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2 J. Differential Equations ) Contents lists available at ScienceDirect Journal of Differential Equations Periodic solutions for doubly nonlinear evolution equations Goro Akagi a,ulissestefanelli b, a Shibaura Institute of Technology, 37 Fukasaku, Minuma-ku, Saitama-shi, Saitama , Japan b Istituto di Matematica Applicata e Tecnologie Infromatiche, Consiglio Nazionale delle Ricerche, v. Ferrata 1, I-271 Pavia, Italy article info abstract Article history: Received 24 November 21 Revised 4 March 211 Available online 3 April 211 MSC: 35K55 Keywords: Doubly nonlinear evolution equations Subdifferential Periodic problem Periodic solution p-laplacian We discuss the existence of periodic solution for the doubly nonlinear evolution equation Au t)) + φut)) f t) governed by a maximal monotone operator A and a subdifferential operator φ in a ilbert space. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may genuinely fail. In order to overcome this difficulty, we firstly address some approximate problems relying on a specific approximate periodicity condition. Then, periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure. As a by-product, a structural stability analysis is presented for the periodic problem and an application to nonlinear PDEs is provided. 211 Elsevier Inc. All rights reserved. 1. Introduction This paper is concerned with the existence of periodic solutions for the abstract doubly nonlinear equation A u t) ) + φ ut) ) f t). 1.1) ere, u : t [, T ] is a trajectory in the ilbert space and u = du/dt, A is a maximal monotone operator in, φ denotes the subdifferential of a proper, lower-semicontinuous, and convex functional φ : [, ], and f L 2, T ; ). The abstract equation 1.1) stems as a suitable variational formulation of doubly nonlinear PDEs of the form * Corresponding author. addresses: g-akagi@shibaura-it.ac.jp G. Akagi), ulisse.stefanelli@imati.cnr.it U. Stefanelli) /$ see front matter 211 Elsevier Inc. All rights reserved. doi:1.116/j.jde

3 G. Akagi, U. Stefanelli / J. Differential Equations ) γ u t ) u p 2 u ) = f 1.2) where γ is maximal monotone in R, p > 1 and f is given. In case γ is linearly growing and p is sufficiently apart from 1, our analysis entails in particular the existence of periodic solutions to the latter. The reader finds some details in this concern in Section 6. Eq. 1.1) is well studied from the point of view of existence for the related Cauchy problem. Indeed, results in this direction can be traced back at least to Senba [37] and Arai [7]. Later on the problem has been considered also by Colli and Visintin [2] in ilbert spaces and Colli [18] in Banach spaces. Besides existence, the Cauchy problem has also been considered from the point of view of structural stability [2], perturbations and long-time behavior [3,4,8,34 36], and variational characterization of solutions [6,5,38,32,4]. The interest in the study of periodic solutions is in particular to be considered as a further step toward the comprehension of long-time dynamics and bifurcation phenomena. The aim of this paper is to address Eq. 1.1) under the periodic boundary condition u) = ut ). To the best of our knowledge, this periodic problem has never been solved before. Moreover, it is clearly quite more delicate with respect to the correspondent Cauchy problem. Consider for instance the ordinary differential equation u t) + I [1,2] u t) ) + ut) f t) in R where I [1,2] is the classical indicator function of the interval [1, 2] namely I [1,2] x) = ifx [1, 2] and I [1,2] = elsewhere). This equation is reduced to the abstract form of 1.1) in = R, andthen, one can check all the assumptions for A and φ of this paper except only a linear boundedness of A see A2) later). As u is constrained to be greater than 1 for all times, no periodic solution may exist. On the other hand, the Cauchy problem admits a unique solution for any given initial datum. A second difficulty arises as, being given the solvability of the Cauchy problem with initial datum u,thestandardapproachtoperiodicitybasedonfindingafixedpointforthepoincarémap P : u ut ) seems here of little use as Eq. 1.1) is known to show genuine non-uniqueness. Even by resorting to fixed point tools for multivalued applications, one has to be confronted with the fact that Pu cannot be generally expected to be convex. Our strategy in order to prove existence of periodic solutions to Eq. 1.1) is that of tackling an approximating equation possessing a unique Cauchy solution. This is obtained by replacing A with the strongly monotone operator εi d + A I d being the identity in ) andφ with its Moreau Yosida regularization φ ε.inparticular,thelatterapproximatingproblemfeaturesanapproximate periodicity condition of the form u) = J ε ut ) where J ε is the classical resolvent of φ at level ε. Then,periodic solutions for Eq. 1.1) are obtained by passing to the limit as ε. Let us note that we move in the exact same assumption frame as in the existence theory for the Cauchy problem from [2] plus an extra coercivity assumption for φ usually harmless with respect to applications). As a by-product of our existence analysis, we devise a structural stability result for the periodic problem. More precisely, by letting φ n and A n be sequences of convex functionals and maximal monotone operators, respectively, such that φ n and A n are convergent as n in some suitable sense, we prove that the periodic solutions for Eq. 1.1) with A,φ) replaced by A n,φ n ) converge to a periodic solution for A,φ) as n. Before moving on, we shall remark that the second type of abstract doubly nonlinear equation, namely, Au) ) + φu) f, 1.3) has been already considered by from the point of view of the existence of periodic solutions in [1,25]. See also [26,27,33,41] for A = id in the perturbation case f = f t, u). In[1,25] the authors cannot

4 1792 G. Akagi, U. Stefanelli / J. Differential Equations ) exploit directly the Poincaré map within a fixed point procedure and resort in proving the existence of periodic solutions for suitably regularized problems. Their argument is then completed by means of a limit passage. This is the plan of the paper. Section 2 is devoted to the statement of the main existence result for periodic solutions. The mentioned ε-approximating problem is discussed in Section 3 and a first passage to the limit for ε under a stronger coercivity assumption on φ is provided in Section 4. Then, Section 5 brings to a general structural stability result from which one can eventually conclude the proof of the existence of periodic solutions in the most general setting. The application of our abstract result to the nonlinear PDE 1.2) is given in Section 6. Finally, Appendix A contains a technical Gronwall-like) lemma on differential inequalities which is used for the estimates. 2. Main result and preliminary facts 2.1. Main result Let be a real ilbert space with norm and inner product, ). Let A be a maximal monotone operator in and let φ be a proper i.e., φ ) lowersemicontinuousconvexfunctional from into [, ] with the effective domain Dφ) := {u ; φu) < }. Thegraphofamaximal monotone operator will always be tacitly identified with the operator itself so that, for instance, the positions [u,ξ] A and u DA), ξ Au) are equivalent. The reader shall be referred to the classical monographs [1,16,42] as well to the recent [11] for a comprehensive discussion on maximal monotone operator techniques and applications. Let us consider the periodic problem P) given by A u t) ) + φ ut) ) f t) in, < t < T, 2.1) u) = ut ), 2.2) where φ denotes the subdifferential of φ see Section 2.2 below for the definition) and f is a given function from, T ) into. WeareconcernedwithsolutionsofP)giveninthefollowingsense: Definition 2.1 Strong solutions). Afunctionu C[, T ]; ) is said to be a strong) solution of P) if the following conditions are all satisfied: i) u W 1,2, T ; ) and u) = ut ); ii) ut) D φ) and u t) DA) for a.e. t, T ); iii) there exist η,ξ L 2, T ; ) such that ηt) A u t) ), ξt) φ ut) ), ηt) + ξt) = f t) for a.e. t, T ). 2.3) In order to discuss the existence of solutions for P), let us set up our assumptions. A1) There exist constants C 1 > andc 2 suchthat C 1 u 2 η, u) + C 2 for all [u,η] A. A2) There exists a constant C 3 suchthat η C 3 u + 1 ) for all [u,η] A. A3) For any λ R, theset{u Dφ); φu) + u λ} is compact in.

5 G. Akagi, U. Stefanelli / J. Differential Equations ) A4) φ is coercive in the following sense: there exists z Dφ) such that lim inf u [u,ξ] φ ξ, u z ) u =. 2.4) A5) f L 2, T ; ). Our main result reads, Theorem 2.2 Existence of periodic solutions). Assume that A1) A5) are satisfied. Then P) admits at least one solution. Let us provide some remarks on the coercivity condition A4). At first, note that the condition A4) can be equivalently rewritten as the following: there exists z Dφ) such that for any δ>itfollows that u δξ,u z ) + C δ for all [u,ξ] φ 2.5) with some constant C δ seepropositionb.1).moreover,letusstressthatassumptiona4)follows when there exist p > 1andC 4 > suchthat C 4 u p φu) + 1 for all u Dφ). 2.6) Indeed, let z Dφ) and [u,ξ] φ.thenbythedefinitionofsubdifferentials,weobservethat ξ, u z ) φu) φz ) C 4 u p φz ) 1, which implies A4). The Cauchy problem for Eq. 2.1) was studied by Colli and Visintin [2], and the existence of solutions was proved for any initial datum u Dφ) under A1) A3) and A5). Note that the simpler) equation u t) + φ ut) ) f t) in, < t < T which corresponds to Eq. 2.1) in case A is the identity mapping) has been proved to admit periodic solutions under A4) and A5) in [16]. Moreover, no periodic solution may exist when the coercivity A4) does not hold e.g., the ODE given by = R, φu), f t) = t). We close this subsection by a proposition on the uniqueness of periodic solutions in a special case. Proposition 2.3 Uniqueness of periodic solutions). Assume that φ is linear and A is strictly monotone. Then any two solutions u 1, u 2 of P) satisfy u 1 t) = u 2 t) + v for all t [, T ] with some v D φ) satisfying φv).

6 1794 G. Akagi, U. Stefanelli / J. Differential Equations ) Proof. Let u 1 and u 2 be solutions for P). Then, by taking the difference of the equations and testing it by u 1 t) u 2 t) in, weseethat η1 t) η 2 t), u 1 t) u 2 t)) + d dt φ u 1 t) u 2 t) ) =, where η i L 2, T ; ) belongs to Au )) for i = 1, 2, almost everywhere in time. The integration of i both sides over, T ) and the periodicity condition 2.2) yield η1 t) η 2 t), u 1 t) u 2 t)) dt =, which implies η1 t) η 2 t), u 1 t) u 2 t)) = for a.e.t, T ). Since A is strictly monotone, we deduce that u 1 t) = u 2 t) for a.e. t, T ). enceonecanwrite u 1 t) = u 2 t) + v with some constant v D φ). BytakingthedifferencebetweenEq.2.1)foru 1 and u 2 and using the fact that φ is linear, we obtain φv). The assumptions frame of Proposition 2.3 basically corresponds to the uniqueness proof for the Cauchy problem for relation 2.1) from [18]. Some alternative set of sufficient conditions for uniqueness for 2.1) are presented in [34, ] and [35]. The latter conditions seem however not to be directly applicable to the periodic problem. As concerns doubly nonlinear equations of the type of 1.3), uniqueness for the Cauchy problem is already discussed in [17,21]. Moreover, uniqueness of periodic solutions is proved by [27] in a specific setting, where periodic solutions satisfy an order property. enceforth, we shall use the same symbol C in order to denote any non-negative constant depending on data and, in particular, independent of ε. The value of the constant C may change from line to line Preliminaries We refer the reader to [16] for the definition and fundamental properties of maximal monotone operators in ilbert spaces. ere let us give some preliminary materials on subdifferentials, their resolvents, and Yosida approximations and Moreau Yosida regularizations of convex functionals proofs can be found in [16] as well). Let φ be a proper lower semicontinuous convex functional from into [, ] with Dφ) := {u ; φu)< }. Thenthesubdifferentialoperator φ : 2 for φ is defined as follows φu) := { ξ ; φv) φu) ξ, v u) for all v Dφ) } with the domain D φ) := {u Dφ); φu) }. Since φ is maximal monotone in, for ε >, one can define the resolvent J ε : D φ) and the Yosida approximation φ) ε : of φ by J ε := I d + ε φ) 1, φ) ε := I d J ε )/ε, where I d stands for the identity mapping of. Furthermore,forε >, the Moreau Yosida regulariza-

7 G. Akagi, U. Stefanelli / J. Differential Equations ) tion φ ε : [, ) of φ is given by { } 1 φ ε u) := inf v 2ε u v 2 + φv) for all u. 2.7) The following proposition provides some classical properties of φ ε. Proposition 2.4 Moreau Yosida regularization). The Moreau Yosida regularization φ ε is a Fréchet differentiable convex functional from into R. Moreover,theinfimumin2.7) is attained by J ε u, where J ε denotes the resolvent of φ,i.e., Furthermore, the following hold. φ ε u) = 1 2ε u J εu 2 + φ J εu) = ε φ)ε u) φ J εu). i) φ ε ) = φ) ε, where φ ε ) is the subdifferential Fréchet derivative) of φ ε. ii) φ J ε u) φ ε u) φu) for all u and ε >. iii) φ ε u) φu) as ε + for all u. Finally, let us recall the chain rule for subdifferentials. Proposition 2.5 Chain rule for subdifferentials). Let u W 1,2, T ; ) be such that ut) D φ) for a.e. t, T ). Assumethatthereexistsξ L 2, T ; ) such that ξt) φut)) for a.e. t, T ). Thenthe function t φut)) is absolutely continuous on [, T ]. Moreover, the set has full Lebesgue measure and I := { t [, T ]; ut) D φ), uandφ u ) ) are differentiable at t } d dt φ ut) ) = h, u t) ) for every h φ ut) ) and t I. 3. Approximate problems We shall firstly focus on the case that φ satisfies a stronger coercivity requirement, specifically: A6) There exists a constant ρ > suchthatρ u 2 φu) for all u Dφ). Note that the latter is stronger than A4). Moreover, A6) entails ρ 2ρε + 1 u 2 φ εu) for all u and ε >, 3.1) where φ ε denotes the Moreau Yosida regularization of φ. Inthissectionweconstructsolutionsfor the following approximate problems P) ε for ε > : εu t) + A u t) ) + φ ε ut) ) fε t) in, < t < T, 3.2) u) = J ε ut ), 3.3) where φ ε := φ ε ) = φ) ε, J ε stands for the resolvent of φ and f ε ) is an approximate sequence

8 1796 G. Akagi, U. Stefanelli / J. Differential Equations ) in L, T ; ) such that f ε f strongly in L 2, T ; ). 3.4) To this end, we also introduce the corresponding Cauchy problem C) ε, i.e., 3.2) with the initial condition, u) = u. 3.5) The existence and the uniqueness of solutions for C) ε can be proved, since 3.2) is equivalently rewritten by u t) = L ε t, ut) ) := εid + A) 1 f ε t) φ ε ut) )) in, < t < T and L ε t, ) is Lipschitz continuous in. enceonecandefineasingle-valuedmappingp ε : by P ε : u J ε ut ), where u is the unique solution of C) ε with the initial data u. The main result of this section is the following. Theorem 3.1 Existence of approximate solutions). Assume that A1) A3) and A6) are satisfied. Then for each ε >, problem P) ε admits a solution u ε. In order to prove this theorem, it suffices to find a fixed point u of the mapping P ε.indeed,let u = P εu and u be the unique solution for the Cauchy problem C) ε with u = u.then,bythe definition of P ε,weobservethat J ε u T ) = P ε u = u = u ). In order to find a fixed point of P ε,weshallprovethefollowingtwolemmasandemployschauder s fixed point theorem. Lemma 3.2 P ε is a self-mapping). There exists a constant R > such that P ε is a self-mapping on the set that is, P ε B R ) B R. B R := { u Dφ); φu) R }, Proof. By testing 3.2) by u t) and using A1) and Proposition 2.5, we have which implies ε u t) 2 + C 1 u t) 2 C 2 + d dt φ ) ε ut) fε t), u t) ) C fε t) 2 + C 1 u t) 2 2, ε u t) 2 + C 1 u t) d dt φ ) ε ut) C fε t) 2 + 1) 3.6)

9 G. Akagi, U. Stefanelli / J. Differential Equations ) for a.e. t, T ). Moreover,wetest3.2)byut) and exploit A2) and 3.1) in order to get φ ε ut) ) φε ) + φ ε ut) ), ut) ) = φ ε ) + f ε t), ut) ) εu t), ut) ) ηt), ut) ) C ε fε t) 2 + u t) 2 + 1) φ ε ut) ), where η := f ε εu φ ε u )) Au )) almost everywhere in time and C ε is a constant depending on ε. Thusweobtain φ ε ut) ) 2Cε fε t) 2 + u t) 2 + 1) 3.7) for a.e. t, T ). ence,bymultiplying3.7)bysomesuitablysmallconstantandaddingitto3.6), we deduce that d dt φ ) ) ε ut) + αε φ ε ut) βε where the two constants α ε,β ε > possiblydependonε, provided f ε L, T ; ). Therefore,by virtue of Proposition A.1 one can take a constant R β ε /α ε > suchthat φ ε ut ) ) R if φε u ) R, which together with the fact that φ J ε u) φ ε u) φu) see Proposition 2.4) also implies φ J ε ut ) ) R if φu ) R. Consequently, we deduce that P ε maps the set B R into itself. Lemma 3.3 Continuity of P ε in ). The mapping P ε is continuous in. Proof. Let u,n, u be such that u,n u strongly in and let u n and u be the unique solutions for C) ε with initial data u,n and u, respectively. Subtract 3.2) for u n from that for u and put w n := u u n.wehave εw n t) + A u t) ) A u n t)) φ ε ut) ) + φε un t) ). By testing the latter by w n t) and exploiting the 1/ε-Lipschitz continuity of Yosida approximations see [16]), we obtain ε w n t) 2 ) φ ε ut) + φε un t) ), w n t)) 1 wn t) w ε n t) for a.e. t, T ). Notingthatd/dt) w n t) w n t),wededucethat d wn t) dt 1 wn t) ε 2 for a.e. t, T ),

10 1798 G. Akagi, U. Stefanelli / J. Differential Equations ) which together with Gronwall s inequality yields wn T ) u u,n e T /ε2. Therefore u n T ) ut ) strongly in as n, and hence, P ε is continuous in, since non-expansive in. J ε is Now, we are ready to prove Theorem 3.1. Proof of Theorem 3.1. We note by A3) and A6) that the set B R is compact in. Moreover,B R is convex because of the convexity of φ. ThereforecombiningLemmas3.2and3.3andapplyingSchauder s fixed point theorem to P ε : B R B R,wecantakeafixedpointu B R such that P ε u = u. This completes the proof of Theorem Estimates and limiting procedure In this section we establish a priori estimates for solutions u ε of P) ε and finally derive the convergence of u ε to a solution u of P) as ε under the stronger coercivity condition A6). Theorem 4.1 Existence of periodic solutions under A6)). Assume that A1) A3), A5) A6) are satisfied. Then problem P) admits at least one solution. Let u ε be a solution of P) ε.weshallfirstlypresentausefulinequalitystemmingfromthefact that u ε ) = J ε u ε T ). Lemma 4.2. It holds that φ ε u ε )) φ ε u ε T )). Proof. By ii) of Proposition 2.4, since u ε ) = J ε u ε T ), itfollowsthat φ ε uε ) ) φ u ε ) ) = φ J ε u ε T ) ) φ ε uε T ) ). Next, we are in the position of establishing the following estimate. Lemma 4.3. There exists a constant C independent of ε such that u ε t) 2 dt C. 4.1) Proof. Test 3.2) by u ε t). Then,inequality3.6)followswithu = u ε.ence,byintegratingbothsides over, T ), wededucethat ε u ε t) 2 dt + C 1 2 u ε t) 2 dt + φ ε uε T ) ) φ ε uε ) ) + C fε t) 2 dt + 1 ), which, together with Lemma 4.2, implies 4.1).

11 G. Akagi, U. Stefanelli / J. Differential Equations ) Moving from estimate 4.1) we deduce by A2) and a comparison in Eq. 3.2) that η ε t) 2 dt C, 4.2) φ ε uε t) ) 2 dt C, 4.3) where η ε := f ε εu ε φ εu ε )) Au ε )) almost everywhere in time. Moreover, we also note that, by 4.1), d dt J εu ε t) 2 dt u ε t) 2 dt C, 4.4) since J ε is non-expansive in see [16]). Let us get to a crucial estimate, namely an ordinary differential inequality for φ ε u ε t)). Lemma 4.4. Let y ε t) := φ ε u ε t)). Then, dy ε dt t) + y εt) g ε t) for a.e. t, T ) where the functions g ε are uniformly bounded in L 1, T ) for ε, 1). Proof. Recall again that ε u ε t) 2 + C 1 u 2 ε t) 2 + d dt φ ε uε t) ) C fε t) 2 + 1). 4.5) On the other hand, by testing 3.2) by u ε t) v with any v Dφ) and using A2) and 3.1), we have, for any ε, 1), φ ε uε t) ) φ ε v ) + φ ε uε t) ), u ε t) v ) φv ) + f ε t) εu ε t) η εt), u ε t) v ) C fε t) 2 + ε2 u ε t) 2 + u ε t) 2 + 1) φ ε uε t) ), which yields φ ε uε t) ) 2C fε t) 2 + ε2 u ε t) 2 + u ε t) 2 + 1) 4.6) with some constant C independentofε. Byadding4.6)to4.5),wehave dy ε dt t) + y εt) C fε t) 2 + u ε t) 2 + 1) =: g ε t)

12 18 G. Akagi, U. Stefanelli / J. Differential Equations ) with some constant C > independent ofε. Moreover, by 3.4) and 4.1), the functions g ε are bounded in L 1, T ) for all ε, 1). ence, by Proposition A.2 and Lemma 4.2 the following estimates follow immediately. Lemma 4.5. There exists a constant C independent of ε such that By A6), it also holds that sup φ ε uε t) ) C. 4.7) t [,T ] sup Jε u ε t) C/ρ. 4.8) t [,T ] From these a priori estimates, by extracting a sequence ε n, which will be also denoted by ε below, we can derive convergences of u ε. Lemma 4.6. There exist u W 1,2, T ; ) and η,ξ L 2, T ; ) such that Moreover, [ut), ξt)] φ for a.e. t, T ). J ε u ε u strongly in C [, T ]; ), 4.9) u ε u strongly in C [, T ]; ), 4.1) weakly in W 1,2, T ; ), 4.11) η ε η weakly in L 2, T ; ), 4.12) φ ε uε ) ) ξ weakly in L 2, T ; ). 4.13) Proof. By A3) and Lemma 4.5, the family J ε u ε t)) ε,1) is precompact in for each t >. By estimate 4.4), the function t J ε u ε t) is equicontinuous in on [, T ] for all ε >. Therefore Ascoli s compactness lemma yields the uniform convergence 4.9). Moreover, by exploiting the bound 4.7), we deduce that sup uε t) J ε u ε t) 2 2ε sup φ ε uε t) ) Cε, t [,T ] t [,T ] which leads us to obtain the strong convergence 4.1). Furthermore, the weak convergences 4.11), 4.12) and 4.13) follow from the estimates 4.1), 4.2) and 4.3), respectively. Finally, from the demiclosedness of maximal monotone operators, one can infer from the convergence 4.9) and 4.13) that ut) D φ) and ξt) φut)) for a.e. t, T ). Next, let us prove the periodicity of u. Lemma 4.7. It holds that u) = ut ). Proof. Since both u ε t) and J ε u ε t) converge to ut) strongly in uniformly) for all t [, T ], we deduce by u ε ) = J ε u ε T ) that u) = ut ). We finally check that ηt) Au t)) for a.e. t, T ).

13 G. Akagi, U. Stefanelli / J. Differential Equations ) Lemma 4.8. It holds that [u t), ηt)] Afora.e.t, T ). Proof. Test η ε t) Au ε t)) by u ε t) and integrate this over, T ). Weobtain T ηε t), u ε t)) dt = T fε t), u ε t)) dt ε u ε t) T 2 dt φε uε t) ), u ε t)) dt fε t), u ε t)) dt, since we deduce by Lemma 4.2 that φε uε t) ), u ε t)) dt = φ ε uε T ) ) + φ ε uε ) ). Therefore, as φu)) = φut )), itholdsbyconvergence3.4)that lim sup ε T ηε t), u ε t)) dt f t) ξt), u t) ) dt. Consequently, by Proposition 2.5 of [16], we conclude that u t) DA) and ηt) Au t)) for a.e. t, T ). Combining these lemmas, we have proved the conclusion of Theorem Structural stability In the last section, we proved the existence of solutions for P) under A6), a stronger coercivity condition of φ. In order to replacea6)by the weaker conditiona4),we shall establish a structural stability result for solutions of P). Indeed, for n N, defineafunctionalφ n : [, ] by φ n u) := φu) + 1 2n u 2 for u, 5.1) which converges to φ precisely, in the sense of Mosco on ) asn.then,φ n complies with A6) as well as all assumptions of Theorem 4.1. ence, we have the existence of solutions u n for P) with φ replaced by φ n.ifonecanobtainastructuralstabilityresult,i.e.,theconvergenceofu n to a solution of P) as n,ourproofoftheorem2.2willbecompleted. We shall work here in a more general setting of possibly independent interest. Let A n ) be a sequence of maximal monotone operators in and let φ n ) be a sequence of proper lower semicontinuous convex functionals from into [, ]. Assume that A n A in the sense of graph on, 5.2) φ n φ in the sense of Mosco on 5.3) as n.erewerecallthedefinitionsofgraph-convergence and Mosco-convergence.

14 182 G. Akagi, U. Stefanelli / J. Differential Equations ) Definition 5.1 Graph-convergence and Mosco-convergence). Let be a ilbert space. Let A n ) be a sequence of maximal monotone operators in and let φ n ) be a sequence of proper lower semicontinuous convex functionals from into [, T ]. i) The sequence A n ) is said to graph-converge to a maximal monotone operator A : or A n A in the sense of graph on ) ifforany[u,ξ] A and n N, thereexists[u n,ξ n ] A n such that u n u strongly in, ξ n ξ strongly in as n. ii) The sequence φ n ) is said to Mosco-converge to a proper lower semicontinuous convex functional φ : [, ] φ n φ in the sense of Mosco on ) ifthefollowinga),b)hold: a) Liminf condition) Let u n u weakly in as n.then lim inf n φn u n ) φu). b) Existence of recovery sequences) For every u Dφ), there exists a recovery sequence u n ) in such that u n u strongly in and φ n u n ) φu). Remark 5.2. Let φ,φ n : [, ] be proper lower-semicontinuous and convex. Then it is known that φ n graph-converges to φ if φ n Mosco-converges to φ. WereferthereadertoTheorem3.66 of [9]. Now, let us consider the following periodic problems P) n : A n u t) ) + φ n ut) ) f n t) in, < t < T, 5.4) u) = ut ), 5.5) where f n ) is a sequence in L 2, T ; ) such that f n f strongly in L 2, T ; ) as n. 5.6) The main result of this section is concerned with the asymptotic behavior of solutions u n for P) n as n. Theorem 5.3 Structural stability). In addition to 5.2), 5.3), and 5.6), assume that A1) and A2) are satisfied with A = A n and constants independent of n. Moreover, suppose that A3) every sequence u n ) is precompact in whenever φ n u n ) + u n is bounded as n, A4) for each n N large enough, there exists z,n Dφ n ) such that for any δ> sufficiently small it holds that u δξ,u z,n ) + C δ for all [u,ξ] φ n with some constant C δ independent of n. Moreover, z,n B 1 for all n N with some constant B 1.

15 G. Akagi, U. Stefanelli / J. Differential Equations ) Let u n be a solution of P) n. Then, there exist a subsequence n ) of n) and a solution u of P) such that u n u weakly in W 1,2, T ; ), strongly in C [, T ]; ) as n. Before proceeding to a proof, we set up a couple of lemmas. Lemma 5.4. Assume that A4) is satisfied and set := L 2, T ; ) with u := ut) 2 dt)1/2 for u. Letu n ) be a sequence in W 1,2, T ; ) such that u n t) D φ n ) for a.e. t, T ). Letξ n ) be a sequence in such that ξ n t) φ n u n t) ) for a.e. t, T ) and ξ n B 2 for all n N with some constant B 2. Then there exists a constant C independent of n such that u n C u n + 1 ) for all n N. Proof. By A4),itfollowsthat un t) δ ξ n t), u n t) z,n ) + C δ for a.e. t, T ). Integrating this over, T ), wefindthat un t) dt δ ξn t), u n t) z,n ) dt + C δ T δ ξ n un + z,n T 1/2) + C δ T δb 2 un + B 1 T 1/2) + C δ T. 5.7) On the other hand, by ölder s and Sobolev s inequalities for vector-valued functions, there is a constant M suchthat u M ut) dt + u ) for all u W 1,2, T ; ). 5.8) ence combining this with 5.7) and taking δ> sufficiently small, one can deduce that un t) dt C u n + ) 1 with some constant C independent of n. The next lemma is well known one can prove this lemma as in Proposition 3.59 of [9] with slight modifications).

16 184 G. Akagi, U. Stefanelli / J. Differential Equations ) Lemma 5.5. Let A n ) be a sequence of maximal monotone operators in such that A n graph-converges to a maximal monotone operator A. Let [v n, η n ] A n be such that Then [v, η] A and η n, v n ) η, v). v n v and η n η weakly in, lim supη n, v n ) η, v). n Now, we are ready to prove Theorem 5.3. Proof of Theorem 5.3. By testing equation 5.4) by u n t) and using A1) with C 1, C 2 independent of n, we deduce that C 1 2 u n t) 2 + d dt φn u n t) ) C 2 + C fn t) 2 for a.e. t, T ). 5.9) Integrating this over, T ) and using periodicity 5.5), we have u n t) 2 dt C, 5.1) which, together with A2) with C 3 independent of n, implies η n t) 2 dt C, 5.11) where η n t) denotes a section of A n u n t)) as in Eq. 2.3). Moreover, by comparison in Eq. 5.4), it follows that ξ n t) 2 dt C, 5.12) where ξ n := f n η n is a section of φ n u n )). Therefore,byLemma5.4,wehave un t) 2 dt C, 5.13) which together with the estimate 5.1) entails that u n ) is bounded in W 1,2, T ; ). ence, sup t [,T ] u n t) C. We next prove the uniform boundedness of φ n u n t)) on [, T ]. Letv Dφ). Then,onecantake v,n Dφ n ) such that φ n v,n ) φv ) and v,n v strongly in, sinceφ n φ in the sense of Mosco. We observe that

17 G. Akagi, U. Stefanelli / J. Differential Equations ) φ n u n t) ) φ n v,n ) + ξ n t), u n t) v,n ) = φ n v,n ) + f n t), u n t) v,n ) η n t), u n t) v,n ) φ n v,n ) + C fn t) 2 + u n t) 2 + un t) 2 + v,n 2 + 1) with a constant C independent of n. ence,byaddingthelattertoinequality5.9)wededucethat d dt φn u n t) ) + φ n u n t) ) g n t) := φ n v,n ) + C fn t) 2 + u n t) 2 + un t) 2 + v,n 2 + 1) with a constant C independent of n. From already obtained estimates we notice thatg n ) is uniformly bounded in L 1, T ). Therefore,byPropositionA.2weconcludethat sup φ n u n t) ) ) T 1 t [,T ] T + 1 gn t) dt C. 5.14) By A3), we note thatu n t)) is precompact in for all t [, T ]. ence one can derive the following convergences: u n u weakly in W 1,2, T ; ), strongly in C [, T ]; ), ξ n ξ η n η weakly in L 2, T ; ), weakly in L 2, T ; ). ere we also obtain u) = ut ), sinceu n ) = u n T ). FromLemma5.5,onecanproveξt) φut)) and ηt) Aut)) for a.e. t, T ) by a similar argument to that of Section 4. Thus u solves P) and it completes our proof. Eventually, let us complete the proof of Theorem 2.2. Proof of Theorem 2.2. Define a sequence φ n ) of functionals from into [, ] by φ n u) := φu) + 1 2n u 2 for u with Dφ n ) = Dφ). Thenφ n satisfies A6). ence by Theorem 4.1, one deduce that P) with φ replaced by φ n admits a strong solution u n.wecaneasilycheckthatφ n φ in the sense of Mosco on as n,andmoreover,wederivea3) and B4) with z,n z,sinceφ complies with A3) and A4). Therefore due to Theorem 5.3, u n converges to u as n and the limit u solves P).

18 186 G. Akagi, U. Stefanelli / J. Differential Equations ) Application to PDEs This section is devoted to a typical application of the preceding abstract theory. Let Ω be a bounded domain in R N with smooth boundary Ω. Weareconcernedwiththefollowingperiodic problem: γ u t ) p u f in Ω, T ), 6.1) u = on Ω, T ), 6.2) u, ) = u, T ) in Ω, 6.3) where u t = u/ t, γ is a maximal monotone graph in R 2, f = f x, t) is a given function, and p is the so-called p-laplace operator given by p φx) := φx) p 2 φx) ), 1 < p <. Inclusions of the type of 6.1) may arise in connection with phase transitions [12,15,19,24,28,29], gas flow through porous media [39], and damage processes [13,14,22,23,31]. In the limiting case of graphs α being -homogeneous which is however not covered by our analysis) this kind of equation may stem in elastoplasticity, brittle fractures, ferroelectricity, and general rate-independent systems [3]. Let us remark that relation 6.1) stems as the gradient flow in = L 2 Ω) of the complementary) energy functional φ given by φu, t) := { Ω 1 p ux) p f x, t)ux)) dx if u W 1,p Ω), otherwise with respect to the metric structure induced by the dissipation functional F ux) ) { := Ω αux)) dx if αu )) L1 Ω), otherwise for α = α. Indeed,bytakingvariationsin L 2 Ω), inclusion6.1)isequivalenttothekineticrelation F u t ) + u φu, t) which represents the balance between the system of conservative u φu, t), respectively)anddissipative actions F u t ),respectively)inthephysicalsystem.thequestionoftheperiodicsolvability of inclusion 6.1) has hence a clear applicative interest, especially in connection with the study of long-time behavior of the above-mentioned physical systems in case of periodic external actions. In order to state our result, we assume that: 1) There exist constants C 5 >, C 6 suchthat C 5 s 2 gs + C 6 for all [s, g] γ. 2) There exists a constant C 7 suchthat g C 7 s +1 ) for all [s, g] γ.

19 G. Akagi, U. Stefanelli / J. Differential Equations ) Remark 6.1. Assumptions 1) and 2) allow γ to be degenerate and multivalued. Indeed, γ 1 and γ 2 given below satisfy the assumptions: s if s <, γ 1 s) = if s 1, s 1 if1< s, s if s <, γ 2 s) = [, 1] if s =, s + 1 if < s. Furthermore, one can check 1) and 2) if γ satisfies these assumptions only for s R with some constant R > andγ W 1, R, R). Our result reads, Theorem 6.2 Existence of periodic solutions for a nonlinear PDE). Assume that f L 2, T ; L 2 Ω)) and 1), 2) are satisfied. Moreover, suppose that p > 2N/N + 2). Then problem 6.1) 6.3) admits at least one solution. Proof. We set = L 2 Ω) with the norm := L 2 and define φ by φu) := Moreover, we let A : be given by { 1 p Ω ux) p dx if u W 1,p Ω), else. Au) := { η ; ηx) γ ux) ) for a.e. x Ω } with the domain DA) := {u ; Au) }. Then,weobservethat φu) = p u with the homogeneous Dirichlet boundary condition in, and A is maximal monotone in. Thus 6.1) 6.3) is reduced to P). Now, A1) and A2) follow immediately from 1) and 2). Moreover, since W 1,p Ω) is compactly embedded in L 2 Ω) by 2N/N + 2) <p, every sublevel set of φ is compact in. ence A3) holds true. Furthermore, thanks to the Sobolev Poincaré inequality, we see that u C p u L p Ω) for all u W 1,p Ω), provided that p 2N/N + 2), φu), u) u = u p L p Ω) u C p p u p 1 for all u D φ), which implies A4) with z =. Therefore, by applying Theorem 2.2, we conclude that the system 6.1) 6.3) admits at least one solution. As for uniqueness we have the following. Proposition 6.3. In case p = 2 and γ is strictly monotone, the periodic solution for 6.1) 6.3) is unique. Proof. Along with the same positions as in the proof of Theorem 6.1, if p = 2, then φ = is linear and φ) 1 ) =. Furthermore, A is strictly monotone, since γ is so. ence by Proposition 2.3 the solution of the system 6.1) 6.3) is unique.

20 188 G. Akagi, U. Stefanelli / J. Differential Equations ) The above well-posedness results can be generalized in a number of different directions. At first, the results for Eq. 6.1) can be extended to more general elliptic operators by replacing the p-laplacian p u by mx, u) + I κ u) where the function m = mx, p) : Ω R N R N is measurable in x and differentiable and maximal monotone in p and the indicator function I κ over some closed interval κ R. Indeed,thestatement of Theorem 2.2 can be extended by assuming that mx, ) admits a strongly coercive primitive function Px, ) and by setting φu) := Ω P x, ux) ) dx + I K u), where I K denotes the indicator function over the set K := {u L 2 Ω); ux) κ for a.e. x Ω}. Furthermore, we can treat doubly nonlinear systems such as αu t ) p u f in Ω, T ), 6.4) where u = u 1, u 2,...,u m ) : Ω, T ) R m and α is a maximal monotone operator in R m with linear growth. ere, the vectorial p-laplacian p u is defined by p u = p u 1, p u 2,..., p u m ). Again, the results of Theorem 2.2 can be extended in order to cover 6.4) by letting = L 2 Ω)) m and φu) := 1 p m i=1 Ω u i x) p dx. Acknowledgments The first author gratefully acknowledges the kind hospitality and support of the IMATI-CNR in Pavia, Italy, where this research was initiated, and the supports of Short-Term Mobility Program 21 of CNR, of the SIT grant and of the Grant-in-Aid for Young Scientists B) # , MEXT. The second author is partly supported by FP7-IDEAS-ERC-StG Grant # 2947 BioSMA). Appendix A. Some tools on ordinary differential inequalities In this appendix we provide two types of estimates for solutions to ordinary differential inequalities. Let us start by recalling without proof an elementary estimate, for the sake of completeness. Proposition A.1. Let T > and let y :[, T ] R be an absolutely continuous function satisfying dy t) + α yt) β dt for a.e. t, T ) A.1) with constants α > and β.thenitfollowsthat

21 G. Akagi, U. Stefanelli / J. Differential Equations ) sup yt) R t [,T ] if y) R for any constant R β/α. The following proposition is exploited in the limiting procedure for solutions of P) ε as ε in Section 5 see also [33]). Proposition A.2. Let T > and let y :[, T ] [, ) be an absolutely continuous function satisfying dy t) + α yt) gt) for a.e. t, T ), A.2) dt y) yt ), A.3) where α is a positive constant and g L 1, T ). Thenitfollowsthat ) T 1 sup yt) t [,T ] αt + 1 gt) dt. Proof. By integrating the ordinary differential inequality A.2) and using condition A.3), we have yt ) + α yτ ) dτ y) + T gτ ) dτ yt ) + gτ ) dτ, which implies yτ ) dτ 1 α gτ ) dτ =: M. Let t min and t max be a minimizer and a maximizer of y = yt), respectively.thenwefindthat yt min )T yτ ) dτ M, which gives yt min ) M/T. In case t min t max,byintegratinginequalitya.2)overt min, t max ),weobservethat yt max ) yt min ) + gτ ) M dτ T + In case t min > t max,theintegrationofa.2)over, t max ) yields gτ ) dτ. yt max ) y) + t max gτ ) dτ.

22 181 G. Akagi, U. Stefanelli / J. Differential Equations ) Moreover, the integration of A.2) over t min, T ) gives yt ) yt min ) + gτ ) dτ. t min Since y) yt ), weconcludethat yt max ) yt min ) + gτ ) M dτ T + gτ ) dτ. Thus we have proved this proposition. Remark A.3. By repeating a similar argument to the above, one can also verify the following: if an absolutely continuous function y :[, T ] R satisfies y t) gt) for a.e. t, T ), yτ ) dτ M, y) yt ), it then follows that sup t [,T ] yt) M T T + Appendix B. Equivalence of coercivity conditions gτ ) dτ. In this section we prove the equivalence of two coercivity conditions. Proposition B.1 Equivalence of coercivity conditions). Let φ be a proper lower semicontinuous convex functional from into [, ] and let z Dφ). Then the following two conditions are equivalent: ξ,u z i) lim inf ) u u =. [u,ξ] φ ii) For any δ>,thereexistsaconstantc δ such that u δξ,u z ) + C δ for all [u,ξ] φ. Proof. We first show that i) implies ii). Assume on the contrary that one can take δ > suchthat for all n N there exists [u n,ξ n ] φ satisfying u n >δ ξ n, u n z ) + n. Then it follows that u n δ φz ) + n

23 G. Akagi, U. Stefanelli / J. Differential Equations ) and ξ n, u n z ) u n < 1 δ. ence, by passing to the limit as n,wederive lim inf n ξ n, u n z ) u n 1 δ <, which contradicts i). Let us next prove that ii) implies i). From ii), one can immediately deduce for u that 1 δ ξ, u z ) u + C δ δ u for all δ>and[u,ξ] φ. Taking a liminf in both sides as u,wededucethat < 1 δ ξ, u z ) lim inf. u u [u,ξ] φ ence by letting δ +,wecompleteourproof. References [1] S. Aizicovici, V.-M. okkanen, Doubly nonlinear periodic problems with unbounded operators, J. Math. Anal. Appl ) [2] S. Aizicovici, Q. Yan, Convergence theorems for abstract doubly nonlinear differential equations, Panamer. Math. J ) [3] G. Akagi, Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces, J. Evol. Equ ) [4] G. Akagi, Global attractors for doubly nonlinear evolution equations with non-monotone perturbations, J. Differential Equations ) [5] G. Akagi, U. Stefanelli, Weighted energy-dissipation functionals for doubly nonlinear evolution, J. Funct. Anal ) [6] G. Akagi, U. Stefanelli, A variational principle for doubly nonlinear evolution, Appl. Math. Lett ) [7] T. Arai, On the existence of the solution for ϕu t)) + ψut)) f t), J.Fac.Sci.Univ.TokyoSec.IAMath ) [8] M. Aso, M. Frémond, N. Kenmochi, Phase change problems with temperature dependent constraints for the volume fraction velocities, Nonlinear Anal. 6 25) [9]. Attouch, Variational Convergence for Functions and Operators, Appl. Math. Ser., Pitman, Advanced Publishing Program, Boston, MA, [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, [11] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monogr. Math., Springer, New York, 21. [12] D. Blanchard, A. Damlamian,. Ghidouche, A nonlinear system for phase change with dissipation, Differential Integral Equations ) [13] E. Bonetti, G. Schimperna, Local existence for Frémond s model of damage in elastic materials, Contin. Mech. Thermodyn ) [14] E. Bonetti, G. Schimperna, A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials, J. Differential Equations ) [15] G. Bonfanti, M. Frémond, F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl. 1 2) [16]. Brézis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de ilbert, Math. Stud., vol. 5, North-olland, Amsterdam, New York, [17] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal ) [18] P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math ) [19] P. Colli, F. Luterotti, G. Schimperna, U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA Nonlinear Differential Equations Appl. 9 22)

24 1812 G. Akagi, U. Stefanelli / J. Differential Equations ) [2] P. Colli, A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations ) [21] E. Di Benedetto, R. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal ) [22] M. Frémond, K. Kuttler, M. Shillor, Existence and uniqueness of solutions for a dynamic one-dimensional damage model, J. Math. Anal. Appl ) [23] M. Frémond, K. Kuttler, B. Nedjar, M. Shillor, One-dimensional models of damage, Adv. Math. Sci. Appl ) [24] M. Frémond, A. Visintin, Dissipation dans le changement de phase. Surfusion. Changement de phase irréversible, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre ) [25] V.-M. okkanen, Existence of a periodic solution for implicit nonlinear equations, Differential Integral Equations ) [26] N. Kenmochi, M. Kubo, Periodic solutions to a class of nonlinear variational inequalities with time-dependent constraints, Funkcial. Ekvac ) [27] N. Kenmochi, M. Kubo, Periodic behavior of solutions to parabolic elliptic free boundary problems, J. Math. Soc. Japan ) [28] F. Luterotti, G. Schimperna, U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci ) [29] F. Luterotti, G. Schimperna, U. Stefanelli, Global solution to a phase-field model with irreversible and constrained phase evolution, Quart. Appl. Math. 6 22) [3] A. Mielke, Evolution of rate-independent systems, in: C. Dafermos, E. Feireisl Eds.), andbook of Differential Equations, Evolutionary Equations, vol. 2, Elsevier, 25, pp [31] A. Mielke, T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci ) [32] A. Mielke, U. Stefanelli, A discrete variational principle for rate-independent evolution, Adv. Calc. Var. 1 4) 28) [33] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, periodic problems, J. Differential Equations ) [34] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Internat. Ser. Numer. Math., vol. 153, Birkhäuser Verlag, Basel, 25. [35] G. Schimperna, A. Segatti, U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations, Discrete Contin. Dyn. Syst ) [36] A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst ) [37] T. Senba, On some nonlinear evolution equation, Funkcial. Ekvac ) [38] U. Stefanelli, The Brezis Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim ) [39] A. Visintin, Models of Phase Transitions, Progr. Nonlinear Differential Equations Appl., vol. 28, Birkhäuser, [4] A. Visintin, A new approach to evolution, C. R. Acad. Sci. Paris Sér. I Math ) [41] I.I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc ) [42] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 199.