Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079

Contents lists available at SciVerse ScienceDirect

Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Asymptotic regularity and attractors of the reaction–diffusion equationwith nonlinear boundary conditionLu YangSchool of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, PR China

a r t i c l e i n f o

Article history:Received 9 November 2010Accepted 18 February 2011

Keywords:Reaction–diffusion equationNonlinear boundary conditionAsymptotic regularityAttractorsNonlinear balance conditions

a b s t r a c t

We consider the dynamical behavior of the reaction–diffusion equation with nonlinearboundary condition for both autonomous and non-autonomous cases. For the autonomouscase, under the assumption that the internal nonlinear term f is dissipative and theboundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions isproved. For the non-autonomous case, we obtain the existence of a compact uniformattractor in H1(Ω) with dissipative internal and boundary nonlinearities.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we study the asymptotic behavior of the solution of the following reaction–diffusion equation withnonlinear boundary condition:

ut − ∆u + f (u) = h(x, t), in Ω,∂u∂ν

+ g(u) = 0, on Γ ,(1.1)

where Ω is a bounded domain of RN(N ⩾ 3) with a smooth boundary Γ , and about the external forcing h(x, t), we considertwo cases: the autonomous case h(x, t) = h(x) and the non-autonomous case h(x, t) which will be given precisely later inSections 3 and 4, respectively. The functions f and g ∈ C1(R, R), satisfy the following conditions:

lim|s|→∞

f (s)s|s|p+1

= Cf , (1.2)

lim|s|→∞

g(s)s|s|q+1

= Cg , (1.3)

here p > 1, q > 1.The reaction–diffusion equationwith nonlinear boundary condition has strong background inmathematical physics, and

it is very natural in manymathematical models. This problem arises in hydrodynamics and the heat transfer theory, such asheat transfer in a solid in contact with a moving fluid, thermoelastic distortion, diffusion phenomena, heat transfer in twomedia, problems in fluid dynamics etc. (see, e.g., [1–6]).

In recent years, the reaction–diffusion equation has been studied extensively by many authors (see, e.g., [7–11]). Forthe case of Dirichlet boundary condition, the asymptotic behavior in terms of an attractor has been obtained in [7–14] etc.In [12], the author obtained the existence and regularity of the global attractor in L2(Ω). Some asymptotic regularity of the

E-mail addresses: yanglu@lzu.edu.cn, lzyanglu03@163.com, lzyanglu@gmail.com.

1468-1218/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2011.02.024

1070 L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079

solution has been proved in [13]. In [15], the author studied the invariance principle and application to the reaction–diffusionequations.

As for the reaction–diffusion equation with nonlinear boundary condition, the global existence of the solution wasdiscussed in [16,17,1]. For the autonomous cases, i.e., h(x, t) = h(x), the long-time behavior has been studied in termsof the global attractor in [1–6], blow up of solutions was considered in [18–20] etc. In [4], the authors studied the well-posedness and the asymptotic behavior of the solution, where the nonlinear terms f and g satisfy ‘‘linearization at infinity’’condition. In Rodríguez-Bernal and Tajdine [1], the authors investigated the nonlinear balance conditions on f and g , suchthat solutions of (1.1) are either globally defined or blow up in finite time, to the best of our knowledge, this result is the firstgeneral result about the nonlinear balances between f and g , which holds only under the assumptions (1.2) and (1.3) and foran arbitrary dimension. Later, in [6], the author gave a detailed discussion about the existence of the global attractor underthe balance conditions given in [1]. In [21,22], the authors considered the quasi-linear parabolic equation with dynamicboundary conditions. For the non-autonomous systems, [23] obtained the existence of a uniform attractor in Lp+1(Ω) withcompeting nonlinearities and the nonlinear balance conditions.

As showed in [1,6], generally speaking if the internal nonlinear term f is dissipative and the boundary nonlinear termg is non-dissipative, i.e., Cf > 0, Cg < 0, then there is a real nonlinear competition between the nonlinear terms f and g ,i.e., the behavior of solutions will depend on the specific balance between the growth exponents p and q, and coefficients Cfand Cg , and the solutions are either globally defined or blow up in finite time.

In this paper, we study the long-time behavior of the reaction–diffusion equation with nonlinear boundary condition forboth autonomous and non-autonomous cases.

For the autonomous systems, inspired by the ideas in [12,15,1,6], we obtain some asymptotic regularity of solutionsfor dissipative internal nonlinearity and non-dissipative boundary nonlinearity (i.e., Cf > 0, Cg < 0). As an applicationof the asymptotic regularity results, we not only can obtain the existence of a global attractor in Lp+1(Ω) (obtained in[1,4,6]) immediately, but also can show further that the fractal dimension of the global attractor is finite in Lp+δ(Ω) forany δ ∈ [0, ∞).

For the non-autonomous systems, we prove the existence of a compact uniform attractor in H1(Ω) with dissipativeinternal and boundary nonlinearities. For the existence of a uniform attractor, as in the autonomous case, some kind ofcompactness of the family of processes is a key ingredient. We remark that here the uniform asymptotic compactness inH1(Ω) was testified only by use of the compactness in L2(Ω), and without any compactness in Lp(Ω), p > 2.

The main results of this paper are Theorem 3.5 (asymptotic regularity), and Theorem 4.5 (uniform attractor and itsstructure), note that the existence of the global attractor is a direct corollary of Theorem 3.5; see Corollary 3.6.

Theorem 3.5 shows some asymptotic regularity of the solutions of (1.1) for the autonomous case, where f and g satisfythe nonlinear balance conditions given in [1] to prevent the solution from blow-up. As an immediate result of Theorem 3.5,combining with the L2(Ω)-asymptotic compactness (obtained in [1,6]) and the interpolation inequality, we can obtain theasymptotic compactness in Lp+1(Ω) immediately, and so the existence of a compact global attractor A . Note that althoughA only belongs to H1(Ω) ∩ Lp+1(Ω), the attraction is w.r.t. H1(Ω) ∩ Lp+δ(Ω)-norm. Moreover, Theorem 3.5 also impliesthat: the solution u(t)will be uniformly (w.r.t. time t and initial data) bounded inH1(Ω)∩Lp+δ(Ω) for any δ ∈ [0, ∞) as t issufficiently large. Theorem 4.5 establishes the existence of a compact uniform attractor in H1(Ω) with dissipative internaland boundary nonlinearities for the non-autonomous case. This theorem shows that the compactness of the process inH1(Ω) did not depend on the compactness of the process in Lp(Ω) for arbitrary p > 2; that is, the compactness of theprocess in H1(Ω) did not depend on the growth exponents of nonlinear terms f and g .

This paper is organized as follows: in Section 2, we give some preparations for our consideration; in Section 3, for theautonomous cases, i.e., h(x, t) = h(x), we prove some asymptotic regularity of the solutions; in Section 4, for the non-autonomous cases, the existence of a uniform attractor in H1(Ω) is obtained.

2. Preliminaries

In this section, we give some auxiliary results which will be used later.At first, we will give some basic notation used throughout this paper. Hereafter, we use the notation γ for the trace

operator u → u|Γ , and ‖ · ‖ and ‖ · ‖Γ stand for the norm in L2(Ω) and L2(Γ ), ⟨·, ·⟩ and ⟨·, ·⟩Γ stand for the inner productin L2(Ω) and L2(Γ ), respectively. |e| denotes the Lebesgue measure of e, C, Ci denote general positive constants, i = 1, . . . ,which will be different in different estimates.

Next, we introduce the spaces of time-dependent external forcing h(x, t) to be considered in this paper.

Definition 2.1 ([8]). A function ϕ is said to be translation bounded in L2loc(R; X), if

‖ϕ‖2b = sup

t∈R

∫ t+1

t‖ϕ‖

2X ds < +∞.

Denote by L2b(R; X) the set of all translation bounded functions in L2loc(R; X).

L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079 1071

Definition 2.2. A function ϕ is said to be uniformly bounded in L2(Ω) with respect to t ∈ R, i.e., ϕ ∈ L∞(R; L2(Ω)), if thereexists a positive constant K , such that

supt∈R

‖ϕ‖L2(Ω) ≤ K .

We now introduce a class of functions that was defined first in [24].

Definition 2.3 ([24]). A function ϕ ∈ L2loc(R; X) is said to be normal if for any ε > 0, there exists η > 0 such that

supt∈R

∫ t+η

t‖ϕ‖

2X ds ≤ ε.

Denote by L2n(R; X) the set of all normal functions in L2loc(R; X).

Lemma 2.4 ([24]). If ϕ0 ∈ L2n(R; X), then for any τ ∈ R,

limγ→∞

supt≥τ

∫ t

τ

e−γ (t−s)‖ϕ(s)‖2

X ds = 0,

uniformly (w.r.t. ϕ ∈ H(ϕ0)), where H(ϕ0) = ϕ0(t + h) | h ∈ R.

Obviously, L2n(R; X) ⊂ L2b(R; X), for more details we refer to [24].Then, we will recall the following version of Poincaré’s inequalities, which were proved in [1].

Lemma 2.5 ([1]). There exists a positive constant c0(Ω, 1) such that for every ϕ ∈ W 1,1(Ω)ϕ −1

|Γ |

∫Γ

ϕ

L1(Ω)

≤ c0(Ω, 1) ‖∇ϕ‖L1(Ω) . (2.1)

Lemma 2.6 ([1]). For every u ∈ H1(Ω) and ε > 0, there exists a positive constant Cε , such that∫Γ

u2 dx ≤ ε

∫Ω

|∇u|2 dx + Cε

∫Ω

u2 dx. (2.2)

3. Autonomous cases: h(x, t) = h(x)

In this section, we will consider the autonomous case of (1.1), that isut − ∆u + f (u) = h(x), in Ω,∂u∂ν

+ g(u) = 0, on Γ ,

u(x, 0) = u0(x),

(3.1)

where h(x) ∈ L∞(Ω). The functions f and g ∈ C1(R, R), satisfy the following conditions:

lim|s|→∞

f (s)s|s|p+1

= Cf > 0, (3.2)

lim|s|→∞

g(s)s|s|q+1

= Cg < 0, (3.3)

here p > 1, q > 1.

3.1. Mathematical setting

We start with the following existence and uniqueness results. For more details, we refer to [16,17,1].

Theorem 3.1 ([16,17,1]). Let Ω be a bounded domain of RN(N ⩾ 3) with a smooth boundary Γ , h(x) ∈ L∞(Ω), f and gsatisfy (3.2) and (3.3). If either

(i) p + 1 > 2q, or(ii) p + 1 = 2q and

Cf > C2g q,

1072 L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079

then for any initial data u0 ∈ H1(Ω) and any T > 0, the solution u(t) of problem (3.1) is globally defined and satisfies

u(t) ∈ C([0, T ]; L2(Ω)) ∩ L2(0, T ;H1(Ω)) ∩ Lp+1(0, T ; Lp+1(Ω)).

Observe that: in Theorem 3.1, we are only concerned with the situation p + 1 > 2q or p + 1 = 2q and Cf > C2g q, and

in this case all solutions of (3.1) are globally defined. It is remarkable that this is an optimal case, since if p + 1 < 2q orp + 1 = 2q and Cf < C2

g q, then some solutions will blow up in finite time; see [17,1] for more details.In the remainder of this section, by Theorem 3.1, we denote by S(t)t⩾0 the operator semigroup associated with the

solutions of (3.1)–(3.3). Without loss of generality, we always assume that f and g satisfy the condition p + 1 > 2q orp + 1 = 2q and Cf > C2

g q throughout this section.Next, from [1,23], we have the following dissipative results.

Theorem 3.2 ([1,23]). Let Ω be a bounded domain of RN(N ⩾ 3)with a smooth boundaryΓ , h(x) ∈ L∞(Ω), f and g satisfy (3.2)and (3.3). If either

(i) p + 1 > 2q, or(ii) p + 1 = 2q and

Cf > M2C2g q,

whereM =|Γ |

|Ω|c0(Ω, 1) and c0(Ω, 1) is given in (2.1). Then the semigroup S(t)t⩾0 has a positively invariant bounded absorbing

set B0 in H1(Ω) ∩ Lp+1(Ω).

As an immediate result of Theorem 3.2, we have the following existence of a global attractor in L2(Ω).

Corollary 3.3. Under the assumption of Theorem 3.2, the semigroup S(t)t⩾0 corresponding to (3.1) has a global attractor A inL2(Ω).

Hereafter, from Theorem 3.2, we denote one of the positively invariant absorbing sets by B0 with

B0 ⊂ u ∈ H1(Ω) ∩ Lp+1(Ω) : ‖u‖2H1(Ω)

+ ‖u‖p+1Lp+1(Ω)

⩽ M,

note that here positively invariant means S(t)B0 ⊂ B0 for any t ⩾ 0.Moreover, thanks to Theorem 3.2, without loss of generality, hereafter we assume u0 ∈ B0.

3.2. Asymptotic regularity

In this subsection, our main goal is to study asymptotic regularity of the solution of systems (3.1)–(3.3). The idea of theproof comes from [15,1,6].

At first, we will apply the Moser–Alikakos iteration technique [15] to prove the following induction estimates (e.g., seealso [10,12,13]).

Lemma 3.4. For each k = 0, 1, 2, . . . , there exist two positive constants Tk and Mk, which depend only on k, p, q,N and‖B0‖H1(Ω)∩Lp+1(Ω) (denotes the bounds of bounded absorbing set B0 in H1(Ω) ∩ Lp+1(Ω)), such that if either

(i) p + 1 > 2q, or(ii) p + 1 = 2q and

Cf > M2C2g q

2,

where M =|Γ |

|Ω|c0(Ω, 1), and for any u0 ∈ B0 and t ⩾ Tk, we have∫

Ω

|u(t)|2

NN−2

k⩽ Mk, (Ak)

and ∫ t+1

t

∫Ω

|u(t)|2

NN−2

k+1 N−2N

⩽ Mk. (Bk)

Proof. (i) Initialization of the induction (k = 0).From Theorem 3.2, we can deduce (A0) immediately. To prove (B0), we multiply (3.1) by u and integrate over Ω , we get

that

L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079 1073

12

ddt

∫Ω

|u|2 dx +

∫Ω

|∇u|2 dx +

∫Ω

f (u)u dx +

∫Γ

g(u)u dx =

∫Ω

h(x)u dx. (3.4)

The inequality above can be rewritten as

12

ddt

∫Ω

|u|2 dx +

∫Ω

|∇u|2 dx +

∫Ω

f (u)u +

|Γ |

|Ω|g(u)u

dx

−|Γ |

|Ω|

∫Ω

g(u)u −

1|Γ |

∫Γ

g(u)u

dx =

∫Ω

h(x)u dx, (3.5)

using Lemma 2.5, we have

|Γ |

|Ω|

∫Ω

g(u)u −

1|Γ |

∫Γ

g(u)u

dx ≤ ε‖∇u‖2L2(Ω)

+M2

4ε‖g ′(u)u + g(u)‖2

L2(Ω), (3.6)

where M =|Γ |

|Ω|c0(Ω, 1). Combining (3.5) and (3.6), we obtain that

12

ddt

∫Ω

|u|2 dx + (1 − ε)

∫Ω

|∇u|2 dx +

∫Ω

f (u)u +

|Γ |

|Ω|g(u)u −

M2

4ε(g ′(u)u + g(u))2

dx ≤

∫Ω

h(x)u dx. (3.7)

Following the similar arguments as in [1,6], using Young’s and Poincaré inequalities, we know that either in case (i) or (ii),we obtain

12

ddt

∫Ω

|u|2 dx + (1 − 2ε)∫

Ω

|∇u|2 dx + C1

∫Ω

|u|p+1 dx ≤ C2, (3.8)

where C2 depends on |Ω|, ‖h‖L∞ . Then, for any t ⩾ 0, integrating the above inequality over [t, t+1] and using Theorem 3.2,we deduce that∫ t+1

t

∫Ω

|∇u(x, s)|2 dx ds ⩽ CM for all t ⩾ 0, (3.9)

where CM depends on |Ω|, ‖h‖L∞ ,M . Using the Sobolev embedding (e.g., see Adams and Fourier [25])

H1(Ω) → L2NN−2 (Ω),

from (3.9) we have, for all t ⩾ 0,∫ t+1

t

∫Ω

|u(x, s)|2NN−2 dx

N−2N

ds ⩽ C

∫ t+1

t

∫Ω

|∇u(x, s)|2 dx ds ⩽ CM,N , (3.10)

where C is the constant of embedding H1(Ω) → L2NN−2 (Ω), note that here C depends only on N , and CM,N depends on

|Ω|, ‖h‖L∞ ,M,N . This implies (B0) holds.(ii) The induction argument.We now assume that (Ak) and (Bk) hold for k ⩾ 1, then we need to prove (Ak+1) and (Bk+1) hold.

Multiplying (3.1) by |u|2

NN−2

k+1−1 and integrating over Ω , we obtain that

12

N − 2N

k+1 ddt

∫Ω

|u|2

NN−2

k+1

dx +

2

NN − 2

k+1

− 1

N − 2N

2(k+1) ∫Ω

∇ u NN−2

k+12 dx

+

∫Ω

f (u)|u|2

NN−2

k+1−1 dx +

∫Γ

g(u)|u|2

NN−2

k+1−1 dx

≤

∫Ω

h(x)|u|2

NN−2

k+1−1 dx, (3.11)

then the inequality above can be rewritten as

12

N − 2N

k+1 ddt

∫Ω

|u|2

NN−2

k+1

dx +

2

NN − 2

k+1

− 1

N − 2N

2(k+1) ∫Ω

∇ u NN−2

k+12 dx

+

∫Ω

f (u)|u|2

N

N−2

k+1−1

+|Γ |

|Ω|g(u)|u|2

N

N−2

k+1−1

dx

1074 L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079

−|Γ |

|Ω|

∫Ω

g(u)|u|2

N

N−2

k+1−1

−1

|Γ |

∫Γ

g(u)|u|2

NN−2

k+1−1

dx

≤

∫Ω

h(x)|u|2

NN−2

k+1−1 dx. (3.12)

Using Lemma 2.5, we obtain |Γ |

|Ω|

∫Ω

g(u)|u|2

N

N−2

k+1−1

−1

|Γ |

∫Γ

g(u)|u|2

NN−2

k+1−1

dx

≤M2

4ε

1 NN−2

k+1 g′(u)u +

2 NN−2

k+1− 1 N

N−2

k+1 g(u)

|u|

N

N−2

k+1−1

2

L2(Ω)

+ ε

∇ u NN−2

k+12L2(Ω)

, (3.13)

where M =|Γ |

|Ω|c0(Ω, 1). Combining (3.12) and (3.13), we obtain

12

N − 2N

k+1 ddt

∫Ω

|u|2

NN−2

k+1

dx

+

2

NN − 2

k+1

− 1

N − 2N

2(k+1)

− ε

∫Ω

∇ u NN−2

k+12 dx

+

∫Ω

f (u)u +

|Γ |

|Ω|g(u)u

|u|2

N

N−2

k+1−2 dx

−M2

4ε NN−2

2(k+1)

∫Ω

g ′(u)u +

2

NN − 2

k+1

− 1

g(u)

2

|u|2

NN−2

k+1−2 dx

≤

∫Ω

h(x)|u|2

NN−2

k+1−1 dx. (3.14)

We set

H(u) = f (u)u +|Γ |

|Ω|g(u)u −

M2

4ε NN−2

2(k+1)

g ′(u)u +

2

NN − 2

k+1

− 1

g(u)

2

.

Applying the similar methods as that in [1,6], from the assumptions it is clear that the leading terms for |u| ≫ 1 are

f (u) ∼ Cf |u|p−1u and g(u) ∼ Cg |u|q−1u.

So the leading terms for H(u) are

Cf |u|p+1+

|Γ |

|Ω|Cg |u|q+1

−M2

4ε NN−2

2(k+1)

q + 2

N

N − 2

k+1

− 1

2

C2g |u|2q.

Therefore, we have

(i) When p + 1 > 2q, the coefficient of the highest order term in H(u) is Cf , which is strictly positive.(ii) If p + 1 = 2q and p, q > 1, then the coefficient of the highest order term in H(u) is Cf −

M2

4ε

NN−2

2(k+1)

q + 2

NN−2

k+1− 1

2C2g , we can choose suitable ε such that the coefficient of the highest order term

in H(u) is positive iff Cf > M2C2g q

2.

In any case, the dynamical system defined by (3.1) is dissipative and we have

H(u) ≥ C3|u|p+1− C4, (3.15)

where C3, C4 > 0 are suitable constants. Combining (3.14) and (3.15), we obtain that there exists a constant C5 such that

12

N − 2N

k+1 ddt

∫Ω

|u|2

NN−2

k+1

dx +

2

NN − 2

k+1

− 1

N − 2N

2(k+1)

− ε

∫Ω

∇ u NN−2

k+12 dx

+ C5

∫Ω

|u|p+1|u|2

N

N−2

k+1−2 dx

L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079 1075

≤ C6 +

∫Ω

h(x)|u|2

NN−2

k+1−1 dx, (3.16)

where C6 depends on |Ω|, using Young’s inequality, we have

12

N − 2N

k+1 ddt

∫Ω

|u|2

NN−2

k+1

dx

+

2

NN − 2

k+1

− 1

N − 2N

2(k+1)

− ε

∫Ω

∇ u NN−2

k+12 dx + C5

∫Ω

|u|p+2

NN−2

k+1−1 dx

⩽C + C∫

Ω

|u|2

NN−2

k+1

dx, (3.17)

whereC depends on |Ω|, ‖h‖L∞ . Then, combining with (Bk), applying the uniform Gronwall lemma to (3.17), we can get(Ak+1) immediately. For (Bk+1), we integrate the above inequality over [t, t + 1] and using (Ak+1), we have∫ t+1

t

∫Ω

∇ u NN−2

k+12 dx ds ⩽ Mk+1 for all t ≥ 0, (3.18)

where Mk+1 depends on k, p, q,N,M, ‖h‖L∞ . By the embedding H1(Ω) → L2NN−2 (Ω) again, we have∫

Ω

|u|

NN−2

k+1·

2NN−2 dx

N−2N

⩽ C

∫Ω

∇ u NN−2

k+12 dx, (3.19)

combining (3.18) and (3.19), we deduce (Bk+1) immediately.

The main result of this section is the following.

Theorem 3.5. Let Ω be a bounded smooth domain in RN(N ⩾ 3), f and g satisfy (3.2) and (3.3), h(x) ∈ L∞(Ω), and supposethat S(t)t≥0 is the semigroup generated by the solutions of Eq. (3.1), if either(i) p + 1 > 2q, or(ii) p + 1 = 2q and

Cf > M2C2g q

2,

where M =|Γ |

|Ω|c0(Ω, 1) and c0(Ω, 1) is given in (2.1). Then, for any δ ∈ [0, ∞), there exists a bounded subset Bδ satisfying the

following properties:

Bδ = u ∈ H1(Ω) ∩ Lp+1(Ω) : ‖u‖H1(Ω)∩Lp+δ(Ω) ⩽ Λp,q,N,δ,‖h‖L∞ < ∞, (3.20)

and for any bounded subset B ⊂ H1(Ω), there exists a T = T (‖B‖H1(Ω), δ) such that

S(t)B ⊂ Bδ for all t ⩾ T , (3.21)

where the constant Λp,q,N,δ,‖h‖L∞ depends only on p, q,N, δ, ‖h‖L∞ , and ‖B‖H1(Ω) denotes the bounds of B in H1(Ω).

Proof. Based on Lemma 3.4, since N ⩾ 3, we have NN−2 > 1 and then

NN − 2

k

→ ∞ as k → ∞.

Hence, for any δ ∈ [0, ∞), we can take k ⩾ log NN−2

p+δ

2 , such that

p + δ ⩽ 2

NN − 2

k

,

therefore, (Ak) implies that: for any δ ∈ [0, ∞),

‖u(t)‖Lp+δ(Ω) ⩽ CM,p,q,δ,N,‖h‖L∞ for all t > 0. (3.22)

Then, combining with (Ak) and Theorem 3.2, we can define Bδ as follows:

Bδ =u ∈ H1(Ω) ∩ Lp+1(Ω) : ‖u‖H1(Ω)∩Lp+δ(Ω) ⩽ CM,p,q,δ,N,‖h‖L∞ < ∞

, (3.23)

so we can verify (3.21) immediately.

1076 L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079

This theorem shows some asymptotic regularity of the solutions of (3.1). Consequently, as a direct result of Theorem 3.5,combining with the L2(Ω)-asymptotic compactness (obtained in [1,6]) and the interpolation inequality, we can obtainimmediately the following results.

Corollary 3.6. Under the assumptions of Theorem 3.5, the semigroup S(t)t⩾0 has a compact global attractor A in Lp+1(Ω).

Remark 3.7. Applying the general framework devised in [26], it is easy to prove that the fractal dimension of A is finite inL2(Ω) (or the existence of an exponential attractor); then, from Theorem 3.5, we know that the fractal dimension of A isfinite in Lp+δ(Ω) for any δ ∈ [0, ∞) by using the interpolation inequality.

Remark 3.8. Note that: the assumption h(x) ∈ L∞(Ω) is only needed for the asymptotic regularity of the solutions, as forthe well-posedness and the existence of the global attractor, the assumption h(x) ∈ L2(Ω) would be sufficient; see [1,6,23]for more details.

Remark 3.9. In this section, we assume that Cf > 0, Cg < 0, in fact, if we consider the case of dissipative nonlinear termsf and g , i.e., Cf > 0, Cg > 0, then the assumption h(x) ∈ L∞(Ω) can be replaced by the weaker assumption: h(x) ∈ L2(Ω),and following the method of [13], we can obtain the better regularity of the solutions than mentioned in Theorem 3.5, wereport these analysis in another paper.

4. Non-autonomous case

In this section, we will discuss the non-autonomous case of (1.1), that is,ut − ∆u + f (u) = h(x, t), in Ω,∂u∂ν

+ g(u) = 0, on Γ ,

u(x, τ ) = uτ (x), in Ω,

(4.1)

where Ω is a bounded domain of RN(N ⩾ 3) with a smooth boundary Γ , h(x, t) ∈ L2b(R; L2(Ω)). The functions f andg ∈ C1(R, R), satisfy the following conditions:

lim|s|→∞

f (s)s|s|p+1

= Cf > 0, (4.2)

lim|s|→∞

g(s)s|s|q+1

= Cg > 0, (4.3)

andf ′(s) ≥ −l, g ′(s) ≥ −m, (4.4)

where p > 1, q > 1, l,m > 0.

4.1. Mathematical setting

At first, we define the symbol space Σ for (4.1). Taking a fixed symbol σ0(s) = h0(s), h0(s) ∈ L2b(R; L2(Ω)). We denoteby L2,wloc (R; L2(Ω)) the space L2loc(R; L2(Ω)) endowed with local weak convergence topology. Set

Σ0 = h0(s + h)|h ∈ R, (4.5)and let

Σ be the closure of Σ0 in L2,wloc (R; L2(Ω)). (4.6)The systems (4.1) can be rewritten in the operator form:

∂ty = Aσ(t)(y), y|t=τ = yτ , (4.7)where σ(t) = h(t) is the symbol of Eq. (4.7).

Next, similar to the autonomous cases (e.g., see [16,17,1]), for each h ∈ Σ , we can also easily obtain the following wellposedness result and the time-dependent terms make no essential complications.

Theorem 4.1. Let Ω be a bounded domain of RN(N ⩾ 3) with a smooth boundary Γ , h(t) be translation bounded inL2loc(R; L2(Ω)), f and g satisfy (4.2)–(4.4). Then for any initial data uτ ∈ H1(Ω), and any τ , T ∈ R, T > τ , the solutionu(t) of problem (4.1) is globally defined and satisfies

u(t) ∈ C([τ , T ]; L2(Ω)) ∩ L2(τ , T ;H1(Ω)) ∩ Lp+1(τ , T ; Lp+1(Ω)).

From Theorem 4.1, we know that problem (4.1)–(4.4) is well posed for all σ(s) ∈ Σ and generates a family of processesUσ (t, τ ), σ ∈ Σ given by the formula Uσ (t, τ )yτ = y(t). The y(t) is the solution of (4.1)–(4.4).

L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079 1077

Moreover, similar to the autonomous case (e.g., Theorem 3.2), we can get the following results about the boundeduniformly (w.r.t. σ ∈ Σ) absorbing set for the family of processes Uσ (t, τ ), σ ∈ Σ .

Theorem 4.2 ([1,23]). Assume that h(t) is translation bounded in L2loc(R; L2(Ω)), f and g satisfy (4.2)–(4.4). Then the family ofprocesses Uσ (t, τ ), σ ∈ Σ corresponding to (4.1) has a bounded uniformly (w.r.t.σ ∈ Σ) absorbing set B1 in H1(Ω)∩Lp+1(Ω).That is, for any τ ∈ R and any bounded subset B ⊂ H1(Ω), there exist two positive constants T = T (B, τ ) > τ and ρ , such that∫

Ω

|∇u(t)|2 dx +

∫Ω

|u(t)|p+1 dx ⩽ ρ ∀ t ≥ T , uτ ∈ B, σ ∈ Σ,

where ρ depends on |Ω|, |Γ |, ‖h(t)‖2b .

As an immediate result of Theorem 4.2, we have the following result.

Corollary 4.3. Under the assumption of Theorem 4.2, the family of processes Uσ (t, τ ), σ ∈ Σ corresponding to (4.1) has acompact uniform (w.r.t. σ ∈ Σ) attractor AΣ in L2(Ω).

4.2. Existence of a uniform attractor in H1(Ω)

In this subsection, we prove the existence of a uniform attractor in H1(Ω). In what follows, we always denote the weakdifferential of h(t) with respect to t by h′(t).

First, we will give some a priori estimates about ut .

Lemma 4.4. Assume that h(t) and h′(t) are translation bounded in L2loc(R; L2(Ω)), f and g satisfy (4.2)–(4.4), then for any τ ∈ Rand any bounded subset B ⊂ H1(Ω), there exist two positive constants T = T (B, τ ) > τ and M2, such that∫

Ω

|ut(s)|2 dx ⩽ M2 ∀ s ≥ T , uτ ∈ B, σ ∈ Σ, (4.8)

where ut(s) =ddt (Uσ (t, τ )uτ ) |t=s and M2 is a positive constant which depends on |Ω|, |Γ |, ρ, ‖h(t)‖2

b, ‖h′(t)‖2

b .

Proof. Multiplying (4.1) by u and integrating by parts, we get

12

ddt

∫Ω

|u|2 dx +

∫Ω

|∇u|2 dx +

∫Ω

f (u)u dx +

∫Γ

g(u)u dx =

∫Ω

h0(t)u dx. (4.9)

Combining with assumptions (4.2)–(4.4), Young’s inequality and Poincaré inequality, and letting F(s) = s0 f (τ ) dτ , G(s) = s

0 g(τ ) dτ , we obtain

ddt

∫Ω

|u|2 dx +

∫Ω

|∇u|2 dx + C1

∫Ω

F(u) dx + C2

∫Γ

G(u) dx ≤ Ck1,k3,|Ω|,|Γ | + C‖h0(t)‖2. (4.10)

Integrating the inequality above from t to t + 1, and using Theorem 4.2, we can find T0 > τ , such that for any t ≥ T0,∫ t+1

t

∫Ω

|∇u|2 dx + C1

∫Ω

F(u) dx + C2

∫Γ

G(u) dx

ds ≤ Ck1,k3,|Ω|,|Γ |,ρ,‖h(t)‖2b. (4.11)

On the other hand, multiplying (4.1) by ut , we deduce that∫Ω

|ut |2 dx +

12

ddt

∫Ω

|∇u|2 dx +ddt

∫Ω

F(u) dx +

∫Γ

G(u) dx

=

∫Ω

h0(t)ut dx. (4.12)

Combining with (4.11) and (4.12), and using the uniformly Gronwall lemma, we obtain

‖∇u‖2+ 2

∫Ω

F(u) dx + 2∫

Γ

G(u) dx ≤ Ck1,k3,|Ω|,|Γ |,ρ,‖h(t)‖2b, ∀ t ≥ T0 + 1, uτ ∈ B, σ ∈ Σ . (4.13)

By differentiating (4.1) with external force h0(t) in the time and denoting v = ut , we have

vt − ∆v + f ′(u)v = h′

0(t), x ∈ Ω, (4.14)

∂v

∂ν+ g ′(u)v = 0, x ∈ Γ . (4.15)

Multiplying (4.14) by v, we obtain that

12

ddt

∫Ω

|v|2 dx +

∫Ω

|∇v|2 dx +

∫Ω

f ′(u)v2 dx +

∫Γ

g ′(u)v2 dx =

∫Ω

h′

0(t)v dx. (4.16)

1078 L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079

From (4.4), and using Lemma 2.6, we obtain that

12

ddt

∫Ω

|v|2 dx +

∫Ω

|∇v|2 dx ≤ l

∫Ω

|v|2 dx + m

∫Γ

|v|2 dx +

12

∫Ω

|v|2 dx +

12‖h′

0(t)‖2

≤ (l + Cεm)

∫Ω

|v|2 dx + εm

∫Ω

|∇v|2 dx +

12

∫Ω

|v|2 dx +

12‖h′

0(t)‖2, (4.17)

so we have

ddt

∫Ω

|v|2 dx +

∫Ω

|∇v|2 dx ≤ (l + Cεm)

∫Ω

|v|2 dx + ‖h′

0(t)‖2. (4.18)

On the other hand, integrating (4.12) from t to t + 1, and noticing (4.13), we have∫ t+1

t

∫Ω

|v|2 dx

ds ≤C, (4.19)

whereC depends on |Ω|, |Γ |, ρ, ‖h(t)‖2b . Combining (4.18) and (4.19), and using the uniform Gronwall lemma, we get∫

Ω

|ut(s)|2 dx ⩽ M2 ∀ s ≥ T , uτ ∈ B, σ ∈ Σ, (4.20)

whereM2 depends on |Ω|, |Γ |, ρ, ‖h(t)‖2b, ‖h

′(t)‖2b .

From now on, we assume that the external force h(x, t) ∈ L∞(R; L2(Ω)).Now, we prove the existence of a uniform attractor in H1(Ω).

Theorem 4.5. Assume that h(t) ∈ L∞(R; L2(Ω)) and h′(t) is translation bounded in L2loc(R; L2(Ω)), f and g satisfy (4.2)–(4.4).Then the family of processes Uσ (t, τ ), σ ∈ Σ corresponding to (4.1) has a compact uniform (w.r.t. σ ∈ Σ) attractor AΣ inH1(Ω) and AΣ satisfies:

AΣ = ω0,Σ (B1) =

σ∈Σ

Kσ (s), ∀ s ∈ R,

where Kσ (s) is the section at t = s of the kernel Kσ of the process Uσ (t, τ ) with symbol σ .

Proof. Let B1 be aH1(Ω) ∩ Lp+1(Ω)

-uniformly (w.r.t. σ ∈ Σ) bounded absorbing set obtained in Theorem 4.2, then we

only need to show that

for any uτn ⊂ B1, σn ⊂ Σ and tn → ∞,

Uσn(tn, τn)uτn∞

n=1 is precompact in H1(Ω). (4.21)

In fact, from Theorem 4.2, by applying the compact embedding H1(Ω) → L2(Ω), we know that Uσn(tn, τn)uτn∞

n=1 isprecompact in L2(Ω). Without loss of generality, we assume that Uσn(tn, τn)uτn

∞

n=1 is a Cauchy sequence in L2(Ω).Now, we prove that Uσn(tn, τn)uτn

∞

n=1 is a Cauchy sequence in H1(Ω).Denote by uσn

n (tn) := Uσn(tn, τn)uτn , after standard transformations and by use of Lemma 2.6, we obtain

‖∇(uσnn (tn) − uσm

m (tm))‖2≤

−

ddt

uσnn (tn) + σn +

ddt

uσmm (tm) − σm, uσn

n (tn) − uσmm (tm)

+ l‖uσn

n (tn) − uσmm (tm)‖2

+ m‖uσnn (tn) − uσm

m (tm)‖2Γ

≤

∫Ω

ddt uσnn (tn) −

ddt

uσmm (tm)

|uσnn (tn) − uσm

m (tm)| +

∫Ω

|σn − σm| |uσnn (tn) − uσm

m (tm)|

+ (l + Cεm)‖uσnn (tn) − uσm

m (tm)‖2+ εm‖∇(uσn

n (tn) − uσmm (tm))‖2, (4.22)

this yields that

(1 − εm)‖∇(uσnn (tn) − uσm

m (tm))‖2≤

ddt

uσnn (tn) −

ddt

uσmm (tm)

‖uσnn (tn) − uσm

m (tm)‖

+ ‖σn − σm‖ ‖uσnn (tn) − uσm

m (tm)‖ + (l + Cεm)‖uσnn (tn) − uσm

m (tm)‖2, (4.23)

combining with Theorem 4.2 and Lemma 4.4, and by applying the embedding H1(Ω) → L2(Ω) again, we have (4.21)immediately. Then, by using the similar argument as in [27] (see Pata and Zelik [28] for autonomous case), we use theclosed process to obtain the structure of AΣ in H1(Ω).

L. Yang / Nonlinear Analysis: Real World Applications 13 (2012) 1069–1079 1079

Remark 4.6. In Theorem 4.5, the assumption h(x, t) ∈ L∞(R; L2(Ω)) is only needed to guarantee the uniform asymptoticcompactness in H1(Ω). In fact, if we are only concerned with the existence of the uniform attractor in Lp+1(Ω), then thisassumption can be replaced by the weaker assumption: h(x, t) ∈ L2n(R; L2(Ω)), i.e., h(x, t) is normal (see Definition 2.3);for more details we refer to [23,24].

Remark 4.7. The technique (scheme) used in Section 4 is also applicable to the quasi-linear parabolic equationwith dynamicboundary conditions. e.g., the model and nonlinearity considered in [29].

Acknowledgments

The author would like to express sincere thanks to the referee for his/her valuable comments and suggestions, also toProf. Ciprian G. Gal for many useful comments and discussions. This work was supported by the NSFC Grant 11101404, theFundamental Research Funds for the Central Universities Grant lzujbky-2011-45 and the Fund of Physics & Mathematics ofLanzhou University Grant LZULL200903.

References

[1] A. Rodríguez-Bernal, A. Tajdine, Nonlinear balance for reaction diffusion equations under nonlinear boundary conditions: dissipativity and blow-up,J. Differential Equations 169 (2001) 332–372.

[2] J. Arrieta, A.N. Carvalho, A. Rodríguez-Bernal, Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds, Comm. PartialDifferential Equations 25 (1/2) (2000) 1–37.

[3] J. Arrieta, A.N. Carvalho, A. Rodríguez-Bernal, Upper semi-continuity of attractors for parabolic problems with localized large diffusion and nonlinearboundary condition, J. Differential Equations 168 (2000) 33–59.

[4] A.N. Carvalho, S.M. Oliva, A.L. Pereira, A. Rodríguez-Bernal, Attractors for parabolic problems with nonlinear boundary conditions, J. Math. Anal. Appl.207 (1997) 409–461.

[5] M. Chipot, P. Quittner, Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions, J. Dyn.Differential Equations 16 (2004) 91–138.

[6] A. Rodríguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data, J. DifferentialEquations 181 (2002) 165–196.

[7] A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.[8] V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc, Providence, RI, 2002.[9] J.W. Cholewa, T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000.

[10] J.C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, 2001.[11] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.[12] M. Marion, Attractors for reactions-diffusion equations: existence and estimate of their dimension, Appl. Anal 25 (1987) 101–147.[13] C.Y. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations 248 (2010) 342–362.[14] C.K. Zhong, M.H. Yang, C.Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear

reaction–diffusion equations, J. Differential Equations 223 (2006) 367–399.[15] A.D. Alikakos, An application of the invariance principle to reaction-diffustion equations, J. Differential Equations 33 (1979) 201–225.[16] J. Arrieta, A.N. Carvalho, A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential

Equations 165 (1999) 376–406.[17] J. Arrieta, On boundedness of solutions of reaction–diffusion equations with nonlinear boundary conditions, Proc. Amer. Math. Soc. 136 (2008)

151–160.[18] J. Arrieta, A. Rodríguez-Bernal, Localizationnear the boundary of blow-up for reaction–diffusion equationswithnonlinear boundary conditions, Comm.

Partial Differential Equations 29 (2004) 1127–1148.[19] F. Quirós, J.D. Rossi, J.L. Vázquez, Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions, Comm. Partial

Differential Equations 27 (2002) 395–424.[20] P. Quittner, A. Rodríguez-Bernal, Complete and energy blow-up in parabolic problems with nonlinear boundary conditions, Nonlinear Anal. 62 (2005)

863–875.[21] C.G. Gal, M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst. 12 (2008) 1009–1040.[22] C.G. Gal, M. Warma, Well-posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions,

Differential Integral Equations 23 (2010) 327–358.[23] L. Yang, M.H. Yang, Attractors of non-autonomous reaction–diffusion equation with nonlinear boundary condition, Nonlinear Anal. Real World Appl.

11 (2010) 3946–3954.[24] S.S. Lu, H.Q. Wu, C.K. Zhong, Attractors for nonautonomous 2D Navier–Stokes equations with normal external forces, Discrete Contin. Dyn. Syst. 13

(2005) 701–719.[25] R.A. Adams, J.J.F. Fourier, Sobolev Spaces, 2nd ed. Academic Press, 2003.[26] M. Efendiev, A. Miranville, S.V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Roy. Soc.

Edinburgh 135A (2005) 703–730.[27] L. Yang, Uniform attractors for the closed process and applications to the reaction–diffusion equation with dynamical boundary condition, Nonlinear

Anal. 71 (2009) 4012–4025.[28] V. Pata, S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal 6 (2007) 481–486.[29] L. Yang, M.H. Yang, On uniform attractor for non-autonomous parabolic equation with dynamic boundary condition (submitted for publication).