aa r X i v : . [ m a t h . A P ] D ec Some simple problems for the next generations.
Alain HARAUXSorbonne Universit´es, UPMC Univ Paris 06, CNRS, UMR 7598,Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, [email protected]
Abstract
A list of open problems on global behavior in time of some evolution systems, mainly governedby P.D.E, is given together with some background information explaining the context in whichthese problems appeared. The common characteristic of these problems is that they appeareda long time ago in the personnal research of the author and received almost no answer till thenat the exception of very partial results which are listed to help the readers’ understanding ofthe difficulties involved.
AMS classification numbers:
Keywords:
Evolution equations, bounded solutions, compactness, oscillation theory, al-most periodicity, weak convergence, rate of decay, .0 ntroduction
Will the next generations go on studying mathematical problems? This in itself is an openquestion, but the growing importance of computer’s applications in everyday’s life together withthe fundamental intrication of computer science, abstract mathematical logic and the developmentsof new mathematical methods makes the positive answer rather probable.This text does not comply with the usual standards of mathematical papers for two reasons:it is a survey paper in which no new result will be presented and the results which we recall tomotivate the open questions will be given without proof.It is not so easy to introduce an open question in a few lines. Giving the statement of thequestion is not enough, we must also justify why we consider the question important and explainwhy it could not be solved until now. Both points are delicate because the importance of a problemis always questionable and the difficulty somehow disappears when the problem is solved.The questions presented here concern the theory of differential equations and mostly the caseof PDE. They were encountered by the author during his research and some of them are already 40years old. They might be considered purely academical by some of our colleagues more concernedby real world applications, but they are selected, among a much wider range of open questions,since their solution probably requires completely new approaches and will likely open the doortowards a new mathematical landscape.
Throughout this section, the terms “maximal monotone operator” and “almost periodic function”will be used without having been defined. Although both terms are by now rather well known, thedefinitions and main properties of these objects will be found respectively in the reference texts [8]and [2].One of my first fields of investigation was, in connection with the abstract oscillation theory,the relationship between (pre-)compactness and asymptotic almost periodicity for the trajectoriesof an almost periodic contractive process. The case of autonomous processes (contraction semi-groups on a metric space) had been studied earlier in the Hilbert space framework by Dafermosand Slemrod [15], the underlying idea being that on the omega-limit set of a precompact trajectory,the semi-group becomes an isometry group. Then the situation resembles the simpler case of theisometry group generated on a Hilbert space H by the equation u ′ + Au ( t ) = 0with A ∗ = − A for which almost periodicity of precompact trajectories was known already from L. Amerio quitea while ago (the case of vibrating membranes and vibrating plates with fixed bounded edge arespecial cases of this general result). 1he case of a non-autonomous process, associated with a time-dependent evolution equation ofthe form u ′ + A ( t ) u ( t ) ∋ periodic contraction process on a complete metric space, and in the same paper I exhibiteda simple almost periodic (linear) isometry process on R , generated by an equation of the form u ′ + c ( t ) J u ( t ) = 0with J a π - rotation around 0, for which no trajectory except 0 is almost periodic.Actually, while writing my thesis dissertation, I was specifically interested in the so-called“quasi-autonomous” problem, and I met the following general question Problem 1.1. (1977) Let A be a maximal monotone operator on a real Hilbert space H, let f : R −→ H be almost periodic and let u be a solution of de u ′ + Au ( t ) ∋ f ( t )on [0 , + ∞ ) with a precompact range . Can we conclude that u is asymptotically almost periodic?After studying a lot of particular cases in which the answer is positive ( A = L linear, A asubdifferential ∂ Φ and some operators of the form L + ∂ Φ ) , I proved in [19] that the answeris positive if H = R N with N ≤
2. But the answer is unknown for general maximal monotoneoperators even if H = R . Remark 1.1.
In [28] it is stated that the answer is positive for all N , but there is a mistake in theproof, relying on a geometrical property which is not valid in higher dimensions, more specificallyin 3D the intersection of the (relative) interiors of two arbitrarily close isometric proper trianglescan be empty. Therefore the argument from [19] cannot be used in the same way for N ≥ Remark 1.2.
The problem is also open even when A ∈ C ( H, H ), in which case the monotonicityjust means ∀ u ∈ H, ∀ v ∈ H, ( A ′ ( u ) , v ) ≥ . Remark 1.3.
The answer is positive if f is periodic, as a particular case of the main result of [17].Since an almost periodic function has precompact range, studying the existence of almost peri-odic solutions requires some criteria for precompactness of bounded orbits. In the case of evolutionPDE, precompactness is classically derived from higher regularity theory. For parabolic equationsthe smoothing effect provides some higher order regularity for t > R + . In the hyperbolic case, although there is no smoothing effect in finite time, precompactnessof orbits was derived by Amerio and Prouse [1] from higher regularity of the source and strongcoercivity of the damping operator g in the case of the semilinear hyperbolic problem u tt − ∆ u + g ( u t ) = f ( t, x ) in R + × Ω , u = 0 on R + × ∂ Ω2here Ω be a bounded domain of R N . But this method does not apply even in the simple case g ( v ) = cv for c > , N ≤
3, a case where boundedness of all trajectories is known. The following questionmakes sense even when the source term is periodic in t and g is globally Lipschitz continuous. Problem 1.2. (1978) Let Ω be a bounded domain of R N and g a nonincreasing Lipschitzfunction. We consider the semilinear hyperbolic problem u tt − ∆ u + g ( u t ) = f ( t, x ) in R + × Ω , u = 0 on R + × ∂ ΩWe assume that f : R −→ L (Ω) is continuous and periodic in t. Assuming u ∈ C b ( R + , H (Ω)) ∩ C b ( R + , L (Ω))can we conclude that [ t ≥ { ( u ( t, . ) , u t ( t, . )) } is precompact in H (Ω) × L (Ω)? Remark 1.4.
The answer is positive in the following extreme cases1) If g = 0 (by Browder-Petryshyn’s theorem, there is a periodic solution, hence compact, andall the others are precompact by addition.)2) If g − is uniformly continuous, cf. [22] , the result does not require Lipschitz continuity of g and applies for instance to g ( v ) = cv for c > , N ≤ g ( v ) = v + and N = 1 already seemsto be non-trivial. Remark 1.5.
The same question is of course also relevant when f is almost periodic, and the resultof [22] is true in this more general context. Moreover precompactness of bounded trajectories when g = 0 is also true when f is almost periodic. This is related to a fundamental result of Ameriostating that if the primitive of an almost periodic function: R −→ H is bounded, it is also almostperiodic. More precisely, if H is a Hilbert space and L is a (possibly unbounded ) skew-adjointlinear operator with compact resolvent, let us consider a bounded solution (on R with values in H )of the equation U ′ + AU = F where F : R −→ H is almost periodic . Then exp( tA ) U := V is a bounded solution of V ′ = exp( tA ) F and, since exp( tA ) ψ is almost periodic as well as exp( − tA ) ψ for any ψ ∈ H , by a density argumenton generalized trigonometric polynomials, it is immediate to check that a function W : R −→ H is almost periodic if and only if exp( tA ) W : R −→ H is almost periodic. Then Amerio’s Theoremapplied to V gives the result, and this property applies in particular to the wave equation writtenas a system in the usual energy space. Then starting from a solution bounded on R + , a classicaltranslation-(weak)compactness argument of Amerio gives a solution bounded on R of the sameequation. We skip the details since this remark is mainly intended for experts in the field.3 Oscillation theory
Apart from the almost periodicity of solutions which provides a starting point to describe preciselythe global time behavior of vibrating strings and membranes with fixed edge, it is natural to try adescription of sign changes of the solutions on some subset of the domain. Let us first consider thebasic equation u ′′ + Au ( t ) = 0 , (1)where V is a real Hilbert space, A ∈ L ( V, V ′ ) is a symmetric, positive, coercive operator and thereis a second real Hilbert space H for which V ֒ → H = H ′ ֒ → V ′ where the imbedding on the left iscompact. In this case it is well known that all solutions u ∈ C ( R , V ) ∩ C ( R , H ) of (1) are almostperiodic : R → V with mean-value 0. Then for any form ζ ∈ V ′ , the function g ( t ) := h ζ, u ( t ) i is a real-valued continuous almost periodic function with mean-value 0. It is then easy to showthat either g ≡
0, or there exists
M > J with | J | ≥ M , g takes bothpositive and negative values. We shall say that a number M > g ∈ L loc ( R ) if the following alternative holds: either g ( t ) = 0 almost everywhere,or for any interval J with | J | ≥ M , we have meas { t ∈ J, f ( t ) > } > meas { t ∈ J, f ( t ) < } > . As a consequence of the previous argument , under the above conditions on
H, V and A , for anysolution u ∈ C ( R , V ) ∩ C ( R , H ) of (1) and for any ζ ∈ V ′ , the function g ( t ) := h ζ, u ( t ) i has somefinite strong oscillation length M = M ( u, ζ ).In the papers [9, 10, 26] the main objective was to obtain a strong oscillation length independentof the solution and the observation in various cases, including non-linear perturbations of equation(1). A basic example is the vibrating string equation u tt − u xx + g ( t, u ) = 0 in R × (0 , l ) , u = 0 on R × { , l } (2)where l > g ( t, . ) is an odd non-decreasing function of u for all t . Here the function spaces are H = L (0 , l ) and V = H (0 , l ). Since any function of V is continous, a natural form ζ ∈ V ′ is theDirac mass δ x for some x ∈ (0 , l ) . It turns out that 2 l is a strong oscillation length independentof the solution and the observation point x , exactly as in the special case g = 0, the ordinaryvibrating string. Since in this case all solutions are 2 l -periodic with mean-value 0 functions withvalues in V , it is clear that 2 l is a strong oscillation length independent of the solution and theobservation point x . The slightly more complicated g ( t, u ) = au with a > t . Thetime-periodicity is too unstable and for an almost periodic function, the determination of strongoscillation lengths is not easy in general, as was exemplified in [26]. The oscillation result of [9, 10]is consequently not so immediate even in the linear case. In the nonlinear case, it becomes evenmore interesting because the solutions are no longer known to be almost periodic.In dimensions N ≥
2, even the linear case becomes difficult. It has been established in [26] thateven for analytic solutions of the usual wave equation in a rectangle, there is no uniform pointwiseoscillation length common to all solutions at some points of the domain. One would imagine thatit becomes true if the point is replaced by an open subset of the domain, but apparently nobodyknows the answer to the exceedingly simpler following question:4 roblem 2.1. (1985) Let Ω = (0 , l ) × (0 , l ) ⊂ R . We consider the linear wave equation u tt − ∆ u = 0 in R × Ω , u = 0 on R × ∂ ΩGiven
T > u for which ∀ ( t, x ) ∈ [0 , T ] × (0 , l ) × (0 , l ) , u ( t, x ) > T large enough?Another simple looking intriguing question concerns the pointwise oscillation of solutions tosemi linear beam equations, since the solutions of the corresponding linear problem oscillate atleast as fast as those of the string equation: Problem 2.2. (1985) We consider the semilinear beam equation u tt + u xxxx + g ( u ) = 0 dans R × (0 , , u = u xx = 0 on R × { , } with g odd and nonincreasing with respect to u. Is it possible for a solution u ( t, . ) to remain positiveat some point x on an arbitrarily long (possibly unbounded) time interval?Finally, let us mention a question on spatial oscillation of solutions to parabolic problems. Sincethe heat equation has a very strong smoothing effect on the data, and all solutions are analyticinside the domain for t >
0, it seems natural to think that they do no accumulate oscillations andfor instance in 1D, the zeroes of u ( t, . ) will be isolated for t > . A very general result of this type,valid for semi linear problems as well has been proved by Angenent [3]. But as soon as N ≥
2, eventhe linear case is not quite understood. The answer to the following question seems to be unknown:
Problem 2.3. (1997) Let Ω ⊂ R N be a bounded open domain. We consider the heat equation u t − ∆ u = 0 in R × Ω , u = 0 on R × ∂ ΩFor t > E = { x ∈ Ω , u ( t, x ) = 0 } Is it true that E has a finite number of connected components? Remark 2.1.
The solutions u of the elliptic problem − ∆ u + f ( u ) = 0 in Ω , u = 0 on ∂ Ωare such that { x ∈ Ω , u ( x ) = 0 } has a finite number of connected components for a large class offunctions f , cf. e.g. [14]. Hence stationary solutions cannot provide a counterexample.5 A semi-linear string equation
There are in the Literature a lot of results on global behavior of solutions to Hamiltonian equationsin finite and infinite dimensions. Apart from Poincar´e’s recurrence theorem and the classical re-sults of Liouville on quasi-periodicilty for most solutions of completely integrale finite dimensionalhamiltonians, none of the recent results is easy and there is essentially nothing on PDE exceptin 1D. Even the case of semi linear string equations is not at all well-understood. While lookingfor almost-periodic solutions (trying to generalize the Rabinowitz theorem on non-trivial periodicsolution) I realized that even precompactness of general solutions is unknown for the simplest semilinear string equation in the usual energy space :
Problem 3.1. (1976) For the simple equation u tt − u xx + u = 0 in R × (0 , , u = 0 on R × { , } the following simple looking questions seem to be still openQuestion 1. Are there solutions which converge weakly to 0 as time goes to infinity?Question 2. If ( u (0 , . ) , u t (0 , . )) ∈ H ((0 , ∩ H ((0 , × H ((0 , V , does ( u ( t, . ) , u t ( t, . ))remain bounded in V for all times? Remark 3.1.
To understand the difficulty of the problem, let us just mention that the equation iu t + | u | u = 0 in R × (0 , , u = 0 on R × { , } has many solutions tending weakly to 0 and, although the calculations are less obvious, the samething probably happens to u tt + u = 0 in R × (0 , , u = 0 on R × { , } Hence the problem appears as a competition between the “good” behavior of the linear stringequation and the bad behavior of the distributed ODE associated to the cubic term.
Remark 3.2.
If the answer to question 2 is negative, it means that, following the terminologyof Bourgain [7], the cubic wave equation on an interval is a weakly turbulent system. Besides,weak convergence to 0 might correspond to an accumulation of steep spatial oscillations of weakamplitude, not contradictory with the energy conservation of solutions.
Remark 3.3.
In [11]-[13], the authors investigated the problem u tt − u xx + u Z l u ( t, x ) dx = 0 in R × (0 , l ) , u = 0 on R × { , l } (3)which can be viewed as a simplified model to understand the above equation. In this case, there isno solution tending weakly to 0, and the answer to question 2 is positive. Interestingly enough, inthis case the distributed ODE takes the form u tt + c ( t ) u = 0 , so that the solution has the form a ( x ) u ( t ) + b ( x ) u ( t ) and remains in a two-dimensional vector space! This precludes both weakconvergence to 0 and weak turbulence. 6 Rate of decay for damped wave equations
Let us consider the semilinear hyperbolic problem u tt − ∆ u + g ( u t ) = 0 in R + × Ω , u = 0 on R + × ∂ Ωwhere Ω be a bounded domain of R N and g is a nondecreasing function with g (0) = 0. Undersome natural growth conditions on g , the initial value problem is well-posed and can be put in theframework of evolution equations generated by a maximal monotone operator in the energy space H (Ω) × L (Ω)An immediate observation is the formal identity ddt [ Z Ω ( u t + |∇ u | ) dx ] = − Z Ω g ( u t ) u t dx ≤ g ( s ) = cs with c >
0, one can provethe exponential decay of the energy by a simple calculation involving a modified energy function E ε ( t ) = Z Ω ( u t + |∇ u | ) dx + ε Z Ω uu t dx The exponential decay is of course optimal since ddt [ Z Ω ( u t + |∇ u | ) dx ] = − Z Ω cu t dx ≥ − c Z Ω ( u t + |∇ u | ) dx A similar calculation can be performed if 0 < c ≤ g ′ ( s ) ≤ C , and the result is even still validfor g ( s ) = cs + a | s | α s under a restriction on α > g ( s ) = a | s | α s, a > , α > α and N , various authors (cf. e.g. [30], [27] and the referencestherein) obtained the energy estimate Z Ω ( u t + |∇ u | ) dx ≤ C (1 + t ) − α But now the energy identity only gives ddt [ Z Ω ( u t + |∇ u | ) dx ] = − Z Ω a | u t | α +2 dx while to prove the optimality of the decay we would need something like ddt [ Z Ω ( u t + |∇ u | ) dx ] ≥ − C ( Z Ω u t dx ) α u t in L α +2 cannot be controlled in terms of the L norm, even if strongrestrictions on u t are known. If u t is known to be bounded in a strong norm, let us say an L p normwith p large, we can derive a lower estimate of the type[ Z Ω ( u t + |∇ u | ) dx ] ≥ δ (1 + t ) − β for some β > α . But even p = ∞ does not allow to reach the right exponent.In 1994, using special Liapunov functions only valid for N = 1, the author ( cf. [20]) showedthat for all sufficiently regular non-trivial initial data, we have the estimate Z Ω ( u t + |∇ u | ) dx ≥ C (1 + t ) − α In general, for
N > Z Ω ( u t + |∇ u | ) dx ≥ C (1 + t ) − K will be obtained if the initial data belong to D ( − ∆) × H (Ω) and α < N − . But we shall have inall cases K > α and K tends to infinity when α approaches the value N − . Remark 4.1.
It is perfectly clear that none of the above partial results is satisfactory, since foranalogous systems in finite dimensions, of the type u ′′ + Au + g ( u ′ )with A symmetric, coercive , ( g ( v ) , v ) ≥ c | v | α +2 and | g ( v ) | ≤ C | v | α +1 , the exact asymptotics of anynon-trivial solution is | u ′ | + | u | ∼ (1 + t ) − α Moreover, an optimality result of the decay estimate has been obtained in 1D by P. Martinez andJ. Vancostenoble [33] in the case of a boundary damping for which the same upper estimate holds.The difference is that inside the domain, an explicit formula gives a lot of information on thesolution.
Problem 4.1.
For the equation u tt − ∆ u + g ( u t ) = 0 in R + × Ω , u = 0 on R + × ∂ Ωwith g ( s ) = a | s | α s, a > , α > u for which | Z Ω ( u t + |∇ u | ) dx ∼ (1 + t ) − α ?Question 2. Can we find a solution u for which the above property is not satisfied? Remark 4.2.
Both questions seem to be still open for any domain and any α >
The resonance problem for damped wave equations with sourceterm
To close this short list, we consider the semilinear hyperbolic problem with source term u tt − ∆ u + g ( u t ) = f ( t, x ) in R + × Ω , u = 0 on R + × ∂ Ωwhere Ω be a bounded domain of R N . We assume that the exterior force f ( t, x ) is bounded withvalues in L (Ω), In this case, all solutions U = ( u, u t ) are locally bounded on (0 , T ) with values inthe energy space H (Ω) × L (Ω). The question is what happens as t tends to infinity.When g ( s ) behaves like a super linear power | s | α s for large values of the velocity, it followsfrom a method introduced by G. Prouse [32] and extended successively by many authors, amongwhich M. Biroli [4], [5] and the author of this survey, that the energy of any weak solution remainsbounded for t large, under the restriction α ( N − ≤ f bounded in stronger norms, f anti-periodic, higher growths for N ≤
2, cf e.g. [23], [24], [16] ). Butthe following basic question remains open:
Problem 5.1.
Assume N ≥ g ( s ) = a | s | α s, a > , α > N − f ( t, x ) boundedwith values in L (Ω)? Remark 5.1.
The positive boundedness results require a weaker boundedness condition on f , itis sufficient that it belongs to a Stepanov space S p ( R , L (Ω) with p >
1. The first results in thedirection were actually published by G. Prodi in 1956, so that the problem is about 60 years old...
References [1] L. Amerio and G. Prouse,
Uniqueness and almost-periodicity theorems for a non linear waveequation , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. , 46 (1969) 1–8.[2] L. Amerio and G. Prouse, Almost-periodic functions and functional equations , Van NostrandReinhold Co., New York-Toronto, Ont.-Melbourne 1971 viii+184 pp.[3] S. Angenent,
The zero set of a solution of a parabolic equation , J. Reine Angew. Math. (1988), 79–96.[4] M. Biroli ,
Bounded or almost periodic solution of the non linear vibrating membrane equation ,Ricerche Mat. (1973), 190–202.[5] M. Biroli and A. Haraux, Asymptotic behavior for an almost periodic, strongly dissipative waveequation , J. Differential Equations (1980), no. 3, 422–440.96] S. Bochner and J. Von Neumann, On compact solutions of operational-differential equations ,Ann. of Math. , 36 (1935), 255–291.[7] Jean Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamil-tonian , Internat. Math. Res. Notices (1996), 277–304.[8] H. Brezis, Op´erateurs maximaux monotones et semi-groupes de contractions dans les espaces deHilbert (French) North-Holland Mathematics Studies, No. 5., Amsterdam-London; AmericanElsevier Publishing Co., Inc., New York, 1973. vi+183 pp.[9] T. Cazenave and A. Haraux,
Propri´et´es oscillatoires des solutions de certaines ´equations desondes semi-lin´eaires , C. R. Acad. Sci. Paris Ser. I Math. (1984), no. 18, 449–452.[10] T. Cazenave and A. Haraux,
Oscillatory phenomena associated to semilinear wave equationsin one spatial dimension , Trans. Amer. Math. Soc. (1987), no. 1, 207–233.[11] T. Cazenave, A. Haraux and F.B. Weissler,
Une ´equation des ondes compl`etement int´egrableavec non-lin´earit´e homog`ene de degr´e trois , C. R. Acad. Sci. Paris Ser. I Math. (1991),no. 5, 237–241.[12] T. Cazenave, A. Haraux and F.B. Weissler,
A class of nonlinear, completely integrable abstractwave equations , J. Dynam. Differential Equations (1993), no. 1, 129–154.[13] T. Cazenave, A. Haraux and F.B. Weissler, Detailed asymptotics for a convex Hamiltoniansystem with two degrees of freedom. , J. Dynam. Differential Equations (1993), no. 1, 155–187.[14] M. Comte, A. Haraux and P. Mironescu, Multiplicity and stability topics in semilinear parabolicequations , Differential Integral Equations (2000), no. 7-9, 801–811.[15] C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups , J.Functional Analysis (1973), 97–106.[16] T. Gallouet, Sur les injections entre espaces de Sobolev et espaces d’Orlicz et application aucomportement `a l’infini pour des ´equations des ondes semi-linaires (French), Portugal. Math. (1983/84), no. 1, 97– 112 (1985).[17] A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic pro-cesses , J. Diff. Eq. (1983), no. 3, 473–483.[18] A. Haraux, A simple almost-periodicity criterion and applications , J. Diff. Eq. (1987), no.1, 51–61.[19] A. Haraux, Asymptotic behavior for two-dimensional, quasi-autonomous, almost periodic evo-lution equations , J. Diff. Eq. (1987), no. 1, 62–70.[20] A. Haraux, L p estimates of solutions to some non-linear wave equations in one space dimen-sion , Int.J. Math. Modelling and Numerical Optimization (2009), Nos 1-2, p. 146– 154.[21] A. Haraux, On the strong oscillatory behavior of all solutions to some second order evolutionequations , Port. Math. (2015), no. 2, 193–206.1022] A. Haraux, Almost-periodic forcing for a wave equation with a nonlinear, local damping term ,Proc. Roy. Soc. Edinburgh Sect. A (1983), no. 3–4, 195–212.[23] A. Haraux, Nonresonance for a strongly dissipative wave equation in higher dimensions ,Manuscripta Math. (1985), no. 1-2, 145–166.[24] A. Haraux, Anti-periodic solutions of some nonlinear evolution equations , Manuscripta Math.63 (1989), no. 4, 479–505.[25] A. Haraux,
Semi-linear hyperbolic problems in bounded domains , Math. Rep. 3 (1987), no. 1,i–xxiv and 1–281.[26] A. Haraux and V. Komornik,
Oscillations of anharmonic Fourier series and the wave equation ,Rev. Mat. Iberoamericana (1985), no. 4, 57–77.[27] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems ,Arch. Rational Mech. Anal. (1988), no. 2, 191–206.[28] Z. Hu and A.B. Mingarelli,
Almost periodicity of solutions for almost periodic evolution equa-tions , Differential Integral Equations (2005), no. 4, 469–480.[29] C. F. Muckenhoupt, Almost periodic functions and vibrating systems , Journal of MathematicalPhysics (1928–1929), 163–199.[30] M.Nakao, Asymptotic stability of the bounded or almost periodic solution of the wave equationwith nonlinear dissipative term , J. Math. Anal. Appl. (1977), no. 2, 336–343.[31] G. Prodi, Soluzioni periodiche di equazioni a derivate parziali di tipo iperbolico non lineari (Italian), Ann. Mat. Pura Appl. (4) (1956), 25–49.[32] G. Prouse, Soluzioni limitate dell’equazione delle onde non omogenea con termine dissipativoquadratico (Italian), Ricerche Mat. (1965), 41–48.[33] J. Vancostenoble and P. Martinez, Optimality of energy estimates for the wave equation withnonlinear boundary velocity feedbacks , SIAM J. Control Optim. 39 (2000), no. 3, 776–797(electronic).[34] S. Zaidman,
Solutions presque-p´eriodiques des ´equations diff´erentielles abstraites (French),Enseign. Math. (2)24