nonhomogeneous equilibria for a class of evolution equation with non local terms

Existence and continuity of global attractors and of

nonhomogeneous equilibria for a class of evolution equation with non local terms

Flank D. M. Bezerra^3∗ Antˆonio L. Pereira^1†, Severino H. da Silva^2‡

E-mail: alpereir@ime.usp.br, horacio@dme.ufcg.edu.br and flank@mat.ufpb.br

Abstract

In this paper we are concerned with some aspects of the asymptotic behavior of the dynamical system generated by evolution equations with nonlocal terms of the type

∂u(x,t)

∂t =−u(x, t) +g(β(Ku)(x, t)), x∈Ω, t >0

where Ω⊂R^N,β >0,N ≥1 is a bounded smooth domain andK is an integral operator with symmetric kernel

(Ku)(x) :=

Z

R^N

J(x, y)u(y)dy.

AMS subject classification: 34G20,47H15.

1 Introduction

A class of problems that have been considered recently are differential equations with non local diffusion terms. Equations of this type appear, for example, in modeling population dynam- ics, propagation of nerve impulses and problems of phase transition in ferromagnetic models.

The mathematical treatment of these equations is generally simpler than similar models based on parabolic partial equations, from the standpoint of the basic theory of existence and well posedness. On the other hand, the qualitative behavior of solutions can be much richer. Thus, many of the issues that arise for models based on parabolic semilinear equations can also be considered in the new context but often require new techniques and ideas.

Our goal here is to consider some of these issues for a non linear Dirichlet problem with non local terms, namely,

∗†Departamento de Matem´atica/UFPB, Cidade Universit´aria-Campus I, 58051-900, Jo˜ao Pessoa-PB, Brazil.

Research partially supported by grant FAPESP 11/04166-5.

†Instituto de Matem´atica e Estat´ıstica-Universidade de S˜ao Paulo, Rua do Mat˜ao, 1010, Cidade Universit´aria, 05508-090, S˜ao Paulo-SP, Brazil. Partially supported by CNPq-Brazil grants 2003/11021-7, 03/10042-0.

‡Unidade Acadˆemica de Matem´atica e Estat´ıstica UAME/CCT/UFCG, Avenida Apr´ıgio Veloso, 882, Bairro Universit´ario, Caixa Postal: 10.044, 58109-970, Campina Grande-PB, Brazil. Partially supported by CNPq- Brazil grant Casadinho 620150/2008 and INCTMat.







∂u(x,t)

∂t =−u(x, t) +g(β(Ku)(x, t)), x∈Ω, t >0

u(x,0) =u0(x), x∈Ω

u(x, t) = 0, x6∈Ω, t >0

(1.1) where Ω ⊂ R^N, N ≥ 1 is a bounded smooth domain, u(x, t) is a real function on R^N ×[0,+∞),β >0 and K is an integral operator with symmetric kernel

(Ku)(x) :=

Z

R^N

J(x, y)u(y)dy, Here,J is an even non negative function of classC² withR

R^NJ(x, y)dy= 1, andg:R→R is a non linear real function of classC¹ withg(0) = 0.

The equation in (1.1) is called a nonlocal diffusion equation since the diffusion of the density u at a point x and time t depends on the values of u in a neighborhood of x in R^N through the termKu.

The asymptotic behavior of solutions of nonlocal diffusion equations has been extensively studied over the past ten years (see [1, 2, 3, 6, 7, 8, 9, 14, 22]). Often, the non local term in (1.1)-like problems is given by convolution. We extend some of these results to (1.1), and consider also some different aspects of the asymptotic behavior of the infinite dimensional dynamical system it generates. In particular, we discuss the existence and characterization of global attractors and the existence of non constant equilibria in the phase space L²(R^N), generalizing results of [4], [20] and [21].

For N = 1 there are several works in the literature dedicated to the analysis of similar models (see, [19, 20, 21], where existence and continuity of global attractors is proved) and, in particular, for the case of (1.1) withg≡tanh, and J(x, y) =J(x−y) that is

∂u(x, t)

∂t =−u(x, t) + tanh (β(J∗u)(x, t)), t >0 (1.2) (see [4, 15, 16, 17, 19]). In this case, ifβ≤1, equation (1.2) has only one (stable) equilibrium, (see [17] and [19]). If β >1, the equation

m_β = tanh(βm_β). (1.3)

has three roots m⁻_β, m⁰_β, m⁺_β,which are then spatially homogeneous equilibria of the equation (1.3). The existence and uniqueness (modulo translation) of a travelling front connecting the equilibria m⁻_β and m⁺_β is proved in [15]. The existence and uniqueness (modulo translation) of a solution tending asymptotically to ±m⁺_β, referred to as the “instanton”is proved in [16]

and [18].

This paper is organized as follows. In Section 2 we prove that (1.1) generates a flow in L²(R^N). Section 3 is dedicated to the proof of existence of the global attractor (for the case N ≥ 1), generalizing results of [4] and [20]. In Section 4, we prove a comparison result, generalizing Theorem 2.7 of [16] and Theorem 4.2 of [20]. In Section 5, we exhibit a Lyapunov functional for the flow of (1.1), and used it to prove the existence of non trivial equilibrium.

We collect here the conditions on g which will be used as hypotheses when needed.

(H1) The function g : R → R, is globally Lipschitz, with g(0) = 0. That is, there exists a positive constant k₁ such that

|g(x)−g(y)| ≤k1|x−y|, ∀x, y∈R.

In particular,

|g(x)| ≤k₁|x|, ∀x∈R. (1.4) (H2)The function g∈C¹(R) and g⁰ is Lipschitz with constant k2. In particular

|g⁰(x)| ≤k2|x|+k3 ∀x∈R, for somek3>0.

(H3)The function g has positive derivative. In particular it is strictly increasing.

(H4)There exists a >0 such that|g(x)|< a <∞, for all x∈R. In this paper, |Ω|denotes the Lebesgue measure of Ω in R^N.

2 Estimates and well posedness

In this section, to obtain well posedness of (1.1), we initially consider the following Cauchy problem in the space L²(R^N)

_∂u(x,t)

∂t = (F u)(x, t)

u(x,0) =u₀ (2.5)

whereF :L²(R^N)→L²(R^N) is given by [F(u)](x) =

−u(x) +g(β(Ku)(x)), x∈Ω

0, x /∈Ω. (2.6)

Proposition 2.1. Suppose that the hypothesis (H1) holds. Then the function F given above is uniformly Lipschitz in L²=L²(R^N).

Proof. From Generalized Young’s Inequality, (see [11]), kK(m−u)k_L2(Ω)≤ km−uk_L2(Ω). Therefore,

kF(m)−F(u)k_L2 = k −(m−u) +g(βKm+βh)−g(βKu)k_L2(Ω)

≤ km−uk_L2(Ω)+k1kβ(Km)−β(Ku)k_L2(Ω)

= km−uk_L2(Ω)+k1βkK(m−u)k_L2(Ω)

≤ (1 +k₁β)km−uk_L2, which concludes the proof.

From Proposition 2.1, it follows that the Cauchy problem (1.1) is well posed in L²(R^N) with a unique global solution, (see [5] and [10]). More precisely, we have

Corollary 2.2. The problem (1.1) has a unique solution for any initial condition in L² = L²(R^N), which is globally defined.

Remark 2.3. Note that, if u∈L²(R^N) then

|(Ku)(x)| ≤ kuk_L2, ∀x∈Ω. (2.7) and

|(K⁰u)(x)| ≤ kJ⁰k_L1kuk_L2, ∀ x∈Ω. (2.8)

The following result is very similar to Proposition 2.6 of [20] and therefore its prove will be omited.

Proposition 2.4. Assume that the hypotheses (H1) and (H2) hold. Then the function [F(u)](x) =

−u(x) +g(β(Ku)(x)), x∈Ω 0, x /∈Ω.

is continuously Frechet differentiable in L²(R^N) with derivative given by [DF(u)v](x) =

−v(x) +g(β(Ku)(x))βK(v)(x), x∈Ω 0, x /∈Ω.

Remark 2.5. Consider the subspace X of L²(R^N) given by X=

u∈L²(R^N) |u(x) = 0, if x6∈Ω.

Since the range of F is X, this is an invariant subspace for the flow generated by (2.5). In the following sections, we always consider the flow restricted to X, which is an abstract way to impose Dirichlet boundary conditions.

3 Existence of a global attractor

We prove, in this section, the existence of a global maximal invariant compact set A in X ⊂ L²(R^N) for the flow of (1.1), which attracts each bounded set of X (the global attractor, see [12] and [24]).

We recall that a set B ⊂ X is an absorbing set for the flow T(t) (here, T(t) denotes the global semi-flow generated by (1.1) in X)) if, for any bounded set C ⊂ X, there is a t1 >0 such thatT(t)C⊂ B for any t≥t1.

The following result is proved in [24].

Theorem 3.1. Let X be a Banach space andT(t) a semigroup onX. Assume that, for every t, T(t) =T₁(t) +T₂(t) where the operators T₁(·) are uniformly compact for tlarge, that is, for every bounded set B there exists t₀, which may depend on B, such that

[

t≥t0

T1(t)B

is relatively compact inX and T2(t) is a continuous mapping from X into itself such that the following holds: For every bounded set C⊂X,

r_c(t) = sup

ϕ∈C

kT₂(t)ϕk_X →0 as t→ ∞.

Assume also that there exists an open setU and bounded subsetBofU such thatBis absorbing in U. Then the ω-limit set of B, A=ω(B), is a compact attractor which attracts the bounded sets ofU. It is the maximal bounded attractor in U (for the inclusion relation). Furthermore, if U is convex and connected, thenA is connected.

Lemma 3.2. Assume that the hypotheses (H1) and (H4) hold and let R=ap

|Ω|. Then, for any ε > 0, the ball of radius R+ε in X is an absorbing set for the flow T(t) generated by (1.1).

Proof. Letu(x, t) be a solution of (1.1) with initial condition u(x,0). Then, ifx6∈Ω, then u(x, t) = 0 and, ifx∈Ω we have, by the variation of constants formula

u(x, t) =e^−tu(x,0) + Z t

0

e^−(t−s)g(β((Ku)(x, s)))ds.

Using hipothesis (H4) it follows that

|u(x, t)| ≤ e^−t|u(·,0)|+ Z t

0

e^−(t−s)|g(β((Ku)(x, s)))|ds

≤ e^−t|u(x,0)|+a.

Hence,

ku(·, t)k_L2 ≤ ke^−t|u(·,0)|+ak_L2

≤ e^−tku(·,0)k_L2 +ap

|Ω|.

Therefore, u(·, t)∈B(0, R+ε) for t >ln(^ku(·,0)k_ε ^L2), and the result is proved.

The next result is a extension of Theorem 3.3 of [4].

Theorem 3.3. Suppose that (H1), (H2) and (H4) hold. Then there exists a global attractorA for the flowT(t) generated by (1.1)in X, which is contained in the ball of radiusR=ap

|Ω|.

Proof. If u(x, t) is the solution of (1.1) with initial conditionu(x,0) we have, if x∈Ω, by the variation of constants formula

u(x, t) =e^−tu(x,0) + Z t

0

e^s−tg(β(Ku(x, s))ds. (3.9) Write

T1(t)u(x) = Z t

0

e^s−tg(β((Ku)(x, s)))ds and

T₂(t)u(x) =e^−tu(x,0)

and supposeu(·,0)∈C,whereC is a bounded set inX. We may suppose thatC is contained in the ball of radius ρ. Then

kT₂(t)uk_L2 →0 as t→ ∞, uniformly in u.

Also, using (3.9), we have that ku(·, t)k_L2 ≤ M, for t ≥ 0, where M = max{ρ, ap

|Ω|}.

Therefore, for t≥0 we have

∂T1(t)u(x)

∂x =

Z t 0

e^s−t ∂

∂xg(β((Ku)(x, s)))ds

= β

Z t 0

e^s−tg⁰(β((Ku)(x, s)))(K⁰u)(x, s)ds.

Thus

∂T1(t)u(x)

∂x

≤ β Z t

0

e^s−t|g⁰(β(Ku)(x, s))||(K⁰u)(x, s)|ds.

Using (H2) and the Remark 2.3, we obtain

|g⁰(βKu(x, s))||(K⁰u)(x, s)| ≤ [k₂|β(Ku)(x, s)|+k₃]|(K⁰u)(x, s)|

≤ [k₂|β(Ku)(x, s)|+k₃]|(K⁰u)(x, s)|

≤ [k2βku(·, s)k_L2 +k3]kJ⁰k_L1ku(·, s)k_L2

≤ k2βkJ⁰k_L1M²+k3)kJ⁰k_L1M.

Hence

∂T₁(t)u(x)

∂x

≤ β Z t

0

e^s−t

k₂βkJ⁰k_L1M²+k₃)kJ⁰k_L1M ds

=

k2β²kJ⁰k_L1M²+ (k3β)kJ⁰k_L1M Z t

0

e^s−tds

≤ k2β²kJ⁰k_L1M²+k3β)kJ⁰k_L1M.

It follows that, for t > 0 and any u ∈ C, the value of k^∂T_∂x^1(t)u_|Ωk_L2(Ω) is bounded by a constant (independent of t and u ). Thus, for all u ∈ C,we have that T₁(t)u_|Ω belongs to a ball of W^1,2(Ω). From Sobolev’s Imbedding Theorem, it follows that S

t≥0

S

u∈CT₁(t)u_|Ω is relatively compact inL²(Ω). SinceX andL²(Ω) are isometric spaces, it follows that

[

t≥0

T1(t)C

is also relatively compact inX. Therefore, the result follows from Theorem 3.1, the attractor A being the setω-limit of the ballB(0, R+ε) inL²(R^N) for all >0.

4 Comparison and boundedness results

The following comparison result has been proved in [20] for the case N = 1 (see also [16] for particular case of g≡tanh). Its extension for N ≥1 is straightforward.

Theorem 4.1. (Comparison Theorem) Assume hypotheses (H1) and (H3) hold and letv(x, t), [V(x, t)]be a sub solution [super solution] of the Cauchy problem of (1.1)with initial condition u(·,0). Then

v(x, t)≤u(x, t)≤V(x, t), almost everywhere.

Theorem 4.2. Assume the hypotheses (H1) and (H4). Then the attractor A belongs to the ball k · k_∞≤ain L^∞(R^N).

Proof. From Theorem 3.3 it follows that the attractor is contained in the ballB[0, ap

|Ω|]

inL²(R^N).

Letu(x, t) be a solution of (1.1) inA. Then,if x∈Ω, by the variation of constants formula u(x, t) =e^−(t−t0)u(x, t0) +

Z t t0

e^−(t−s)g(β(Ku)(x, s))ds.

Letting t₀→ −∞ we obtain, for all (x, t)∈Ω×R⁺ u(x, t) =

Z _t

−∞

e^−(t−s)g(β(Ku)(x, s))ds,

where the equality above is in the sense of L²(R^N).

Thus, using (H4) again, we have

|u(x, t)| ≤ Z t

−∞

e^−(t−s)|g(β(Ku)(x, s))|ds

≤ Z t

−∞

ae^−(t−s)ds

≤ a.

as claimed.

5 Existence of nonhomogeneous equilibria

In this section we exhibit a Lyapunov’s functional that decreases along the solutions of (1.1), and use it to show the existence of nonhomogeneous equilibria for (1.1), via La Salle’s Invariance Principle (see [13]).

Remark 5.1. Since g(0) = 0, under the hypothesis (H1) above, it is easy to see that u≡0 is equilibrium solutions of (1.1).

Proposition 5.2. Assume that k1β < 1. Then u ≡ 0 is the unique equilibrium solution of (1.1).

Proof. Consider the map Ψ :X →X given by Ψ(u) :=g(β(Ku))).

Then u is an equilibrium if and only if is fixed point of Ψ.

Note that, using (H1) and Generalized Young’s Inequality, obtain kΨ(u)−Ψ(v)k_L2 ≤ k1βkKk_L1k(u−v)k_L2

= k1βk(u−v)k_L2.

Thus, sincek₁β <1, it follows that Ψ is a contraction. Hence the result follows from Remark 5.1.

Suppose now that k₁β >1. X Letf be given by f(m) =−1

2m²−β⁻¹i(m),

where

i(m) =− Z _m

0

g⁻¹(s)ds.

It is easy to see that f has two local minimum m− <0< m₊, one of which, say ¯m=m₊, is the global minimum of f.

Define m^∗ :R^N →Rby

m^∗(x) =

m, x¯ ∈Ω 0, x /∈Ω.

Motivated by [15, 16, 19, 20, 21] we define the following functional in L²(R^N) F(u) =

Z

Ω

[f(u(x))−f(m)]dx+1 4

Z

Ω

Z

Ω

J(x, z)[u(x)−u(z)]²dxdz (5.10) wheref is given in the hypothesis (H5).

Note that the functional given in (5.10) is lower bounded, namely, F(u)≥0, for allu . Remark 5.3. It is easy to see that, under hypotheses (H1), (H4) and (H5), the functional F is continuous with respect to the topology of L²(R^N).

The following result is proved in [20] using a comparison Theorem similar to (4.1)

Theorem 5.4. Suppose that the hypotheses (H1)-(H5). Let u(·, t) be a solution of (1.1)with u(·, t)≤a. Then F(u(·, t)) is differentiable with respect to tfor t >0 and

d

dtF(u(·, t)) =−I(u(·, t))≤0, where, for any u∈L² =L²(R^N),

I(u(·, t)) = Z

Ω

[(Ku)(x, t)−β⁻¹g⁻¹(u(x, t))][−u(x, t) +g(β(Ku)(x, t))]dx.

Consequently, I(u) = 0 if and only if u is equilibrium point.

Now, considerm(·, t) the solution of (1.1) withm(·,0) =m^∗. SinceF(0)>F(m^∗), it follows from Theorem 5.4 above that the Ω-limit ofm^∗ does not contain the null stationary solution.

Furthermore, from existence of a global compact attractor we have precompacity of the orbits ofT(t). It follows then by La Salle’s Invariance Principle (see [13]) that m(·, t) tends to a non constant equilibrium. Using the Comparison Theorem 4.1, we can show that this equilibrium is positive.

6 Continuity of the global attractors

In this section we study the continuity of the global attractors with respect to the parameter β atβ=β₀.

By Proposition 2.4 the map F given in (2.6) is continuously Frechet differentiable in L²(R^N). Therefore, the problem (2.5) generates a C¹ flow in X which depends on the pa- rameterβ. From now on we denote this flow byT_β(t) orT(β, t). As proved in Section 3 the flowT(t) =T_β(t) admits a global compact attractor, which denote byA_β. We will study the dependence of this attractor with respect to the parameterβ.

Let us recall that a family of subsets {A_β}, is upper semicontinuous atβ0 if dist(A_β, A_β0)−→0, asβ →β₀,

where

dist(A_β, A_β0) = sup

x∈Aβ

dist(x, A_β0) = sup

x∈Aβ

y∈Ainf_β

0

kx−yk_L2. (6.11) Analogously,{A_β} is lower semicontinuous atβ₀ if

dist(Aβ0, Aβ)−→0, asβ →β0. 6.1 Upper semicontinuity

The upper semicontinuity of the global attractors is an immediated consequence of the conti- nuity of the flow.

Lemma 6.1. Under the assumptions (H1) and (H4), the flowT_β(t)is continuous with respect to β, uniformly foru in bounded sets andt∈[0, b]with b <∞.

Proof. The solutions of (1.1) satisfy the ‘variations of constants formula’, forx∈Ω T_β(t)u=e^−tu+

Z t 0

e^−(t−s)g(βK(T_β(s)u)ds.

Letβ₀ fixed, b >0 andC a bounded set inL²(R^N). Given ε >0, we want to findδ >0 such that|β−β₀|< δimplies

kT_β(t)u−Tβ0(t)uk_L2 < ε,

fort∈[0, b] and u inC. Sinceg is globally Lipschitz, for anyt >0 andu∈C, it follows that kT_β(t)u−T_β0(t)uk_L2 ≤

Z t 0

e^−(t−s)kg(βK(T_β(s)u))−g(β₀K(T_β0(s)u))k_L2ds

≤ Z t

0

e^−(t−s)k1[kβK(T_β(s)u)−β0(K(T_β0(s)u)k_L2ds.

Subtracting and summing the termβ0K(Tβ(s)u) and using Young’s inequality, we obtain kT_β(t)u−T_β0(t)uk_L2 ≤

Z t 0

e^−(t−s)k₁|β−β₀|kJk_L1kT_β(s)uk_L2ds +

Z t 0

e^−(t−s)k1β0kJk_L1kT_β(s)u−T_β0(s)uk_L2ds.

From Theorem 3.3 in [20] it follows that, for all t∈[a, b],kT_β(t)uk_L2 is bounded by a positive constantL depending only ofC. Thus, since kJk_L1 = 1, we obtain

kT_β(t)u−T_β0(t)uk_L2 ≤ {Lk₁|β−β₀|}

+ Z t

0

e^−(t−s)k₁β₀kT_β(s)u−T_β0(s)uk_L2ds

= C(β) + Z _t

0

k₁β₀kT_β(s)u−T_β0(s)uk_L2ds,

whereC(β) ={Lk₁|β−β0|}. Therefore, by Gronwall’s Lemma, it follows that kT_β(t)u−T_β0(t)uk_L2 ≤ C(β)e^k1β0t.

From this, the results follows immediately.

The following result is an immediate consequence of the Lemma 6.1 and therefore its proof will be omitted.

Theorem 6.2. Assume the hypotheses (H1) and (H4). Then, the family of global attractors A_β is upper semicontinuous with respect to β.

6.2 Existence and continuity of the local unstable manifolds

In order to obtain theExistence and continuity of the local unstable manifolds we will need the following additional hypotheses:

(H5)For eachβ0≥1, the setE_β0, of the equilibria ofT_β0(t), has only hyperbolic equilibria;

(H6)The function g∈C²(R). (Observe that (H6) implies (H2)).

We need to assume that the equilibrium points of (2.5) with β0 are stable under pertur- bation. This stability under perturbation will follow from the hyperbolicity of the equilibrium points.

Lemma 6.3. Suppose that the hypotheses (H1)-(H5) hold. Then the set E_β of the equilibria of T_β(t) is continuous with respect toβ at β =β0.

Proof. The upper semicontinuity of the equilibria is an consequence of the upper semicon- tinuity of the global attractors. The lower semicontinuity follows from the Implicit Function Theorem since the equilibria are all hyperbolic.

Remark 6.4. Any hyperbolic equilibrium pointu of (2.5) withβ₀ is isolated. The result above guarantees the continuity of equilibrium points. In fact, small neighborhoods of equilibrium points of the problem (2.5) with β0, have a unique equilibria of the problem (2.5). Therefore, there are only a finite number of hyperbolic equilibrium points of (2.5) for β near of β₀.

Let us return to equation (2.5). Recall that theunstable setW_β^u =W_β^u(u_β) of an equilibrium uβ is the set of initial conditions ϕ of (2.5), such that Tβ(t)ϕ is defined for all t ≤ 0 and T_β(t)ϕ→u_β ast→ −∞. For a given neighbourhoodV ofu_β, the setW_β^u∩V is called a local unstable set of u_β.

Using results of [23] we now show that the local unstable sets are actually Lipschitz man- ifolds in a sufficiently small neighbourhood and vary continuously withβ. More precisely, we have

Lemma 6.5. If u0 is a fixed equilibrium of(2.5)for β =β0 then there is a δ >0such that, if

|β−β0|+ku₀−u_βk_L2 < δ and

U_β^δ :={u∈W_β^u(u_β) : ku−u_βk_L2 < δ}

thenU_β^δ is a Lipschitz manifold and

dist(U_β^δ, U_β^δ₀) +dist(U_β^δ₀, U_β^δ)→0, as |β−β0|+ku₀−u_βk_L2 →0, with dist defined as in (6.11).

Proof. As already mentioned, assuming (H1) and (H2), the mapF :L²(R^N)×R→X, F(u, β) =−u+g(β(K(u))),

defined by the right-hand side of (2.5) is continuously Frechet differentiable. Let uβ be an equilibrium of (2.5). Writingu=u_β+v, it follows thatu is a solution of (2.5) if and only ifv satisfies

∂v

∂t =L(β)v+r(u_β, v, β), (6.12) where L(β)v = _∂u^∂F(u_β, β) = −v+g⁰(β(K(u_β)))βK(v) and r(u_β, v, β) = F(u_β +v, β) − F(u_β, β)−L(β)v.We rewrite equation (6.12) in the form

∂v

∂t =L(β0)v+f(v, β), (6.13)

where f(v, β) = [L(β)−L(β₀)]v+r(u_β, v, β) is the “non linear part” of (6.13). Observe that now the “linear part” of (6.13) does not depend on the parameter β, as required by theorems Theorems 2.5 and 3.1 from [23].

To obtain the needed estimates we first observe that, by H¨older inequality

|(K(v))(x)| ≤√

2τkJk_∞kvk_L2, ∀x∈R^N (6.14) for anyv∈X.Therefore, sincegis of classC²,g⁰(βK(u_β)(x)+βK(v)(x)) andg⁰⁰(βK(u_β)(x)+

βK(v)(x)) are bounded by a constantM, for anyβ in a neighbourhood ofβ₀ and||v||_L2(R^N)≤ 1.We then obtain

k g⁰(βK(u_β))β(K(v))−g⁰(β₀K(u_β0))β(K(v))k²_L2

= Z

R^N

|g⁰(βK(uβ)(x))β−g⁰(β0K(uβ0)(z))|²β²|K(v)(x)|²dx

≤ Z

R^N

M²|[|βK(u_β)(x)−β0K(u_β0)(x)|]²β²|K(v)(x)|²dx

≤ Z

R^N

M²[|βK(u_β)(x)−β0K(u_β0)(x)|]²β²2τkJk²_∞kvk²_L2dx

≤ Z

R^N

M²[|βK(u_β)(x)−βK(u_β0)(x)|+|βK(u_β0)(x)−β₀K(u_β0)(x)|]²β²2τkJk²_∞kvk²_L2dx

≤ Z

R^N

M²[β√

2τkJk_∞ku_β−u_β0k_L2

+ |β−β0|√

2τkJk∞ku_β0k_L2]²β²2τkJk²_∞kvk²_L2dx

= d₁(β)kvk²_L2,

withd₁(β)→0, asβ →β₀. Analogously

k g⁰(β₀K(u_β0))(β−β₀)K(v)k²_L2

≤ Z

R^N

kg⁰(β₀K(u_β0))k²_∞|β−β₀|²|K(v)(x)|²dx

≤ Z

R^N

kg⁰(β₀K(u_β0))k²_∞|β−β₀|²2τkJk²_∞kvk²_L2dx

= d2(β)kvk²_L2.

withd2(β)→0, asβ →β0. It follows that

k(L(β)−L(β0))vk_L2

≤ kg⁰(βK(uβ))βK(v)−g⁰(β0K(uβ0))βK(v)k_L2

+kg⁰(β0K(uβ0))(β−β0)K(v)k_L2

≤C₁(β)kvk_L2, (6.15)

withC₁(β) =p

d₁(β) +p

d₂(β)→0, asβ →0.

Observe now that, for any x∈R^N, we get

F(u_β(x) +v(x), β)−F(u_β0(x) +v(x), β₀)

= [g(β₀K(u_β0)(x))−g(β₀K(u_β0)(x) +β₀K(v)(x))]

−[g(βK(u_β)(x))−g(βK(u_β)(x) +βK(v)(x))]

=g⁰(β0K(uβ0)(x) +β0K(¯v(x)))β0K(v)(x)]

−g⁰(βK(u_β)(x) +βK(¯¯v)(x))βK(v)(x),

for some ¯vin the segment defined byK(u_β0) andK(u_β0+v) and some ¯v¯in the segment defined by K(uβ) and K(uβ +v). Then

|F(u_β(x) +v(x), β)−F(u_β0(x) +v(x), β₀)|

≤[|g⁰(β₀K(u_β0)(x) +β₀K(¯v)(x))β₀−g⁰(β₀K(u_β0)(x) +β₀K(¯v)(x))β|

+β|g⁰(β0K(u_β0)(x) +β0K(¯v)(x))

−g⁰(βK(uβ)(x) +βK(¯v)(x))|]|βK(v)(x)|¯

≤[M|β−β0|+βM|β₀K(uβ0)(x)−βK(uβ)(x)|

+βM|β₀K(¯v)(x)−βK(¯v)(x)|]¯ √

2τkJk_∞kvk_L2

≤[M|β−β0|+βM|β−β0||K(u_β0)(x)−K(u_β)(x))|+βM(|β−β0||K(¯v)(x)|

+β|K(¯v)(x)−K(¯v)(x)|)]¯ √

2τkJk∞kvk_L2

≤[M|β−β₀|+βM|β−β₀|√

2τkJk_∞ku_β−u_β0k_L2+βM|β−β₀|√

2τkJk_∞k¯vk_L2

+β²M√

2τkJk_∞k¯v−¯¯vk_L2]√

2τkJk_∞kvk_L2. Therefore, since k¯v−¯¯vk_L2 →0, as β→β0,

kF(u_β+v, β)−F(u_β0 +v, β₀)k_L2 ≤C₂(β)kvk_L2, (6.16) withC₂(β)→0, as β→0.

Since r(uβ, v, β) =F(uβ+v, β)−L(β)v, we obtain from (6.15) and (6.16) that

kr(u_β, v, β)−r(uβ0, v, β0)k ≤C3(β)kvk_L2. (6.17) From (6.15) and (6.17), it follows that

kf(v, β)−f(v, β₀)k ≤C₄(β)kvk_L2, whereC4(β)→0 as β →β0.

We also obtain which, as kvk_L2,kwk_L2 ≤ρ,

kr(u_β(x), v(x), β)−r(uβ(x), w(x), β)k_L2 ≤ν1(ρ)kv−wk_L2, withν(ρ)→0, asρ→0. Furthermore

k[L(β)−L(β₀)]v−[L(β)−L(β₀)]wk_L2 ≤C₁(β)kk(v−w)k_L2. Thus

kf(v, β)−f(w, β)k_L2 ≤(ν(ρ) +C₁(β))kv−wk_L2, (6.18) whereν(ρ)→0, asρ→0 and kvk_L2,kwk_L2 ≤ρ, and C1(β)→0, asβ →β0.

Therefore, the conditions of Theorems 2.5 and 3.1 from [23] are satisfied and we obtain the existence of locally invariant sets for (6.13) near the origin, given as graphs of Lipschitz func- tions which depend continuously on the parameter β near β0. Using uniquennes of solutions, we can easily prove that these sets coincide with the local unstable manifolds of (6.13).

Observing now that the translation

u→(u−u_β)

sends an equilibriumu_β of (2.5) into the origin (which is an equilibrium of (6.13)), the results claimed follow immediately.

Using the compacity of the set of equilibria, one can obtain an ‘uniform version’ of Lemma 6.5 that will be needed later.

Lemma 6.6. Let β =β₀ be fixed. Then, there is a δ >0 such that, for any equilibrium u₀ of (P)β0, if|β−β0|+ku₀−uβk_L2 < δ and

U_β^δ :={u∈U_β(u_β) : ||u−u_β||_L2(S¹)< δ}

thenU_β^δ is a Lipschitz manifold and sup

u0∈E_β0

dist(U_β^δ, U_β^δ₀) +dist(U_β^δ₀, U_β^δ)→0 as |β−β₀|+ku₀−u_βk_L2 →0, with dist defined as in (6.11)

Proof. From Lemma 6.5, we know that, for any u₀ ∈E_β0, there is a δ =δ(u₀) such that U_β^δis a Lipschitz manifold, if|β−β0|+ku₀−uβk_L2 <2δ. Thus, in particular,U_β^δ is a Lipschitz manifold, if|β−β0|+ku˜0−u_βk_L2 < δ. for any ˜u0 ∈E_β0 withk˜u0−u0k_L2 < δ. Taking a finite subcovering of the covering ofE_β0 by balls B(u₀, δ(u₀)), withu₀ varying inE_β0, the first part of the result follows withδ chosen as the minimum of thoseδ(u0).

Now, if ε >0 andu0∈E_β0, there exists, by Lemma 6.5,δ =δ(u0) such that, if |β−β0|+ ku₀−u_βk_L2 <2δ, then

dist(U_β^δ, U_β^δ₀) +dist(U_β0^δ , U_β^δ)< ε/2.

If ˜u₀∈E_β0 is such thatku˜₀−u₀k_L2 < δand |β−β₀|+k˜u₀−u_βk_L2 < δthen, since |β−β₀|+ ku₀−u_βk_L2 <2δ

dist(U_β^δ(u_β), U_β^δ

0(˜u₀)) + dist(U_β^δ

0(˜u₀), U_β^δ(u_β))

<dist(U_β^δ(u_β), U_β^δ

0(u₀)) + dist(U_β^δ

0(u₀), U_β^δ(u_β)) + dist(U_β^δ

0(˜u₀), U_β^δ

0(u₀)) +dist(U_β^δ₀(u0), U_β^δ₀(˜u0))< ε

By the same procedure above of taking a finite subcovering of the covering of E_β0 by balls B(u₀, δ(u₀)), and δ the minimum of those δ(u₀), we conclude that

dist(U_β^δ(u_β), U_β^δ₀(˜u0)) + dist(U_β^δ₀(˜u0), U_β^δ(u_β))< ε

if|β−β0|+k˜u0−uβk_L2 < δ, for any ˜u0 ∈Eβ0. This proves the result claimed.

6.3 Lower semicontinuity

Now we are ready to prove the lower semicontinuity of the global attractors.

Theorem 6.7. Under hypotheses (H1)-(H7), the family of attractors {A_β} is lower semicon- tinuous at β₀.

Proof. Using Theorems 3.3, 5.4, Lemmas 6.1, 6.3, 6.5 and 6.6, the result follows from Theorem 4.10.8 of [12].

References

[1] F. Andreu, J. M. Maz´on, J. D. Rossi and J. Toledo. The Neumann problem for nonlocal nonlinear diffusion equation. J. Evol. Equations,8(1), (2008), 189-215.

[2] F. Andreu, J. M. Maz´on, J. D. Rossi and J. Toledo. A nonlocal p−Laplacian evolution equation with Neumann boundary conditions. J. Math. Pures Appl. (9) 90(2), (2008), 201-227.

[3] F. Andreu, J. M. Maz´on, J. D. Rossi and J. Toledo. The limit as p → ∞ in a nonlocal p−Laplacian evolution equation. A nonlocal approximation of a model for sandpiles. Calc.

Var. Partial Differential Equations.35, (2009), 279-316.

[4] S. R. M. Barros, A. L. Pereira, C. Possani and A. Simonis, Spatial Periodic Equilibria for a Non local Evolution Equation. Discrete and Continuous Dynamical Systems 9 N. 4, (2003), 937-948.

[5] H. Brezis,Analisis funcional, teoria y aplicaciones. Alianza, Madrid, 1984.

[6] E. Chasseigne, M. Chaves and J. D. Rossi, Asumptotic behavior for nonlocal diffusion equations.J. Math. Pures Appl., 86, (2006), 271-291.

[7] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffu- sion equations, SIMUAMAT summer school 2009. Departamento de Matematica, FCEyN UBA, Buenos Aires- Argentina, (2009).

[8] C. Cortazar, M. Elgueta and J. D. Rossi, A non-local diffusion equation whose solutions develop a free boundary.Annales Henri Poincar´e,6(2), (2005), 269-281.

[9] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Rat.

Mech. Anal,187(1), (2008), 137-156.

[10] J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces.American Mathematical Society, Providence, Rhode Island, 1974.

[11] G. Folland,Introduction to partial differential equations. Princeton Univ. Press, 1976.

[12] J. K. Hale, Asymptotic behavior of dissipative Systems. American Surveys and Mono- graphs, N. 25, 1988.

[13] D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math- ematics N. 840, Springer-Verlag, 1981.

[14] L. I. Ignat and J. D. Rossi. A nonlocal convection-diffusion equation. J. Functional Anal- ysis,251(2), (2007), 399-437.

[15] A. Masi, T. Gobron, E. Presutti, Traveling fronts in non local evolution equations. Arch.

Rational Mech. Anal.132 (1995), 143-205.

[16] A. Masi, E. Orland, E. Presutti and L. Triolo, Uniqueness and global stability of the instanton in non local evolution equations. Rendiconti di Matematica, Serie VII, Vol. 14, (1994), 693-723.

[17] A. Masi, E. Oliveri and E. Presutti, Critical droplet for a non local mean field equation.

Markov Processes Relat. Fields 6(2000), 439-471.

[18] A. Masi, E. Orland, E. Presutti and L. Triolo, Stability of the interface in a model of phase separation. Proc. Royal Society of Edinburgh 124A, (1994), 1013-1022.

[19] A. L. Pereira, Global attractor and nonhomogeneous equilibria for a non local evolution equation in an unbounded domain. J. Diff. Equations226 (2006) 352-372.

[20] A. L. Pereira and S. H. da Silva, Existence of global attractors and gradient property for a class of non local evolution equations, S˜ao Paulo Journal of Mathematical Sciences, v.

2, p. 1-20, 2008.

[21] A. L. Pereira and S. H. da Silva, Continuity of global attractors for a class of non local evolution equations, Discrete and Continuous Dynamical Systems, v. 26, p. 1073-1100, 2010.

[22] J. D. Rossi, Asymptotic behaviour of solutions to evolution problems with non local dif- fusion, SIMUAMAT summer school 2009. Departamento de Matematica, FCEyN UBA, Buenos Aires-Argentina, (2009).

[23] S. H. da Silva and A. L. Pereira, Exponential trichotomies and continuity of invariant manifolds, S˜ao Paulo Journal of Mathematical Sciences 5, no. 2 (2011), 124.

[24] R. Teman,Infinite Dimensional Dynamical Systems in Mechanics and Physics.Springer, 1988.