Z

Γ

e^λt(λ+A)⁻¹dλ, for all∈[0,1],

where Γ is a contour in−S_δ,ω such thatarg(λ)→ ±θ as |λ| → ∞ for some θ∈(^π₂, π).

Proof: See Theorem 1.3.4 of [17].

Corollary 2.21. Given ω₁ ∈(0, ω), there exists constant M_ω1 >1 such that kA(λ− A)⁻¹k_L(X×X)6Mω1, for all λ∈Sω1,ϕ and ∈[0,1].

Proof: We have that A(λ− A)⁻¹ =λ(λ− A)⁻¹−I and hence

kA(λ− A)⁻¹k_L(X×X)6|λ|k(λ− A)⁻¹k_L(X×X)+ 16M |λ|

|λ−ω|+ 1.

Now, for eachω1∈(0, ω), the map Sω1,ϕ3λ7→ _λ−ω^λ is bounded, hence there existsMω1 such that kA(λ− A)⁻¹k_L(X×X) 6M_ω1, for all λ∈S_ω1,ϕ.

To obtain the uniform convergence of resolvents, forλin a sector ofC, we will need the following result:

Proposition 2.22. If A is a positive type operator and Λ its Yosida approximation then, for all α∈[0,¹₂),

kΛ^−1/2 −Λ^−1/2₀ k_L(X)6C^α. Proof: See Appendix A.

With this result and Corollary 2.21 we can prove:

Corollary 2.23. Given ω1 ∈ (0, ω) we have that (λ− A)^{−1 L(X×X}−→ ⁾ (λ− A₀)⁻¹ as → 0⁺, uniformly for λ∈S_ω1,ϕ.

Proof: We have that

A⁻¹ − A⁻¹₀ =





0 Λ⁻

1

2 −Λ⁻

1 2

0

Λ⁻

1 2

0 −Λ⁻

1

2 0



, and

(λ− A)⁻¹−(λ− A₀)⁻¹ =A(λ− A)⁻¹(A⁻¹ − A⁻¹₀ )A₀(λ− A₀)⁻¹.

ThereforeProposition 2.22 and Corollary 2.21 we have, givenω1 ∈(0, ω) andα∈[0,¹₂), that k(λ− A)⁻¹−(λ− A₀)⁻¹k_L(X×X)6M_ω²₁C^α.

Remark 2.24. If Ais the negative Laplacian with Dirichlet boundary conditions, then we can take α= ¹₂ (see Remark A.1).

Let w₁ ∈ (0, ω). Given r > 0, Corollary 2.20 implies that we can choose the curve Γ given by Γ = Γ1∪Γr∪Γ1, where

Γ1 ={λ∈C:λ=−ω₁+se^i(π−ϕ), s>r}, Γr={λ∈C:λ=−ω₁+re^iξ, ξ∈[π−ϕ, ϕ−π]}, such that

e^−At= 1 2πi

Z

Γ

e^λt(λ+A)⁻¹dλ, for all ∈[0,1] andt >0.

Proof of Theorem 1.4:We have that e^−At−e^−A0t= 1 2πi

Z

Γ

e^λt

(λ+A)⁻¹−(λ+A₀)⁻¹ dλ,

thus, if α∈[0,¹₂), then

ke^−At−e^−A0tk_L(X×X)6 C^α 2π

Z

Γ

e^Reλt|dλ|

= C^α 2π e^−ω1t

2

Z ∞ r

e^−stcosϕds+r Z π−ϕ

ϕ−π

e^rtcosξdξ

6 C^α 2π e^−ω1t

2e^−rtcosϕ

tcosϕ + 2r(π−ϕ)e^rt

6 C^α π

e^−ω1t

tcosϕ+ C^α

π e^−ω1tr(π−ϕ), for any 0< r < ω₁ and therefore makingr→0⁺, we obtain

ke^−At−e^−A0tk_L(X×X)6 C

πcosϕ^αt⁻¹e^−ω1t. But ke^−At−e^−A0tk_L(X×X)62M e^−ω1t and hence, for γ ∈[0,1] we have

ke^−At−e^−A0tk_L(X×X)6(2M)^1−γ C

πcosϕ γ

^αγt^−γe^−ω1t.

Remark 2.25. Again, if A is the negative Laplacian with Dirichlet boundary conditions, we can take α= ¹₂.

2.2. Fractional powers of A. In this subsection we are interested in some properties of the fractional powers of the operators A. We know that for >0 we are always working withX×X with an equivalent norm, but again, we are concerned about the uniformity in ∈ [0,1] for the problems(1.7), and it will be useful to have some additional properties of the fractional powers of A.

Proposition 2.26. A is a positive type operator for some constant C>1.

Proof: We know thatδI− A is dissipative inX×X with the normk · k_,β1, byProposition 2.5, and ρ(δI− A)∩(0,∞)6=∅, and thus by Lumer’s Theorem, we have

k(λ+ (A−δI))⁻¹k_{L(X×X),k·k,β}

1 6 1

λ, for allλ >0.

Therefore, if µ=λ−δ,

k(µ+A)⁻¹k_L(X×X),k·k

,β1 6 1

µ+δ, for all µ >−δ, and thus if µ >0 we have that

(1 +µ)k(µ+A)⁻¹k_{L(X×X),k·k,β}

1 6 µ+ 1 µ+δ,

and since the map [0,∞)3µ7→ ^µ+1_µ+δ is bounded and the norms k · k_,β1 andk · k_X×X are uniformly equivalent, the result follows.

Now, for τ ∈[0,∞), we have that s∈ρ(−A)

(τ +A)⁻¹= 1 τ + 1





τ + Λ Λ^1/2

−Λ^1/2 τ



 τ²

τ+ 1+ Λ

−1

, and hence for α∈(0,1) we have (see Theorem 1.4.2 of[17]) that

A^−α = sinπα π

Z ∞ 0

τ^−α τ + 1





τ + Λ Λ^1/2

−Λ^1/2 τ



 τ²

τ+ 1+ Λ ⁻¹

dτ.

If we set

A^−α =





P_1,1(, α) P_1,2(, α) P_2,1(, α) P_2,2(, α)



,

we have that

P_1,1(, α) =sinπα π

Z ∞ 0

τ^−α

τ+ 1(τ + Λ) τ²

τ + 1+ Λ −1

dτ, P_1,2(, α) =sinπα

π

Z ∞ 0

τ^−α τ+ 1Λ

1

2

τ² τ + 1+ Λ

−1

dτ, P2,1(, α) =−B_1,2(, α),

P2,2(, α) =sinπα π

Z ∞ 0

τ^−α+1 τ + 1

τ² τ + 1+ Λ

−1

dτ.

To continue, we will need the following result.

Proposition 2.27. IfA is a positive type operator with constantC then there exists a constantC1

such that, for any β ∈(0,1) and ∈[0,1]

kΛ^β(µ+ Λ)⁻¹k_L(X) 6 C1

(µ+ 1)^1−β, for all µ>0.

Proof: See Appendix A.

And now we can state our result for the fraciontal powers of A.

Proposition 2.28. For each β ∈(0,¹₂) andα∈(β,1), the operatorsΛ^βP1,2(, α) and Λ^βP2,2(, α) are uniformly bounded for ∈[0,1].

Proof: For P_1,2(, α) we have that Λ^βP1,2(, α) = sinπα

π

Z ∞ 0

τ^−α τ+ 1Λ

1 2+β

τ² τ + 1+ Λ

−1

dτ, and thus by Proposition 2.27we have

kΛ^βP_1,2(, α)k_L(X)6 C1sinπα π

Z ∞ 0

τ^−α τ+ 1

τ + 1 τ²+τ + 1

¹₂−β

dτ 6 C1sinπα π

Z ∞ 0

τ^−α τ + 1dτ, and the integral on the right side is convergent, for any α∈(0,1).

For P2,2(, α) we have that

Λ^βP_2,2(, α) = sinπα π

Z ∞ 0

τ^−α+1 τ + 1Λ^β

τ² τ + 1+ Λ

−1

dτ, and thus by Proposition 2.27we have

kΛ^βP_2,2(, α)k_L(X)6 C1sinπα π

Z ∞ 0

τ^−α+1 τ + 1

τ + 1 τ²+τ + 1

1−β

dτ, and the integral on the right side is convergent, provided that α∈(β,1).

3. Local and global well posedness results

3.1. Local well posedness result. To state the results of local well posedness of equations(1.7), and consequently of (1.4), we firstly prove the auxiliary lemma below.

Lemma 3.1. Let f :R→ R, and A is the negative Dirichlet Laplacian in X with domain X¹ = H²(Ω)∩H₀¹(Ω) and consider its closed extension to H^−r = (X^r2)⁰, where Y⁰ represents the dual space of the Banach space Y, (in particular,H⁻¹ =H₀¹(Ω)⁰). Then

f^e(φ)(x) =f(φ(x)), x∈Ω,

defines an operator from X^s2 intoH^−r which is Lipschitz continuous in bounded sets provided that condition(1.2)holds and r∈h_{(ρ−1)(n−2)}

4 ,1i

, s∈[r,1]∩h

n

2 −_ρ−1² ,1i

. If in addition,r can be taken strictly less than 1, then f^e takes bounded sets of X^s2 into relatively compact sets of H⁻¹.

Proof: LetBbe a bounded set inX^s2 and choose arbitraryφ1, φ2∈B. Since condition (1.2) holds we use the Sobolev and H¨older inequalities to get

kf^e(φ1)−f^e(φ2)k_H−r 6Ckf^e(φ1)−f^e(φ2)k

L^n+2r2n (Ω)

6Ckφˆ ₁−φ2k

L^n−2r2n (Ω)

1 +kφ₁k^ρ−1

L

n(ρ−1) 2r (Ω)

+kφ₂k^ρ−1

L

n(ρ−1) 2r (Ω)

6Ckφ₁−φ2k

X^s2

1 +kφ₁k^ρ−1

X^s2 +kφ₂k^ρ−1

X^2s

,

for any s∈[r,1]∩h

n

2 −_ρ−1² ,1 i

. The last statement holds since H^−r is compact embedded inH⁻¹ forr <1.

To continue, let W be the extrapolated space of X ×X - which is the completion of the normed space (X ×X,kA⁻¹ · kX×X) - and we consider the power scale {W^α}_α∈[0,1] generated by (W,kA^α · k_W).

Remark 3.2. Note thatW¹=X×X for all ∈[0,1].

Lemma 3.3. Let s∈h_{(ρ−1)(n−2)}

4 ,1i

andγ ∈(0,1−^s₂), then F (defined in (1.8)) takesW¹ in W^γ and is Lipschitz continuous in bounded sets.