A PARTIALLY HINGED RECTANGULAR PLATE AS A MODEL FOR SUSPENSION BRIDGES

. A plate model describing the statics and dynamics of a suspension bridge is suggested. A partially hinged plate subject to nonlinear restoring hangers is considered. The whole theory from linear problems, through nonlinear stationary equations, ending with the full hyperbolic evolution equation is studied. This paper aims to be the starting point for more reﬁned models.


Introduction. Due to the videos available on the web [34], the Tacoma Narrows
Bridge collapse is certainly the most impressive failure of the history of bridges. But, unfortunately, it is not an isolated event, many other bridges collapsed in the past, see [3,15]. According to [14], around 400 recorded bridges failed for several different reasons and the ones who failed after year 2000 are more than 70. Strong aerodynamic instability is manifested, in particular, in suspension bridges which usually have fairly long spans. Hence reliable mathematical models appear necessary for a precise description of the instability and of the structural behavior of suspension bridges under the action of dead and live loads.
On one hand, realistic models appear too complicated to give helpful hints when making plans. On the other hand, simplified models do not describe with sufficient accuracy the complex behavior of actual bridges. We refer to [10] for a survey of some existing models.
A one-dimensional simply supported beam suspended by hangers was suggested as a model for suspension bridges in [19,27,28]. It is assumed that when the hangers are stretched there is a restoring force which is proportional to the amount of stretching but when the beam moves in the opposite direction, the hangers slacken and there is no restoring force exerted on it. If u = u(x, t) denotes the vertical displacement of the beam (of length L) in the downward direction, the following fourth order nonlinear equation is derived x ∈ (0, L) , t > 0 ,

ALBERTO FERRERO AND FILIPPO GAZZOLA
where u + = max{u, 0}, γu + represents the force due to the hangers, and f represents the forcing term acting on the bridge, including its own weight per unit length.
For time periodic f , McKenna-Walter [27] prove the existence of multiple periodic solutions of (1). Moreover, in [28] they normalize (1) by taking γ = 1 and f ≡ 1: then by seeking traveling waves u(x, t) = 1 + w(x − ct) they end up with the ODE w (s) + kw (s) + ψ(w(s)) = 0 (s ∈ R, k = c 2 ) , a term which takes into account both the restoring force due to the hangers and external forces including gravity. Soon after the Tacoma Narrows Bridge collapse [32,34], three engineers were assigned to investigate and report to the Public Works Administration. The Report [4] considers ...the crucial event in the collapse to be the sudden change from a vertical to a torsional mode of oscillation, see [32, p.63]. But if one views the bridge as a beam as in (1), there is no way to highlight torsional oscillations. A model suggested by McKenna [25] considers the cross section of the bridge as a rod, free to rotate about its center which behaves as a forced oscillator subject to the forces exerted by the two lateral hangers. After normalization, the force is taken again as in (2). In order to smoothen the force by maintaining the asymptotically linear behavior at 0, McKenna-Tuama [26] also consider ψ(w) = c(e aw − 1) for some a, c > 0. Then, after adding some damping and forcing, [25,26] were able to numerically replicate in a cross section the sudden transition from standard and expected vertical oscillations to destructive and unexpected torsional oscillations. More recently, Arioli-Gazzola [5] reconsidered this model and studied its isolated version (energy conservation) with nonlinear restoring forces due to the hangers: they were able to display a sudden appearance of torsional oscillations. This phenomenon was explained using the stability of a fixed point of a suitable Poincaré map. The full bridge was then modeled in [5] by considering a finite number of parallel rods linked to the two nearest neighbors rods with attractive linear forces representing resistance to longitudinal and torsional stretching; this discretization of a suspension bridge is justified by the positive distance between hangers. The sudden appearance of torsional oscillations was highlighted also within the multiple rods model.
The nonlinear behavior of suspension bridges is by now well established, see e.g. [7,10,17,31]. After replacing the specific nonlinear term ψ(w) in (2) by a fairly general superlinear term ψ(w), one sees that traveling waves of (1) display selfexcited oscillations, see [6,12,13]: the solution may blow up in finite time with wide oscillations. So, a reliable model for suspension bridges should be nonlinear and it should have enough degrees of freedom to display torsional oscillations. In this respect, Problem 11] suggest to study the following equation where ψ is "like" ψ in (2). The purpose of the present paper is to set up the full theory for (3) in a bounded domain (representing the roadway) and to study the corresponding evolution problem similar to (1). A long narrow rectangular thin plate hinged at two opposite edges and free on the remaining two edges well describes the roadway of a suspension bridge which, at the short edges, is supported by the ground. Let L denote its length and 2 denote its width; a realistic assumption is that 2 ∼ = L 100 . For simplicity, we take L = π so that, in the sequel, Ω = (0, π) × (− , ) ⊂ R 2 . Our purpose is to provide a reliable model and to study the corresponding Euler-Lagrange equations. Since several energies are involved, we reach this task in several steps. We first recall the derivation of the bending elastic energy of a deflected plate, according to the Kirchhoff-Love [16,22] theory. Then we consider the action of both dead and live loads described by some forcing term f ; the equilibrium position of the plate u is then the minimum of a convex energy functional and is the unique solution of in Ω (4) under suitable boundary conditions. We set up the correct variational formulation of (4) (Theorem 3.1) and when f depends only on the longitude, f = f (x), we are able to determine the explicit form of u by separating variables (Theorem 3.2). In order to analyze the oscillating modes of the bridge, we also consider the eigenvalue problem ∆ 2 w = λw in Ω (5) where λ is the eigenvalue and w = w(x, y) is the eigenfunction. We characterize in detail the spectrum and the corresponding eigenfunctions (Theorem 3.4). The eigenvalues exhibit some weakness on the long edges and manifest a tendency to display a torsional component, see Figure 3.
Then we introduce into the model the elastic restoring force due to the hangers which is confined in a proper subset ω of Ω such as two small rectangles close to the horizontal edges, see Figure 1. The restoring force h = h(x, y, u) is superlinear with respect to u, which yields a superquadratic potential energy ω H(x, y, u). A particular form of h is suggested to describe the precise behavior of hangers, see (15) below. The equilibrium position is then given by the unique solution of in Ω .
Finally, if the force f is variable in time, so is the the equilibrium position and also the kinetic energy of the structure comes into the energy balance. This leads to the fourth order wave equation in Ω × (0, T ) where (0, T ) is an interval of time. Well-posedness of an initial-boundary-valueproblem is shown in Theorem 3.6. Our future target is to reproduce within our plate model the same oscillating behavior visible at the Tacoma Bridge [34]. This paper should be considered as a first necessary step in order to reach more challenging results. This paper is organized as follows. In Section 2 we describe the physical model and we derive the PDE's which have to be solved. In Section 3 we state our main results: existence, uniqueness, and qualitative behavior of the solutions of the PDE's. The remaining sections of the paper are devoted to the proofs of these results.
2. The physical model.

2.1.
A linear model for a partially hinged plate. The bending energy of the plate Ω involves curvatures of the surface. Let κ 1 and κ 2 denote the principal curvatures of the graph of a smooth function u representing the vertical displacement of the plate in the downward direction, then a simple model for the bending energy of the deformed plate Ω is where d denotes the thickness of the plate, σ the Poisson ratio defined by σ = λ

2(λ+µ)
and E the Young modulus defined by E = 2µ(1 + σ), with the so-called Lamé constants λ, µ that depend on the material. For physical reasons it holds that µ > 0 and usually λ > 0 so that Moreover, it always holds true that σ > −1 although some exotic materials have a negative Poisson ratio, see [18]. For metals the value of σ lies around 0.3, see [22, p.105], while for concrete 0.1 < σ < 0.2. For small deformations the terms in (8) are taken as approximations being purely quadratic with respect to the second order derivatives of u. More precisely, for small deformations u, one has (κ 1 + κ 2 ) 2 ≈ (∆u) 2 , κ 1 κ 2 ≈ det(D 2 u) = u xx u yy − u 2 xy , and therefore κ 2 Then, if f denotes the external vertical load (both dead and live) acting on the plate Ω and if u is the corresponding (small) deflection of the plate in the vertical direction, by (8) we have that the total energy E T of the plate becomes By replacing the load f with Ed 3 12(1−σ 2 ) f and up to a constant multiplier, the energy E T may be written as Note that for σ > −1 the quadratic part of the functional (11) is positive. This variational formulation appears in [8], while a discussion for a boundary value problem for a thin elastic plate in a somehow old fashioned notation is made by Kirchhoff [16], see also [11, Section 1.1.2] for more details and references. The unique minimizer u of E T , satisfies the Euler-Lagrange equation (4). We now turn to the boundary conditions to be associated to (4). We seek the ones representing the physical situation of a plate modeling a bridge. Due to the connection with the ground, the plate Ω is assumed to be hinged on its vertical edges and hence u(0, y) = u xx (0, y) = u(π, y) = u xx (π, y) = 0 ∀y ∈ (− , ) .
The deflection of the fully hinged rectangular plate Ω (that is u = u νν = 0 on ∂Ω) under the action of a distributed load has been solved by Navier [30] in 1823, see also [23,Section 2.1]. The general problem of a load on the rectangular plate Ω with two opposite hinged edges was considered by Lévy [21], Zanaboni [35], and Nadai [29], see also [23,Section 2.2] for the analysis of different kinds of boundary conditions on the remaining two edges y = ± . In the plate Ω, representing the roadway of a suspension bridge, the horizontal edges y = ± are free and the boundary conditions there become (see e.g. [33, (2.40 (13) In Section 3.1 we show how these boundary conditions arise. Note that free boundaries yield small stretching energy for the plate; this is the reason why we take c = 0 in (3).
2.2. A nonlinear model for a dynamic suspension bridge. Assume that the bridge is suspended by hangers whose action is concentrated in the union of two thin strips parallel to the two horizontal edges of the plate Ω, i.e. in a set of the type ω : In order to describe the action of the hangers we introduce a continuous function g : R → R satisfying g ∈ C 1 (0, +∞), g(s) = 0 for any s ≤ 0, g (0 + ) > 0, g (s) ≥ 0 for any s > 0 .
Then, the restoring force due to the hangers takes the form h(x, y, u) = Υ(y)g(u + γx(π − x)) (15) where Υ is the characteristic function of (− , − + ε) ∪ ( − ε, ) and γ > 0. This choice of h is motivated by the fact that the action of the hangers is larger around the central part of the bridge x = π/2, than on its sides x = 0 and x = π where the bridge is supported. This parabolic behavior is quite visible in certain bridges such as the Deer Isle Bridge, see Figure 2. More generally we may consider a force h satisfying the following assumptions: and h is locally Lipschitzian with respect to s, i.e.
for any bounded interval I ⊂ R.
Finally, assume that the external force also depends on time, f = f (x, y, t). If m denotes the mass of the plate, then the corresponding deformation u has a kinetic energy given by the integral m 2|Ω| Ω u 2 t dxdy .
By the time scaling t → m|Ω| −1 t, we can set m|Ω| −1 = 1. This term should be added to the nonlinear static energy (19): (21) This represents the total energy of a nonlinear dynamic bridge. As for the action, one has to take the difference between kinetic energy and potential energy and integrate on an interval [0, T ]: The equation of the motion of the bridge is obtained by taking critical points of the functional A: Due to internal friction, we add a damping term and obtain for (x, y)∈ Ω where δ is a positive constant. Notice that this equation also arises in different contexts, see e.g. [9, equation (17)], and is sometimes called the Swift-Hohenberg equation.
3. Main results. Our first purpose is to minimize the energy functional E T , defined in (11), on the space We also define H(Ω) := the dual space of H 2 * (Ω) and we denote by ·, · the corresponding duality. Since we are in the plane, H 2 * (Ω) ⊂ C 0 (Ω) so that the condition on {0, π}×(− , ) introduced in the definition of H 2 * (Ω) is satisfied pointwise and If f ∈ L 1 (Ω) then the functional E T is well-defined in H 2 * (Ω), while if f ∈ H(Ω) we need to replace Ω f u with f, u although we will not mention this in the sequel. The first somehow standard statement is the connection between minimizers of the energy function E T and solutions of (14). It shows that the variational setting is correct and it allows to derive the boundary conditions. Theorem 3.1. Assume (9) and let f ∈ H(Ω). Then there exists a unique u ∈ H 2 * (Ω) such that , and if u ∈ C 4 (Ω) then u is a classical solution of (14).
Since we have in mind a long narrow rectangle, that is π, it is reasonable to assume that the forcing term does not depend on y. So, we now assume that In this case, we may solve (14) following [23, Section 2.2] although the boundary conditions (13) require some additional effort. A similar procedure can be used also for some forcing terms depending on y such as e ±y f (x) or yf (x), see [23]. We extend the source f as an odd 2π-periodic function over R and we expand it in Fourier series so that {β m } ∈ 2 and the series converges in L 2 (0, π) to f . Then we define the constants and we prove Theorem 3.2. Assume (9) and that f satisfies (25)- (26). Then the unique solution of (14) is given by where the constants A and B are defined in (27) and (28).
When → 0, the plate Ω tends to become a one dimensional beam of length π. We wish to analyze the behavior of the solution and of the energy in this limit situation. To this end, we re-introduce the constants appearing in (10) that were normalized in (11). Let f ∈ L 2 (Ω) be as in (25) and let u be a solution of the problem whose total energy is given by (10).
where u is the unique solution of (14) found in Theorem 3.2. If we view the plate as a parallelepiped- Here the forcing term 2 f represents a force per unit of length and I = d 3 6 = (− , )×(−d/2,d/2) z 2 dydz is the moment of inertia of the section of the beam with respect to its middle line parallel to the y-axis. Then (30) reduces to Ed 3 12 ψ = f , the function ψ is independent of but the corresponding total energy of the beam depends on : Then we prove Theorem 3.3. Assume (9) and let f ∈ L 2 (Ω) be a vertical load per unit of surface depending only on x, see (25)- (26). Let u and ψ be respectively as in (29) and (30). Then where E T (u ) is given by (10) and E T (ψ) is given by (31). Theorem 3.3 states that, when → 0, the solution and the energy of the plate are "almost the same" as for the beam. However, one cannot neglect the o( ) term if one wishes to display torsional oscillations.
Next, we study the oscillating modes of the rectangular plate; we consider the eigenvalue problem Similar to (24), problem (33) admits the following variational formulation: a nontrivial function w ∈ H 2 * (Ω) is an eigenfunction of (33) if In such a case we say that λ is an eigenvalue for problem (33). In Section 7 we prove that for all > 0 and σ ∈ (0, 1 2 ) there exists a unique (34) The number λ = µ 2 1 is the least eigenvalue. Theorem 3.4. Assume (9). Then the set of eigenvalues of (33) may be ordered in an increasing sequence {λ k } of strictly positive numbers diverging to +∞ and any eigenfunction belongs to C ∞ (Ω). The set of eigenfunctions of (33) is a complete system in H 2 * (Ω). Moreover, the least eigenvalue of (33) is is the unique solution of (34); the least eigenvalue µ 2 1 is simple and the corresponding eigenspace is generated by the positive eigenfunction In fact, we obtain a stronger statement describing the whole spectrum and characterizing the eigenfunctions, see Theorem 7.6 in Section 7. In Proposition 7.7 we also show that if is small enough ( ≤ 0.44), then the first two eigenvalues are simple. In Figure 3 we display the qualitative behavior of the first two "longitudinal" eigenfunctions and of the first two "torsional" eigenfunctions. It appears that the maximum and minimum of these eigenfunctions are attained on the boundary and that every mode has a tendency to display a torsional behavior: as expected, the "weak" part of the plate are the two long free edges. Note also that in the limit case σ = 0, excluded by assumption (9), the first eigenvalue is λ 1 = 1 and the first eigenfunction is sin x.
We now turn to the nonlinear model. With a simple minimization argument one can prove where E T is the nonlinear static energy defined in (19).
Since the proof of Theorem 3.5 is standard, we omit it. Our last result proves well-posedness for the evolution problem (22). If T > 0 we say that is a solution of (22) if it satisfies the initial conditions and if If T = +∞ then the interval [0, T ] should be read as [0, +∞). Then we have Theorem 3.6. Assume (9), (16)- (18). Let T > 0 (including the case T = +∞), let f ∈ C 0 ([0, T ]; L 2 (Ω)) and let δ > 0; let u 0 ∈ H 2 * (Ω) and u 1 ∈ L 2 (Ω). Then (i) there exists a unique solution of (22); (ii) if f ∈ L 2 (Ω) is independent of t, then T = +∞ and the unique solution u of (22) satisfies On the closed subspace H 2 * (Ω) we may also define a different scalar product. Lemma 4.1. Assume (9). On the space H 2 * (Ω) the two norms are equivalent. Therefore, H 2 * (Ω) is a Hilbert space when endowed with the scalar product Proof. We first get rid of the L 2 -norm. Take any u ∈ H 2 * (Ω) so that u ∈ C 0 (Ω) and for all (x, y) ∈ Ω we have where we used an integration by parts and twice Hölder's inequality. This inequality, readily yields u L 2 ≤ C D 2 u L 2 for some C > 0 and proves that the H 2 (Ω)-norm is equivalent to the norm u → D 2 u L 2 on the space H 2 * (Ω). Next, we notice that so that the norms u → D 2 u L 2 and H 2 * (Ω) are equivalent. These two equivalences prove the lemma.
By combining Lemma 4.1 with the Lax-Milgram Theorem, we infer that for any f ∈ H(Ω) there exists a unique u ∈ H 2 * (Ω) satisfying (24). This proves the first part of Theorem 3.1.
Finally, we show that smooth weak solutions and classical solutions coincide. Note first that, for all u ∈ H 2 * (Ω) we have u(0, y) = u(π, y) = u y (0, y) = u y (π, y) = u yy (0, y) = u yy (π, y) = 0 for any y ∈ (− , ). Then, by adapting the Gauss-Green formula to our situation, and with some integration by parts, we obtain that if u ∈ C 4 (Ω) ∩ H 2 * (Ω) satisfies (24), then (40), then all the boundary terms vanish and we deduce that ∆ 2 u = f in Ω. Hence we may drop the double integral in (40). By arbitrariness of v, the coefficients of the terms v x (π, y), v x (0, y), v(x, − ), v y (x, − ), v y (x, ), and v(x, ) must vanish identically and we obtain (12)-(13); this conclusion may also be reached with particular choices of v but we omit here the tedious computations.
and note that it solves the ODE Moreover, φ ∈ H 2 (0, π) is given by and the series (44) converges in H 2 (0, π) and, hence, uniformly. We now introduce the auxiliary function v(x, y) := u(x, y) − φ(x); if u solves (14), We seek solutions of (45) by separating variables, namely we seek functions for all x ∈ (0, π), and, by (44), , the condition for y = − being automatically fulfilled since Y m is even. By plugging these information into the explicit form (48) of the derivatives we find the system Let us now recall a well-known result about Fourier series which will be repeatedly used in the sequel. Then, by Lemma 6.1, we obtain Since u solves (14), the corresponding energy is given by (11) and hence collecting (51)-(54) we obtain With a direct computation one can see that by (27) and (28) Consider now u and ψ as in (29) and (30); recall that u = 12(1−σ 2 ) Ed 3 u where u solves (14) and that ψ = 12 Ed 3 φ. Then, from (50) we deduce the first of (32). Since u solves (29), the corresponding energy is given by (10) and hence, by (55) and the identity u = 12(1−σ 2 ) and the second of (32) follows.
7. Proof of Theorem 3.4. By Lemma 4.1 the bilinear form (37) is continuous and coercive; standard spectral theory of self-adjoint operators then shows that the eigenvalues of (33) may be ordered in an increasing sequence of strictly positive numbers diverging to +∞ and that the corresponding eigenfunctions form a complete system in H 2 * (Ω). The eigenfunctions are smooth in Ω: this may be obtained by making an odd extension as in Lemma 4.2 and with a bootstrap argument. This proves the first part of Theorem 3.4.
Take an eigenfunction w of (33) and consider its Fourier expansion with respect to the variable x: Since w ∈ C ∞ (Ω), the Fourier coefficients h m = h m (y) are smooth functions and solve the ordinary differential equation for some λ > 0. The eigenfunction w in (56) satisfies (12), while by imposing (13) we obtain the boundary conditions on h m Put µ = √ λ > 0 and consider the characteristic equation α 4 − 2m 2 α 2 + m 4 − µ 2 = 0 related to (57). By solving this algebraic equation we find Three cases have to be distinguished.
Proof. Consider the function η m (t) := For any t > √ σm we have This shows that η m is decreasing in ( √ σm, +∞) and, if β > γ > √ σm then η m (β) < η m (γ) so that (63) cannot hold. We have proved that if γ and β satisfy (63) then necessarily γ ∈ [0, √ σm). Since β = 2m 2 − γ 2 , identity (63) is equivalent to Then we define is nonnegative and decreasing and hence so is its square. It then follows that g m is decreasing in [0, √ σm] and g m ( √ σm) = 0. On the other hand, the map t → t tanh( t) is increasing in [0, √ σm] and vanishes at t = 0. This proves that there exists a unique γ m ∈ (0, √ σm) satisfying (64). The statements of the lemma now follow by putting µ m = m 2 − γ 2 m . In the next result we prove that the sequence {µ m } found in Lemma 7.1 is increasing.
Proof. By (60), the equation (63) reduces to We consider Φ as a function defined in the region of the plane {(m, µ) ∈ R 2 ; (1 − σ)m 2 < µ < m 2 }. In this region, the three maps are all positive, strictly increasing with respect to m, and strictly decreasing with respect to µ. Therefore, the function m → µ m , implicitly defined by Φ(m, µ m ) = 1, is strictly increasing.
Similarly, the second system in (62) has nontrivial solutions (b, d) if and only if
We finally compare µ m with µ m .
Lemma 7.5. Assume (9). Let µ m and µ m be, respectively, as in Lemmas 7.1 and 7.3. Then for any m ≥ m σ we have µ m < µ m .
• The case µ = m 2 . By (59) we infer that possible nontrivial solutions of (57)-(58) have the form By differentiating h m and by imposing the boundary conditions (58) we get (75) The first system in (75) has the unique solution a = c = 0 under the assumption (9). The second system in (75) admits a nontrivial solution (b, d) if and only if By (9)  • The case µ > m 2 . By (59) we infer that Therefore, possible nontrivial solutions of (57) have the form h m (y) = a cosh(βy) + b sinh(βy) + c cos(γy) + d sin(γy) (a, b, c, d ∈ R) .
Differentiating h m and imposing the boundary conditions (58) yields the two systems: Due to the presence of trigonometric sine and cosine, for any integer m there exists a sequence ζ m k ↑ +∞ such that ζ m k > m 2 for all k ∈ N and such that if µ = ζ m k for some k then one of the above systems admits a nontrivial solution. Not only the above arguments prove all the statements of Theorem 3.4, but they also prove the following result.
Note that the eigenfunctions in (iii) are even with respect to y whereas the eigenfunctions in (iv) are odd. In the next result we give a precise description of the first two eigenvalues when is small enough. Proposition 7.7. Assume (9) and consider the eigenvalue problem (33). If ≤ 1 5 then the first two eigenvalues are simple and they coincide with the numbers λ 1 , λ 2 defined by Lemma 7.1. Therefore, Proof. By (9) we know that (67) may hold only if m √ 2 coth( m √ 2) > 9. In turn, since ≤ 1 5 , this necessarily yields m > 31. From Theorem 7.6 we readily obtain (80). In order to prove the statement it is therefore enough to show that all the other eigenvalues found in Theorem 7.6 are larger than or equal to 16 for ≤ 1 5 . We start by showing that for ≤ 1 5 the numbers µ corresponding to the case µ > m 2 are larger than or equal to 4. We take m = 1 since if m ≥ 2 we immediately obtain µ > 4 and we are done.
When m = 1 system (78) admits a nontrivial solution if and only if This may happen only if the two terms sin(γ ) and cos(γ ) have opposite sign: this yields We prove that µ ≥ 4 by showing that It is readily verified that h σ (µ, σ, ) < 0 so that (82) is satisfied provided that By differentiating we obtain h µ, and by the inequality s cosh s > sinh s, valid for any s > 0, we get Since the map x → tan x x is increasing in (0, π/2) and √ µ − 1 < √ 3/5, we have that > 0 for 1 < µ < 4 and ≤ 1 5 so that (83) follows and completes the proof in the case µ > m 2 .
By (9)  Hence, if µ = m 2 is the square root of an eigenvalue, then by (76) we have We have so shown that, in any case, µ > 4; hence, λ > 16.
If ≤ 0.44, (67) implies m > 14. Moreover, numerical computations show that (82), and hence Proposition 7.7, are true for all ≤ 0.44. 8. Proof of Theorem 3.6. In order to prove existence of solutions of (22), we perform a Galerkin-type procedure directly on the nonlinear problem (22). Uniqueness of solutions of (22) is obtained from suitable estimates coming from an energy identity. We start by proving global existence for solutions of (22). Proof. We divide the proof in several steps.
Step 1. We construct a sequence of solutions of approximated problems in finite dimensional spaces. By Theorem 3.4 we may consider an orthogonal complete system {w k } k≥1 ⊂ H 2 * (Ω) of eigenfunctions of (33) such that w k L 2 = 1. Let {λ k } k≥1 be the corresponding eigenvalues and, for any k ≥ 1, put W k := span{w 1 , . . . , w k }. For any k ≥ 1 let for any v ∈ W k and t ∈ (0, T ). If we write u k in the form u k (t) = k i=1 g k i (t)w i and we put g k (t) := (g k 1 (t), . . . , g k k (t)) T then the vector valued function g k solves From (18) we deduce that Φ k ∈ Lip loc (R k ; R k ) and hence (85) admits a unique local solution. We have shown that the function u k (t) = k j=1 g k j (t)w j belongs to C 2 ([0, τ k ); H 2 * (Ω)) is a local solution in some maximal interval of continuation [0, τ k ), τ k ∈ (0, T ], of the problem (Ω) and P k : H 2 * (Ω) → W k is the orthogonal projection onto W k .
Step 2. In this step we prove a uniform bound on the sequence {u k }.
Testing (90) with u n,m and integrating over (0, t), up to enlarging n, we obtain u n,m (t) 2 L 2 + u n,m (t) 2 ). But we have seen before that u k → u in C 0 ([0, T ]; L 2 (Ω)) so that u belongs to the same space and, up to subsequences, thus completing the proof of the claim.
Step 4. We take the limit in (86) and we prove the existence of a solution of (22).
In the next lemma we provide an energy identity for the nonlinear problem (22) and, by exploiting it, we show uniqueness of the solution.
In the last part of this section we consider problem (22) with f ∈ L 2 (Ω) independent of t. We want to study the behavior of the solution u(·, t) of (22) as t → +∞: its global existence and uniqueness is an easy consequence of Lemmas 8.1 and 8.2. Consider the energy function H(x, y, u(x, y, t)) dxdy .
By (91) we have that so that E u is nonincreasing in [0, +∞) and in particular it is bounded from above. On the other hand by Hölder and Young inequalities, continuous embedding H 2 * (Ω) ⊂ L 2 (Ω), (17) it follows the existence of two constants C 1 , C 2 > 0 such that C 1 ( u (t) 2 L 2 + u(t) 2 for any t > 0 .
This bound allows us to study the long-time behavior of the global solution.