TO PARABOLIC EQUATIONS

TO PARABOLIC EQUATIONS

GORO AKAGI

Received 11 December 2003

Asymptotic behavior of solutions of some parabolic equation associated with the p- Laplacian as p→+∞is studied for the periodic problem as well as the initial-boundary value problem by pointing out the variational structure of the p-Laplacian, that is,

∂ϕp(u)= −∆pu, whereϕp:L²(Ω)→[0, +∞]. To this end, the notion of Mosco conver- gence is employed and it is proved thatϕpconverges to the indicator function over some closed convex set onL²(Ω) in the sense of Mosco asp→+∞; moreover, an abstract the- ory relative to Mosco convergence and evolution equations governed by time-dependent subdifferentials is developed until the periodic problem falls within its scope. Further ap- plication of this approach to the limiting problem of porous-medium-type equations, such asut=∆|u|^m−2uasm→+∞, is also given.

1. Introduction

The so-called p-Laplacian∆p given below could be regarded as a nonlinear differential operator generalizing the usual linear Laplacian:

∆pu(x) := ∇ ·∇u(x)^p−2∇u(x), 1< p <+∞. (1.1) This paper is motivated by the following naive question: what is the limit of∆pas p→ +∞? This limiting problem was studied by several authors and their results were applied in various fields; for example, growing sandpile model [2], macroscopic models for type- II superconductors [1,4,13], and so on. In order to figure out the substantial features of this problem, we here recall the variational structure ofp-Laplacian:

−∆pu=dI_p(u), I_p(u) :=1 p

Ω

∇u(x)^pdx ∀u∈W₀^1,p(Ω), (1.2)

wheredI_p(u) denotes the Fr´echet derivative of the functionalI_p atu, and we intend to investigate the limit of the functionalIpinstead of∆pasp→+∞.

Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:11 (2004) 907–933 2000 Mathematics Subject Classification: 34G25, 40A30, 47J35 URL:http://dx.doi.org/10.1155/S1085337504403030

However, it is easily expected that the limit ofI_pmay not belong to the class of Fr´echet differentiable functionals. On the other hand, from the viewpoint of studies on evolution equations, it is convenient for applications to extendIponL²(Ω) as follows:

ϕ_p(u) :=





Ip(u) ifu∈W0^1,p(Ω),

+∞ ifu∈L²(Ω)\W0^1,p(Ω). (1.3) Then it is well known thatϕpis no longer Fr´echet differentiable onL²(Ω), but lower semi- continuous and convex onL²(Ω); moreover, its subdifferential∂L²(Ω)ϕp(u) coincides with

−∆puin the distribution sense.

Several authors also studied the asymptotic behavior of solutions for the following initial-boundary value problem asp→+∞:

∂u

∂t(x,t)−∆pu(x,t)=0, (x,t)∈Ω×(0,T), u(x,t)=0, (x,t)∈∂Ω×(0,T),

u(x, 0)=u0(x), x∈Ω.

(1.4)

Here it is well known that (1.4) can be reduced to the following abstract Cauchy problem:

du

dt(t) +∂L²(Ω)ϕp

u(t) =0 inL²(Ω), 0< t < T, u(0)=u0.

(1.5) According to the previous studies, for example, [2,4], every solutionupof (1.5) con- verges touasp→+∞and the limitugives a solution of the following Cauchy problem:

du

dt(t) +∂_L²_(Ω)ϕ_∞u(t) 0 inL²(Ω), 0< t < T, u(0)=u0,

(1.6) whereϕ_∞is defined onL²(Ω) by

ϕ_∞(u) :=





0 ifu∈H0¹(Ω),|∇u|L^∞(Ω)≤1,

+∞ otherwise. (1.7)

Hence one can easily expect thatϕpconverges toϕ_∞asp→+∞in a certain sense; how- ever, it is not so obvious in what sense it is realized. In this paper, we prove thatϕ_pcon- verges toϕ_∞onL²(Ω) in the sense of Mosco asp→+∞; moreover, we discuss the asymp- totic behavior of solutions for (1.4) as p→+∞in a more general setting. These results will be shown inSection 3.1whereas the definition of Mosco convergence will be given inSection 2. Moreover, our method can also be applied to the porous medium equation

∂u

∂t(x,t)−∆|u|^m−2u(x,t)=0, (x,t)∈Ω×(0,T). (1.8) InSection 3.2, we deal with the asymptotic behavior of solutions for (1.8) asm→+∞.

To formulate our results in an abstract form, we work in a more generalized setting, that is, we consider the following abstract evolution equations in a real Hilbert spaceH governed by time-dependent subdifferential operators∂Hϕ^t_n:

dun

dt (t) +∂_Hϕ^t_nu_n(t) f_n(t) inH, 0< t < T,n∈N, (1.9) where fn∈L²(0,T;H), fn→ f strongly inL²(0,T;H), andϕ^t_nis a time-dependent proper lower semicontinuous convex functional fromHinto (−∞, +∞] such thatϕ^t_n→ϕ^tonH in the sense of Mosco asn→+∞. For the case whereϕ^t does not depend ont, that is, ϕ^t=ϕ, we can find related results in [3]. For the general case, we refer to Kenmochi [10].

However, all of the previous studies were done on the Cauchy problem for (1.9). As for the periodic problem for (1.9), there seems to be no attempt yet. The main objective here is to investigate the periodic problem as well as the Cauchy problem. The Cauchy problem has a unique solution, and the uniqueness of solution plays an essential role in deriving the convergence ofunasn→+∞in [3,10]. On the other hand, in general, periodic solution is not unique. Hence the same procedure as in [3,10] breaks down.

To cope with this difficulty, we introduce a remedy based on a compactness argument under a compactness assumption on the level set of{ϕ^t_n}n∈N. This result will be illustrated in the next section.

2. Evolution equations and Mosco convergence

This section deals with the following evolution equation (E(ϕ^t,f)) in a Hilbert spaceH.

(E(ϕ^t,f))

du

dt(t) +∂_Hϕ^tu(t) f(t) inH, 0< t < T, (2.1) where f _∈L¹(0,T;H) and∂_Hϕ^t is the subdifferential of a proper lower semicontinuous convex functionalϕ^t:H→(−∞, +∞] for everyt∈[0,T].

Throughout this paper, we denote byΨ(X) the set of all proper lower semicontinuous convex functionalsφfrom a Hilbert spaceXinto (−∞, +∞], where “proper” means that φ≡+∞. Moreover, the subdifferential∂_Xφofφ∈Ψ(X) is defined as follows:

∂_Xφ(u) :=

ξ∈X;φ(v)−φ(u)≥(ξ,v−u)_X∀v∈D(φ), (2.2) where (·,·)Xdenotes the inner product ofXandD(φ) is theeffective domainofφgiven by

D(φ) :=

u∈X;φ(u)<+∞

. (2.3)

Moreover, the domainD(∂_Xφ) of∂_Xφis defined by D∂Xφ :=

u∈D(φ);∂Xφ(u)= ∅

. (2.4)

Now solutions of (E(ϕ^t,f)) are defined as follows.

Definition 2.1. A functionu_∈C([0,T];H) is said to be a strong solution of (E(ϕ^t,f)) if the following are both satisfied:

(i)uis anH-valued absolutely continuous function on [0,T];

(ii)u(t)∈D(∂_Hϕ^t) for a.e.t∈(0,T) and there exists a sectiong(t)∈∂_Hϕ^t(u(t)) such that

du

dt(t) +g(t)= f(t) inHfor a.e.t∈(0,T). (2.5) Moreover, a functionu∈C([0,T];H) is said to be a weak solution of (E(ϕ^t,f)) if there exist sequences (fn)⊂L¹(0,T;H) and (un)⊂C([0,T];H) such thatunis a strong solution of (E(ϕ^t,fn)), fn→f strongly inL¹(0,T;H), andun→ustrongly inC([0,T];H).

We next introduce a notion of the convergence of functionals.

Definition 2.2. LetXbe a Hilbert space. Let (ϕn) be a sequence inΨ(X) and letϕ∈Ψ(X).

Thenϕ_n→ϕonX in the sense of Mosco asn→+∞if the following conditions are all satisfied.

(1) For allu∈D(ϕ), there exists a sequence (un) inXsuch thatun→ustrongly inX andϕ_n(u_n)→ϕ(u).

(2) Let (un) be a sequence inXsuch thatun→uweakly inX. Then lim infn→+∞ϕn(un)≥ ϕ(u).

Remark 2.3. The second condition inDefinition 2.2is equivalent to the following.

(2)Let (uk) be a sequence inXsuch thatuk→uweakly inXask→+∞and let (nk) be a subsequence of (n). Then lim infk→+_∞ϕnk(uk)≥ϕ(u).

Indeed, it is easily seen that (2) is derived immediately from (2). Hence it suffices to show that (2) implies (2). Suppose that (2) holds but (2)does not, that is, there exist a sequence (uk) and a subsequence (nk) of (n) such that

u_k−→u weakly inX, lim inf

k→+∞ ϕ_nku_k < ϕ(u). (2.6) Now define the sequence (˜u_n) as follows: ˜u_n=u_kifn∈[n_k,n_k+1) for eachk∈N. It then follows that ˜un→uweakly inXasn→+∞. Moreover, (2.6) yields

ϕ(u)>lim inf

k→+∞ ϕ_nku_{k =} lim

K→+_∞inf

k≥Kϕ_nku_k

≥ lim

K→+∞inf

n≥nKϕn

u˜n =lim inf

n→+_∞ ϕn

u˜n , (2.7)

which contradicts (2). Hence (2) implies (2).

In the following two subsections, we discuss the existence and uniqueness of solutions u_nfor (E(ϕ^t_n,f_n)) and the convergence ofu_nasn→+∞for the periodic problem as well as the Cauchy problem. To this end, we fix notations. From now on, we write{ϕ^t}t∈[0,T]∈ Ψ(α,β) for some functionsα,β: [0, +∞)×[0,T]→Rif the following hold true:

(i)ϕ^t∈Ψ(H) for allt∈[0,T];

(ii) there existsδ >0; for allt0∈[0,T] and allu0∈D(ϕ^t0), there exists a functionu fromIδ(t0) :=[t0−δ,t0+δ]∩[0,T] intoH; for allt∈Iδ(t0) and allr≥ |u0|H,

u(t)−u0

H≤α(r,t)−αr,t0 ϕ^t0u0 + 1^1/2,

ϕ^tu(t) ≤ϕ^t0u0 +β(r,t)−βr,t0 ϕ^t0u0 + 1. (2.8) Moreover, we say{ϕ^t}t∈[0,T]∈B(α,β,C0,{Mr}r≥0) for some functionsα,β: [0, +∞)× [0,T]→Rand constantsC0,{Mr}r≥0if the following are all satisfied.

(i){ϕ^t}t∈[0,T]∈Ψ(α,β).

(ii)ϕ^t(u)≥ −C0(|u|H+ 1) for allu∈Hand allt∈[0,T].

(iii) There exists a functionh: [0,T]→Hsuch that sup

t∈[0,T]

h(t)_H+ϕ^th(t) + _T

0

dh dt(t)

2 H

dt 1/2

≤C0. (2.9) (iv) For everyr∈[0, +∞), it follows that

_T

0

α(r,t)˙ ²dt+ _T

0

β(r,˙ t)dt≤M_r, (2.10)

where ˙αand ˙βdenote∂α/∂tand∂β/∂t, respectively.

Now let{ϕ^t}t∈[0,T]∈Ψ(α,β) be such thatα(r,·)∈W^1,2(0,T) andβ(r,·)∈W^1,1(0,T) for allr∈[0, +∞) and introduce the following functionalΦ^Sdefined onᏴS:=L²(0,S;H) for anyS∈(0,T]:

Φ^S(u) :=





 _S

0ϕ^tu(t) dt if the functiont−→ϕ^tu(t) ∈L¹(0,S),

+∞ otherwise.

(2.11)

Then we see thatΦ^S∈Ψ(ᏴS). Moreover, [9, Proposition 1.1] implies that for anyu,f ∈ ᏴS,

f ∈∂_ᏴSΦ^S(u)⇐⇒f(t)∈∂Hϕ^tu(t) for a.e.t∈(0,S). (2.12) The following proposition plays an important role in investigating the convergence of strong solutionsu_nfor (E(ϕ^t_n,f_n)) asn→+∞. For its proof, we refer to [10, Proposition 2.7.1].

Proposition 2.4 [10]. For every n∈N, let {ϕ^t_n}t∈[0,T]∈B(α_n,β_n,C0,{M_r}r≥0) and {ϕ^t}t∈[0,T]∈Ψ(α,β) be such that α_n(r,·),α(r,·)∈W^1,2(0,T) and β_n(r,·),β(r,·)∈ W^1,1(0,T)for everyr∈[0, +∞). Suppose thatϕ^t_n→ϕ^tonHin the sense of Mosco for every t∈[0,T]asn→+∞.Then for anyS∈(0,T], it follows that

(1)for eachu∈D(Φ^S), there exists a sequence(u_n)inᏴSsuch thatu_n→ustrongly in ᏴSandΦ^S_n(un)→Φ^S(u), whereΦ^S_nis defined by (2.11) withϕ^treplaced byϕ^t_n; (2)let(uk)be a sequence inᏴSsuch that(uk)is bounded inL^∞(0,S;H)anduk(t)→u(t)

weakly inH for a.e.t∈(0,S)ask→+∞and let(n_k)be a subsequence of(n). Then lim infk→+∞Φ^Snk(uk)≥Φ^S(u).

Throughout the present paper, we denote byCorCi (i=1, 2,. . .) nonnegative con- stants which do not depend on the elements of the corresponding space or set.

2.1. Cauchy problem. In this subsection, we consider the following Cauchy problem (CP(ϕ^t,f,u0)) in a Hilbert spaceH.

(CP(ϕ^t,f,u0)) du

dt(t) +∂_Hϕ^tu(t) f(t) inH, 0< t < T, u(0)=u0,

(2.13)

whereϕ^t∈Ψ(H) for allt∈[0,T], f ∈L¹(0,T;H), andu0∈H.

We first give a definition of solutions for (CP(ϕ^t,f,u0)) as follows.

Definition 2.5. A functionu∈C([0,T];H) is said to be a strong (resp., weak) solution of (CP(ϕ^t,f,u0)) ifuis a strong (resp., weak) solution of (E(ϕ^t,f)) such thatu(t)→u0

strongly inHast→+0.

As for the existence of solutions for (CP(ϕ^t,f,u0)), we here employ the following.

Theorem2.6 [10]. Let{ϕ^t}t∈[0,T]∈Ψ(α,β)be such thatα(r,·)∈W^1,2(0,T)andβ(r,·)

∈W^1,1(0,T) for every r∈[0, +∞). Then for all f ∈L¹(0,T;H) and u0∈D(ϕ⁰)^H, (CP(ϕ^t,f,u0)) has a unique weak solutionusuch that the functiont→ϕ^t(u(t)) is inte- grable on(0,T). In particular, if f ∈L²(0,T;H), then the weak solutionusatisfies

√tdu

dt ^∈L²(0,T;H), sup

t∈[0,T]

tϕ^tu(t) <+∞. (2.14) Moreover, if f ∈L²(0,T;H)andu0∈D(ϕ⁰), then the unique weak solutionubecomes a strong solution of(CP(ϕ^t,f,u0))such that

du

dt ^∈L²(0,T;H), sup

t∈[0,T]

ϕ^tu(t) <+∞. (2.15) On account ofProposition 2.4, Kenmochi also proved the following result on the con- vergence of solutionsu_nfor (CP(ϕ^t_n,f_n,u_0,n)) asn→+∞. Its proof can be found in [10, Theorem 2.7.1].

Theorem2.7 [10]. Under the same assumptions as inProposition 2.4, let(f_n)and(u_0,n)be sequences inL²(0,T;H)andD(ϕ⁰_n)^H, respectively, such that fn→ f strongly inL²(0,T;H) andu0,n→u0∈D(ϕ⁰)^Hstrongly inH. Then the unique weak solutionunof(CP(ϕ^t_n,fn,u0,n)) converges touin the following sense:

u_n−→u strongly inC[0,T];H (2.16)

and the limitubecomes the unique weak solution of(CP(ϕ^t,f,u0)). Moreover, _T

0 ϕ^t_nu_n(t) dt−→

_T

0 ϕ^tu(t) dt. (2.17)

In particular, ifϕ⁰_n(u_0,n)is bounded for alln∈N, then the limitubecomes a strong solution of(CP(ϕ^t,f,u0)).

2.2. Periodic problem. In this subsection, we consider the following periodic problem (PP(ϕ^t,f)):

(PP(ϕ^t,f))

du

dt(t) +∂Hϕ^tu(t) f(t) inH, 0< t < T, u(0)=u(T).

(2.18)

We are concerned with strong solutions of (PP(ϕ^t,f)) in the following sense.

Definition 2.8. A functionu∈C([0,T];H) is said to be a strong solution of (PP(ϕ^t,f)) if uis a strong solution of (E(ϕ^t,f)) such thatu(0)=u(T).

To state our results, define Ψπ

α,β,C0 :=





ϕ^t_t∈[0,T]∈Ψ(α,β); ^|u|²_H≤C0

ϕ^t(u) + 1 ∀u∈Dϕ^t ,∀t∈[0,T], Dϕ^T ⊂Dϕ⁰



 (2.19) for any positive constantC0. Moreover, we write{ϕ^t}t∈[0,T]∈B_π(α,β,C0,{M_r}r≥0) if the following hold true.

(i){ϕ^t}t∈[0,T]∈Ψπ(α,β,C0).

(ii) There exists a functionh: [0,T]→Hsuch that (2.9) holds andh(0)=h(T).

(iii) For everyr∈[0, +∞), (2.10) holds.

(iv)ϕ⁰(u)≤ϕ^T(u) for allu∈D(ϕ^T).

Then it is easily seen thatB_π(α,β,C0,{M_r}r≥0)⊂B(α,β,C0,{M_r}r≥0).

The existence of strong solutions for (PP(ϕ^t,f)) is assured by the following.

Theorem 2.9 [10]. Let {ϕ^t}t∈[0,T] ∈Ψπ(α,β,C0) be such that α(r,·)∈W^1,2(0,T), β(r,·)∈W^1,1(0,T)for allr∈[0, +∞). Then for all f ∈L²(0,T;H), (PP(ϕ^t,f)) has at least one strong solutionusatisfying

du

dt ^∈L²(0,T;H), sup

t∈[0,T]

ϕ^tu(t) <+∞. (2.20) In particular, ifϕ^t is strictly convex onH for a.e.t∈(0,T), then every strong solution of (PP(ϕ^t,f))is unique.

We next focus on the convergence of strong solutionsu_nfor (PP(ϕ^t_n,f_n)) whenϕ^t_n→ ϕ^t onH in the sense of Mosco and fn→ f weakly in L²(0,T;H). However, any stud- ies similar toTheorem 2.7 have not been done on the periodic problem (PP(ϕ^t_n,fn)) yet, which would be caused by a difficulty peculiar to the periodic problem. More pre- cisely, by virtue of Theorems2.6and2.7, for any f ∈L²(0,T;H) andu0∈D(ϕ⁰)^H, every unique weak solution of (CP(ϕ^t,f,u0)) becomes the limit of unique weak solutionsu_nfor (CP(ϕ^t_n,f,u0)) asn→+∞. However, in general, periodic solutions could not be unique.

Hence there could exist a strong solutionuof (PP(ϕ^t,f)) such that any strong solutions u_n of (PP(ϕ^t_n,f)) never converge to uas n→+∞. In fact, we can give such a counter example (seeRemark 3.10).

Thus because of the essential difference described above, the strong convergence of solutionsu_nfor (PP(ϕ^t_n,f_n)) inC([0,T];H) cannot be verified by the same manner as in the case of the Cauchy problem (see the proof of [10, Theorem 2.7.1]); so in order to cope with this difficulty, we introduce the following level set compactness assumption on {ϕ^t_n}n∈N.

(A1) For everyλ >0 andt∈[0,T], any sequence (un) inHsatisfying sup_n∈N{ϕ^t_n(un) +

|un|H} ≤λis precompact inH. Then our result can be stated as follows.

Theorem 2.10. For every n ∈N, let {ϕ^t_n}t∈[0,T] ∈Bπ(αn,βn,C0,{Mr}r≥0) and let {ϕ^t}t∈[0,T]∈Ψ(α,β) be such that α_n(r,·),α(r,·)∈W^1,2(0,T) and β_n(r,·),β(r,·)∈ W^1,1(0,T) for everyr∈[0, +∞). Suppose that ϕ^t_n→ϕ^t on H in the sense of Mosco as n→+∞and that (A1) holds. Moreover, let(fn)be a sequence inL²(0,T;H)such that fn→ f weakly inL²(0,T;H)and let(u_n)be a sequence of strong solutions for(PP(ϕ^t_n,f_n)). Then there exists a subsequence(nk)of(n)such thatunkconverges touin the following sense:

u_nk−→u strongly inC[0,T];H ,weakly inW^1,2(0,T;H), (2.21) and the limitubecomes a strong solution of(PP(ϕ^t,f)). Moreover,

_T

0 ϕ^t_nun(t) dt−→

_T

0 ϕ^tu(t) dt. (2.22)

Remark 2.11. (1) In Theorem 2.10, the limit u possibly depends on the choice of the subsequence (n_k).

(2) By virtue of the assumptions on{ϕ^t_n}t∈[0,T]and{ϕ^t}t∈[0,T]inTheorem 2.10, we can verify that{ϕ^t}t∈[0,T]∈Ψπ(α,β,C0). Indeed, letu∈D(ϕ^T). Then we can take a sequence (un) inH such thatun→ustrongly inH andϕ^T_n(un)→ϕ^T(u). Moreover, from the fact thatϕ⁰_n≤ϕ^T_n, it follows that

ϕ⁰_nun ≤ϕ^T_nun −→ϕ^T(u). (2.23) Furthermore, since lim infn→+∞ϕ⁰_n(un)≥ϕ⁰(u), we haveϕ⁰(u)≤ϕ^T(u), which implies D(ϕ^T)⊂D(ϕ⁰). Similarly we can also deduce that|u|²H≤C0(ϕ^t(u) + 1) for allu∈D(ϕ^t) andt∈[0,T].

Proof ofTheorem 2.10. Since {ϕ^t_n}t∈[0,T]∈B_π(α_n,β_n,C0,{M_r}r≥0) for alln∈N, we can take a sequence (hn) such that

hn(0)=hn(T), (2.24)

sup

t∈[0,T]

hn(t)_H+ϕ^t_nhn(t) + _T

0

dh_n dt (t)

2 Hdt

1/2

≤C0. (2.25) Moreover, since{ϕ^t_n}t∈[0,T]∈Ψπ(α_n,β_n,C0) for alln∈N, we see that

|u|²_H≤C0

ϕ^t_n(u) + 1 ∀u∈Dϕ^t_n ,∀t∈[0,T],∀n∈N. (2.26) Now letu_nbe a strong solution of (PP(ϕ^t_n,f_n)) for eachn∈N. Then multiplying the inclusion in (PP(ϕ^t_n,fn)) byun(t)−hn(t), we have

1 2

d

dtun(t)−hn(t)²_H+ϕ^t_nun(t)

≤ϕ^t_nhn(t) +

fn(t)−dh_n

dt (t),un(t)−hn(t)

H

≤C0+

fn(t)_H+dhn

dt (t)

H

un(t)−hn(t)_H.

(2.27)

Now by (2.26) it follows that 1

2 d

dtun(t)−hn(t)²_H+αun(t)−hn(t)²_H

≤Ch_n(t)²_H+ 1 +

f_n(t)_H+dhn

dt (t)

H

u_n(t)−h_n(t)_H

(2.28)

for someα >0. Hence by [11, Lemma 4.2], we get, by (2.25), sup

t∈[0,T]

un(t)−hn(t)_H≤C, (2.29)

which implies

sup

t∈[0,T]

u_n(t)_H≤C1. (2.30)

Furthermore, integrating (2.27) over (0,T), we have _T

0 ϕ^t_nun(t) dt≤C2. (2.31)

Hence by (2.31) there existst_n∈(0,T) such thatϕ^tnⁿ(u_n(t_n))≤C2/T.

Next multiplying the inclusion in (PP(ϕ^t_n,fn)) bydun(t)/dt, we have dun

dt (t)

2 H

+

g_n(t),dun

dt (t)

H

=

fn(t),du_n dt (t)

H≤fn(t)²_H+1 4

du_n dt (t)

2 H

,

(2.32)

wheregn(t) :=fn(t)−dun(t)/dt∈∂Hϕ^t_n(un(t)). Hence putr0=C1. Then by [12, Lemma 2.4], it follows from (2.30) that

gn(t),du_n dt (t)

H− d

dtϕ^t_nun(t)

≤α˙_nr0,t g_n(t)_Hϕ^t_nu_n(t) + 1^1/2+β˙_nr0,t ϕ^t_nu_n(t) + 1 (2.33)

for a.e.t∈(0,T). Thus 3

4 du_n

dt (t)

2 H+ d

dtϕ^t_nun(t)

≤fn(t)²_H+α˙n

r0,t fn(t)−du_n dt (t)

H

ϕ^t_nun(t) + 1^1/2 +β˙n

r0,t ϕ^t_nun(t) + 1

≤5

4fn(t)²_H+1 4

du_n dt (t)

2

H

+2α˙n

r0,t ²+β˙n

r0,t ϕ^t_nun(t) + 1.

(2.34)

Integrating (2.34) over (tn,t) and noting that (2.26) implies|ϕ^t_n(u)| ≤ϕ^t_n(u) + 2 for all u∈D(ϕ^t_n), we observe

1 2

_t

tn

dun

dτ (τ)

2 H

dτ+ϕ^t_nu_n(t)

≤ϕ^t_nⁿun

tn

+5 4

_T

0

fn(τ)²_Hdτ

+ _t

tn

2α˙_nr0,τ ²+β˙_nr0,τ ϕ^τ_nu_n(τ) + 3dτ

(2.35)

for allt∈[t_n,T]. Thus from the fact that _T

0

α˙_nr0,τ ²dτ+ _T

0

β˙_nr0,τ dτ≤M_r0, _T

0

f_n(τ)²_Hdτ≤C, ϕ^t_nⁿu_nt_n ≤C2

T,

(2.36)

by Gronwall’s inequality, it follows that sup

t∈[tn,T]

ϕ^t_nu_n(t) ≤C3. (2.37)

Hence sinceun(0)=un(T) andϕ⁰_n(u)≤ϕ^T_n(u) for allu∈D(ϕ^T_n) andn∈N, we find that ϕ⁰_n(u_n(0))≤ϕ^T_n(u_n(T))≤C3. Moreover, integrating (2.34) over (0,t), we also get

1 2

_t

0

dun

dτ (τ)

2 H

dτ+ϕ^t_nu_n(t)

≤ϕ⁰_nun(0) +5 4

_T

0

fn(τ)²_Hdτ

+ _t

0

2α˙n

r0,τ ²+β˙n

r0,τ ϕ^τ_nun(τ) + 3dτ

(2.38)

for allt∈[0,T]. Thus Gronwall’s inequality implies sup

t∈[0,T]

ϕ^t_nun(t) ≤C. (2.39)

Moreover, it follows from (2.26), (2.38), and (2.39) that _T

0

dun

dt (t)

2

Hdt≤C. (2.40)

Hence by (PP(ϕ^t_n,f_n)), (2.40) implies _T

0

g_n(t)²_Hdt≤C. (2.41)

By virtue of the above a priori estimates, we can take a subsequence (n_k) of (n) such that the following convergences hold true:

unk−→u weakly inW^1,2(0,T;H), (2.42) gnk−→g weakly inL²(0,T;H). (2.43) Moreover, by (A1), it follows from (2.30) and (2.39) that

un(t) is precompact inH ∀t∈[0,T]. (2.44)

Furthermore, by (2.40), we can deduce thatu_nis equicontinuous inC([0,T];H) for all n∈N. Thus Ascoli’s theorem implies

unk−→u strongly inC[0,T];H (2.45) for a suitable subsequence (nk) of (n). Hence sinceun(0)=un(T) for alln∈N, we have u(0)=u(T).

In the rest of this proof, we writensimply forn_k. Now defineΦ^T andΦ^Tn as in (2.11) with obvious replacements and letv∈D(Φ^T) be fixed. Then byProposition 2.4we can take a sequence (vn) inᏴT:=L²(0,T;H) such that

v_n−→v strongly inᏴT, Φ^Tn

v_{n −→}Φ^T(v). (2.46)

Now sincegn∈∂_ᏴTΦ^Tn(un), we have _T

0

fn(t)−du_n

dt (t),un(t)−vn(t)

H

dt

= _T

0

gn(t),un(t)−vn(t) _Hdt

≥Φ^Tn

u_n −Φ^Tn

v_n .

(2.47)

Moreover, byProposition 2.4, it follows from (2.45) that lim inf

n→+∞ Φ^Tn

un ≥Φ^T(u). (2.48)

Thus passing to the limitn→+∞in (2.47), by (2.42) and (2.45), we find _T

0

f(t)−du

dt(t),u(t)−v(t)

H

dt≥Φ^T(u)−Φ^T(v), (2.49) which together with the arbitrariness ofv∈D(Φ^T) impliesu∈D(∂_ᏴTΦ^T) andg=f − du/dt∈∂_ᏴTΦ^T(u). Hence by [9, Proposition 1.1], we deduce thatg(t)∈∂_Hϕ^t(u(t)) for a.e.t∈(0,T). Thereforeuis a strong solution of (PP(ϕ^t,f)).

Finally we prove (2.22). Sinceu∈D(Φ^T), byProposition 2.4, we can take a sequence (w_n) in ᏴT such thatw_n→ustrongly inᏴT and Φ^Tn(w_n)→Φ^T(u). Hence we get, by (2.40) and (2.45),

Φ^T_nun −Φ^T_nwn

≤ _T

0

fn(t)−du_n

dt (t),un(t)−wn(t)

H

dt

≤C _T

0

u_n(t)−u(t)²_Hdt+ _T

0

w_n(t)−u(t)²_Hdt 1/2

−→0,

(2.50)