Let H be a real Hilbert space, in which denotes the norm of an element , and denotes the scalar product of two elements x and y. Let A be a self-adjoint operator on H with dense domain . We assume that H admits a countable orthonormal basis made by eigenvectors of A corresponding to an increasing sequence of positive eigenvalues .
We consider the second-order evolution equation
with initial conditions
All nonzero solutions to (1.1) decay to zero in the energy space , with a decay rate proportional to (see Proposition 3.1). This suggests the introduction and the investigation of the rescaled variable .
The special structure of the damping term guarantees that for any linear subspace such that , the space is positively invariant by the flow generated by (1.1). In particular, equation (1.1) possesses the so-called finite-dimensional modes, namely, solutions whose both components of the initial state are finite combinations of the eigenvectors. Denoting by and the projections of and on the k-th eigenspace, we shall call for simplicity the quantity
the “energy of the k-th Fourier component of ” (standing for total energy) while
will be called the “energy of ” (again meaning total energy). For t large, these quantities are easily seen to be equivalent to and , respectively. Our main results, formally stated as Theorem 2.1 and Theorem 2.5, can be summed up as follows.
The limit of the energy of depends only on the number of Fourier components of that are different from 0. In particular, the limit of the energy can take only countably many values.
If has only a finite number of Fourier components different from 0, then is asymptotic in a strong sense to a suitable solution to the nondissipative linear equation
Moreover, there is equipartition of the total energy in the limit, in the sense that all nonzero Fourier components of do have the same total energy.
If has infinitely many components different from 0, then tends to 0 weakly in the energy space, but not strongly. Roughly speaking, the energy of does not tend to 0, but in the limit there is again equipartition of the energy, now among infinitely many components, and this forces all components of to vanish in the limit.
In other words, the Fourier components of rescaled solutions to (1.1) communicate to each other, and this can result in some sort of energy transfer from lower to higher frequencies, longing for a uniform distribution of the energy among components. In the case of an infinite number of non-trivial Fourier components, the weak convergence to 0 implies non-compactness of the profile in the energy space. In particular, if A has compact resolvent, whenever the initial state belongs to and has an infinite number of elementary modes, the norm of in is unbounded, a typical phenomenon usually called weak turbulence, cf., e.g., [1, 6] for other examples.
Our abstract theory applies for example to wave equations with nonlinear nonlocal damping terms of the form
in a bounded interval of the real line with homogeneous Dirichlet boundary conditions. This is a toy model of the wave equation with local nonlinear damping
which in turn is the prototype of all wave equations with nonlinear dissipation of order higher than one at the origin. This more general problem was the motivation that led us to consider equations (1.4) and (1.1). It is quite easy to prove that all solutions to (1.5) decay at least as . Actually, the more general problem
in any bounded domain with homogeneous Dirichlet boundary conditions and g non-decreasing has been extensively studied under relevant assumptions on the behavior of g near the origin and some conditions on the growth of g at infinity, cf., e.g., [9, 2, 3, 8], in which reasonable energy estimates of the same form as those in the ODE case are obtained. However, the asymptotic behavior of solutions to the simple equation (1.5) is still a widely open problem since, unlike the ODE case, the optimality of this decay rate in unknown: there are neither examples of solutions to (1.5) whose decay rate is proportional to , nor examples of nonzero solutions that decay faster.
It is not clear whether our results shed some light on the local case or not. For sure, they confirm the complexity of the problem. In the case of (1.5), there are no simple invariant subspaces, and the interplay between components induced by the nonlinearity is more involved. Therefore, it is reasonable to guess that at most the infinite-dimensional behavior of (1.4) extends to (1.5), and this behavior is characterized by lack of an asymptotic profile and of strong convergence.
As a matter of fact, the problem of optimal decay rates is strongly related to regularity issues. It can be easily shown that the solutions to (1.5) with initial data in the energy space remain in the same space for all times, and their energy is bounded by the initial energy. But what about more regular solutions? Can one bound higher order Sobolev norms of solutions in terms of the corresponding norms of initial data? This is another open problem whose answer would imply partial results for decay rates, as explained in [5, 6], cf. also  for a partial optimality result in the case of boundary damping. However, the energy traveling toward higher frequencies might prevent the bounds on higher order norms from being true, or at least from being easy to prove.
This paper is organized as follows. In Section 2 we state our main results. In Section 3 we prove the basic energy estimate from above and from below for solution to (1.1), we introduce Fourier components, and we interpret (1.1) as a system of infinitely many ordinary differential equations. In Section 4 we consider a simplified system, obtained from the original one by averaging some oscillating terms. Then we analyze the simplified system, and we discover that it is the gradient flow of a quadratically perturbed convex functional, whose solutions exhibit most of the features of the full system we started with, including the existence of a large class of solutions which die off weakly at infinity. In Section 5 we investigate the asymptotic behavior of solutions to scalar differential equations and inequalities involving fast oscillating terms. Section 6 is devoted to estimates on oscillating integrals. Finally, in Section 7 we put things together and we conclude the proof of our main results.
Moreover, the classical energy
is of class , and its time-derivative satisfies
The following is the main result of this paper.
Let H be a Hilbert space, and let A be a linear operator on H with dense domain . Let us assume that there exist a countable orthonormal basis of H and an increasing sequence of positive real numbers such that
Then the asymptotic behavior of and its energy depends on J as follows.
(Trivial solution) If , then for every and, in particular,
(Finite-dimensional modes) If J is a finite set with j elements, then for every and every . In addition, for every , there exists a real number such that
and, in particular,
(Infinite-dimensional modes) If J is infinite, then
and hence converges to weakly but not strongly.
Let us comment on some aspects of Theorem 2.1 above.
The result holds true also when H is a finite-dimensional Hilbert space, but in this case only the first two options apply.
In the case of finite-dimensional modes, let us set
The assumptions of Theorem 2.1 imply, in particular, that all eigenvalues are simple. Things become more complex if multiplicities are allowed. Let us consider the simplest case where H is a space of dimension 2, and the operator A is the identity. In this case equation (1.1) reduces to a system of two ordinary differential equations of the form
If for some constant c, then for every , hence there is no equipartition of the energy in the limit.
Let H, A, , and J be as in Theorem 2.1. Let us assume in addition that J is infinite and
Then it turns out that
3 Basic energy estimates and reduction to ODEs
In this section we make the first steps toward the proof of Theorem 2.1. In particular, we prove a basic energy estimate, and we reduce the problem to a system of countably many ordinary differential equations.
Proposition 3.1 (Basic energy estimate).
Let H, A and be as in Theorem 2.1. Assume that . Then there exist two positive constants and such that
Integrating this differential inequality, we obtain the estimate from below in (3.1). Since for every , we deduce also that
Let us consider now the modified energy
where ε is a positive parameter. We claim that there exists such that
An integration of this differential inequality gives that
for a suitable constant , and hence the estimate from below in (3.3) implies that
which proves the estimate from above in (3.1).
and hence, from (3.2), we obtain
for a suitable constant depending on the initial data. This guarantees that (3.3) holds true when ε is small enough.
As for (3.4), after some computations, we obtain that it is equivalent to
The left-hand side is a quadratic form in the variables and , and it is nonnegative for all values of the variables, provided that
which is clearly true when ε is small enough. This completes the proof. ∎
Proposition 3.1 suggests that decays as , and motivates the variable change
The energy of is given by
We claim that there exist constants and such that
The upper estimate being quite clear, we just prove the lower bound. To this end, we start by the simple inequality
On the other hand,
is obviously greater than or equal to
By decomposing this expression, we obtain the inequality
and we end up with
which proves the lower bound in (3.7) with
Starting from (1.1), after some computations, we can verify that solves
where and are defined by
Due to (3.7), there exists a constant such that
In the sequel, we interpret and as time-dependent coefficients satisfying this estimate, rather than nonlinear terms.
Let now denote the components of with respect to the orthonormal basis. Then (3.8) can be rewritten as a system of countably many ordinary differential equations of the form
Let us introduce polar coordinates and in such a way that
In these new variables, every second-order equation in (3.10) is equivalent to a system of two first-order equations of the form (for the sake of shortness we do not write explicitly the dependence of and on t)
In particular, since the eigenvalues are bounded from below, from (3.9) it follows that there exists a constant such that
and so from (3.13) it follows, on replacing t by , that there exists a constant such that
We observe that can be factored out in the right-hand side of (3.14), and hence either for every , or for every , where the second option applies if and only if k belongs to the set J defined in (2.3). We observe also that the sequence is square-summable for every , and the square of its norm
In particular, from (3.7), it follows that
for every nontrivial solution.
(Finite-dimensional modes) If J is a nonempty finite set, then, for every , it turns out that
and there exists a real number such that
(Infinite-dimensional modes) If J is infinite, then
and under the additional uniform gap assumption (2.7), it turns out that
In this section we make some drastic simplifications in equations (3.14)–(3.15). These non-rigorous steps lead to a simplified model, which is then analyzed rigorously in Theorem 4.1 below. The result is that solutions to the simplified model exhibit all the features stated in Theorems 2.1 and 2.5 for solutions to the full system. Since the derivation of the simplified model is not rigorous, we can not exploit Theorem 4.1 in the study of (3.14)–(3.15). Nevertheless, the proof of Theorem 4.1 provides a short sketch without technicalities of the ideas that are involved in the proof of the main results.
To begin with, in (3.14) and (3.15) we ignore the terms with and . Indeed, these terms are integrable because of (3.16), and hence it is reasonable to expect that they have no influence on the asymptotic dynamics. Now let us consider (3.15), which seems to suggest that . If this is true, then the trigonometric terms in (3.14) oscillate very quickly, and in turn this suggests that some homogenization effect takes place. Therefore, it seems reasonable to replace all those oscillating terms with their time-averages.
The time-averages can be easily computed to be
A comparison of (4.1) and (4.2) reveals that the two oscillating functions in the integral (4.2) are in some sense independent when , while (4.3) shows that this independence fails when . This lack of independence plays a fundamental role in the sequel.
After replacing all oscillating coefficients in (3.14) with their time-averages, we are left with the following system of autonomous ordinary differential equations:
Quite magically, this system turns out to be the gradient flow of the functional
where ρ belongs to the space of square-summable sequences of nonnegative real numbers . Since is a continuous quadratic perturbation of a convex functional (the sum of the last two terms), its gradient flow generates a semigroup in . Solutions are expected to be asymptotic to stationary points of . In addition to the trivial stationary point with all components equal to 0, all remaining stationary points ρ are of the form
for some finite subset with j elements. Incidentally, it is not difficult to check that any such stationary point is the minimum point of the restriction of to the subset
Now we show that the asymptotic behavior of solutions to the averaged system (4.4) corresponds to the results announced in our main theorems.
Theorem 4.1 (Asymptotics for solutions to the homogenized system).
Let be a solution to system (4.4) in , and let . Then the asymptotic behavior of the solution depends on J as follows.
(Trivial null solution) If , then for every and every .
(Finite-dimensional modes) If J is a finite set with j elements, then for every and every , and
In other words, in this case the solution leaves in the subspace defined by ( 4.5 ), and tends to the minimum point of the restriction of to .
(Infinite-dimensional modes) If J is infinite, then
and, in particular, the solution tends to 0 weakly but not strongly.
First of all, we observe that components with null initial datum remain null during the evolution, while components with positive initial datum remain positive for all subsequent times.
Then we introduce the total energy of the solution, defined as in (3.17). Moreover, for every pair of indices h and k in J, we consider the ratio
which is well defined because the denominator never vanishes.
Simple calculations show that
Now we prove some basic estimates on the energy and the quotients, and then we distinguish the case where all components tend to 0, and the case where at least one component does not tend to 0. Non-optimal energy estimates. We prove that
Indeed, plugging the trivial estimate
into (4.9), we obtain that
Integrating the two differential inequalities, we deduce (4.11). Uniform boundedness of quotients. We prove that for every , there exists a constant such that
We point out that is independent of k, and actually it can be defined as
Therefore, it is enough to remark that the solutions to (4.10) are decreasing as long as they are greater than 1, and observe that the inner maximum in (4.13) is well defined because for every fixed , it turns out that as (because as ). The case where all components vanish in the limit. Let us assume that
In this case, we prove that J is infinite and (4.7) holds true.
Plugging this estimate into (4.9), we deduce that
Since as , these two differential inequalities imply (4.7) (we refer to Proposition 5.3 below for a more general result). The case where at least one component does not vanish in the limit. Let us assume that there exists such that
In this case, we prove that J is finite and (4.6) holds true.
Since is Lipschitz continuous (because its time-derivative is bounded), from (4.16) we deduce that
We are now ready to prove that J is finite. Let us assume on the contrary that this is not the case. Then, for every , there exists a subset with n elements, and hence
When , the last sum tends to n because of (4.17), and hence
which contradicts the estimate from above in (4.11) when n is large enough. To finish the proof, we now observe that the vector is a bounded solution of a first-order gradient system, so that (cf., e.g., [4, Example 2.2.5] or [7, Corollary 7.3.1]) its omega-limit set is made of stationary points only. But the only stationary point satisfying the condition of having all its limiting components positive and equal is the point with all components equal to the right-hand side of (4.6). ∎
5 Estimates for differential inequalities
In this section we investigate the asymptotic behavior of solutions to two scalar differential equations characterized by the presence of fast oscillating terms. Equations of this form are going to appear in the proof of our main results as the equations solved by the energy of the solution and by the ratio between two Fourier components.
Throughout the text, we shall meet oscillatory functions which are not absolutely integrable at infinity but have a convergent integral in a weaker sense.
Definition 5.1 (Semi-integrable function).
A function will be called semi-integrable on if the integral
converges to a finite limit as t tends to . In this case, the limit will be denoted as .
A classical example of a function which is semi-integrable but not absolutely integrable in for is
whenever . Another classical case (Fresnel’s integrals) is
In the second case the integrability comes from fast oscillations at infinity and the convergence of the integral appears immediately by the change of variable , which reduces us to (5.1) with . The semi-integrable functions that we shall handle are closer to in , in which case the integral can be reduced to (5.1) with , by the change of variable .
The first equation we consider is actually a differential inequality which generalizes (4.15). It takes the form
When is a positive constant, and , this inequality reduces to an ordinary differential equation, and it is easy to see that all its positive solutions tend to as . In the following statement we show that the same conclusion is true under a more general assumption on and .
Let be a positive constant, and let be a solution of class to the differential inequality (5.2). Let us assume the following:
The function is continuous and semi-integrable on .
The function is continuous and satisfies
There exists a constant such that
Then it turns out that
For every , let us set
Now (5.2) is equivalent to the two differential inequalities
Assumption (5.4) implies that
and (5.5) is equivalent to
Let us set
and observe that (5.8) implies that is increasing and
Let us concentrate on the differential inequality (5.6). Due to a well-known formula, every solution satisfies
We claim that the three terms in the right-hand side tend to 0 as , and hence
In order to estimate the third term, let us introduce the function
Due to the semi-integrability of , the function is well defined and as . Now an integration by parts gives that
The first two terms tend to 0 when multiplied by . As for the third term, we apply again de L’Hôpital’s rule and conclude that
This completes the proof of (5.11).
In an analogous way, from (5.7), we deduce that
The second equation we consider is a generalization of (4.10). It takes the form
When and , it is easy to see that all positive solutions tend to 1 as . In the following result we prove the same conclusion under more general assumptions on the coefficients.
Let be a positive solution of class to the differential equation (5.13).
Let us assume the following:
The function is bounded and of class , and it satisfies
There exists a constant such that
The functions and are bounded and semi-integrable.
Then it turns out that
Equation (5.13) is a classical Bernoulli equation, and the usual variable change transforms it into the linear equation
In the new setting, conclusion (5.16) is equivalent to proving that
In order to avoid plenty of factors 2, with a little abuse of notation, we replace , , with , , . This does not change the assumptions, but allows us to rewrite (5.17) in the simpler form
Now we introduce the function
and observe that
because of assumption (5.14). We also introduce the functions
which are well defined for every as a consequence of assumption (iii), and satisfy
Every solution to (5.19) is given by the well-known formula
We claim that the first and third term tend to 0 as , while the second term tends to 1. This would complete the proof of (5.18).
The second term can be rewritten as
The factor tends to . Since is increasing and tends to , we can apply de L’Hôpital’s rule to the second factor. We obtain that
and this settles the second term.
In order to compute the limit of the third term, we integrate by parts. We obtain that
When we multiply by , the first two terms in the right-hand side tend to 0, because of (5.20)–(5.22) and the boundedness of the function . Thanks to assumption (5.15), the absolute value of the last integral is less than or equal to
Now we multiply by , we factor out , and we compute the limit of the rest by exploiting de L’Hôpital’s rule, as we did before. From (5.20)–(5.22) and the boundedness of the functions and , we conclude that
This completes the proof of (5.18). ∎
In the third and last result of this section, we consider again equation (5.13). Let us assume for simplicity that for every , and . These assumptions do not guarantee that positive solutions tend to 1 as , but nevertheless they are enough to conclude that all solutions are bounded from above for (because solutions are decreasing as long as they stay in the region ). In the following result we prove a similar conclusion under more general assumptions on the coefficients.
Let be a positive solution of class to the differential equation (5.13). Let us assume the following:
The function is of class .
The functions and are continuous.
There exists a constant such that
There exists a constant such that
Let be any nonnegative real number such that
Then the following implication holds true:
Let us assume that , and set
If , the result is proved. Let us assume by contradiction that this is not the case, and hence . Due to the continuity of and the maximality of , it follows that
Let us set
Then it turns out that and, moreover,
Since in and is positive, (5.13) implies also that
which we can integrate as a linear differential inequality. Taking (5.27) into account, we find that
Now we claim that
In order to estimate the second integral, we introduce the function
This function is well defined because of the first inequality in (5.24) and, for the same reason, it satisfies
Now an integration by parts gives that
6 Estimates on oscillating integrals
In the three results of this section, we prove the convergence of some oscillating integrals and series of oscillating integrals. We need these estimates in the proof of our main result when we deal with the trigonometric terms of (3.14) and (3.15).
Let , let , and let be a function of class such that
Then, for every , it turns out that
We introduce the complex-valued functions
so that, clearly,
Now we have
yielding the immediate estimate
which implies (6.1). ∎
Lemma 6.1 can also be viewed as a special case of the following result.
Let be a continuous function, and let be a function of class . Let us assume that there exist two constants and such that
Then it turns out that
Let us introduce the function
This function is well defined because of assumption (6.2) and, for the same reason, it satisfies
Integrating by parts the left-hand side of (6.3), we find that
At this point, our assumptions imply that
which proves (6.3). ∎
The next lemma extends the previous estimates to some series of functions.
Let be a sequence of continuous functions, and let be a sequence of functions of class . Let us assume that the two series of functions
are normally convergent on compact subsets of , and that there exist three constants , , and such that
Then the series
is normally convergent on compact subsets of , and it satisfies
In analogy with the proof of Lemma 6.2, we introduce the functions
We observe that they are well defined because of assumption (6.5), and they satisfy
From assumption (6.4), it follows that
for every compact set . As a consequence, the normal convergence in K of the series (6.7) follows from the normal convergence in K of the series with general term . Due to normal convergence, we can exchange series and integrals in the left-hand side of (6.8) and deduce that
for every . When we sum over k, from (6.6) we deduce that
As for the sum of integrals, we first observe that the normal convergence, on compact subsets of , of the series with general term implies an analogous convergence of the series
Therefore, we can exchange once again series and integrals. Taking (6.6) into account, this leads to
7 Proof of the main results
7.1 Equations for the energy and quotients
Preliminary estimates on components.
Let us consider the notations introduced in Section 3, where we reduced ourselves to proving (3.19) through (3.21). In this first paragraph we derive some k-independent estimates on and that are needed several times in the sequel. The constants , we introduce hereafter, depend on the solution (as the constants of Section 3), but they do not depend on k. First of all, from (3.17) and (3.18), it follows that
and, in particular, we find
From this estimate and (3.16), it follows that
This implies, in particular, that
Moreover, from (7.3), it follows that
Let us consider now the series
where is a fixed exponent (in the sequel we need only the cases and ). From the previous estimates, we have
where of course the constant depends also on m. Moreover, from (7.5) and the square-integrability of the sequence , it follows that both series are normally convergent on compact subsets of .
We stress that we can not hope that these series are normally convergent in , even when . Indeed, normal convergence would imply uniform convergence, and hence the possibility to exchange the series and the limit as , while the conclusion of Theorem 2.1 says that this is not the case, at least when J is an infinite set.
for a suitable function of class satisfying
Estimates on trigonometric coefficients. For every , we set
and, for every ,
These functions represent the corrections we have to take into account when we approximate the trigonometric functions with their time-average, as we did at the beginning of Section 4.
It is easy to see that
where the supremum is taken over all admissible indices or pairs of indices. Now we claim that
In order to prove (7.10), we just observe that
and hence, by (7.7),
where in the last inequality, we exploited that all eigenvalues are larger than a fixed positive constant.
The proof of (7.11) is analogous, just starting from the trigonometric identity
Also the proof of (7.12) is analogous, but in this case the trigonometric identity is
All the four terms can be treated through Lemma 6.1, but now in the last term the differences between eigenvalues are involved. As a consequence, for the last term, we obtain an estimate of the form
If we want this estimate to be uniform for , we have to assume that the differences between eigenvalues are bounded away from 0, and this is exactly the point where assumption (2.7) comes into play in the proof of Theorem 2.5. Equation for the energy. Let be the total energy as defined in (3.17). We claim that solves a differential equation of the form
where (for the sake of shortness, we do not write the explicit dependence on t in the right-hand sides)
We also claim that satisfies
The verification of (7.13) is a lengthy but elementary calculation, which starts by writing
and by replacing with the right-hand side of (3.14). The crucial point is that when computing the product
one has to isolate the term of the series with . In this way, the product becomes
and now one can express in terms of , and in terms of . The rest is straightforward algebra.
For the term , we choose and . Indeed, the assumptions on follow from (7.6) with and from the normal convergence of the same series on compact subsets of , while the assumptions on follow from (3.16).
Equation for quotients. For every pair of indices h and k in J, we consider the ratio introduced in (4.8). We remind that components with indices in J never vanish, and therefore the quotient is well defined and positive for every . After some lengthy calculations, we obtain
We claim that
where the supremum is taken over all admissible indices or pairs of indices, and that
for every pair of admissible indices and every . We point out that the supremum in (7.19) is finite because the sequence of eigenvalues is increasing.
Finally, in order to verify (7.19), we consider the expression for , and we apply
7.2 Proof of Theorem 2.1
Key estimate for quotients.
We prove that if there exists such that
Indeed, assumption (5.14) is exactly (7.22), assumptions (5.15) follows from (7.3), and the boundedness and semi-integrability of and follow from (7.17)–(7.19). Thus, from Proposition 5.4, we obtain (7.21). The case where J is infinite. In this case, we show that all components tend to 0, which establishes statement (3).
Let us assume that this is not the case. Then there exists for which (7.20) holds true, and hence also (7.21) holds true. At this point, arguing exactly as in the corresponding point in the proof of Theorem 4.1, from (7.20) and (7.21), we deduce that the total energy is unbounded, thus contradicting the estimate from above in (3.18). The case where J is finite. In this case, we prove that (3.19) is true. To begin with, we observe that there exists for which (7.20) holds true, because otherwise the total energy would tend to 0, thus contradicting the estimate from below in (3.18). As a consequence, also (7.21) holds true and, in particular, the limit of is the same for every , provided that this limit exists. At this point, (3.19) is equivalent to showing that
where j denotes the number of elements of J.
To this end, we consider the equalities
From these, we deduce that
hence, by (7.21),
Going back to (7.13), we find that solves a differential equation of the form
Indeed, assumption (5.3) follows from (7.24), while assumption (5.4) follows from the estimate from below in (3.18). It remains to prove that is semi-integrable in . The semi-integrability of is a consequence of (7.15), and the semi-integrability of follows from a finite number of applications of Lemma 6.2 with and (here it is essential that the set J is finite). The required assumptions of and follow from (7.1), (7.4) and (7.12).
Asymptotic behavior of the phase.
It remains to prove (3.20). Actually we need this fact just in the case where J is finite, but the statement is true and the proof is the same even in the general case.
is semi-integrable in for every . First of all, we write the function as
All these oscillating functions can be treated as we did many times before, starting from the trigonometric identities
7.3 Proof of Theorem 2.5
Let us consider again the differential equation (7.13) solved by . We prove that the uniform gap assumption (2.7) implies the semi-integrability of and a uniform bound on the quotients that allows to show that the series of fourth powers is negligible in the limit. At this point, we can conclude by applying Proposition 5.3. Estimate on . We show that
Now we set
Let us check the assumptions. Estimate (5.23) follows from (7.17). Estimates (5.24) follow from (7.18) and (7.19), and the constant is independent of h and k due to the uniform gap assumption (2.7). As a consequence, any satisfying (5.25) is independent of h and k, and ensures that the following implication holds true for every h and k in J:
At this point, we choose any such , and we fix the index (or one of the indices) such that
Such an index exists, even when J is infinite, because for every it turns out that as , due to the square-integrability of the sequence . This choice of implies that for every , and therefore, at this point, (7.29) follows from (7.30) with . Conclusion. To complete the proof, we now observe that
for every . Plugging this estimate into (7.13), we deduce that
We are now (up to a time-translation by ) in the framework of Proposition 5.3 with
Indeed, the semi-integrability of follows from (7.15) and (7.25), assumption (5.3) follows from the boundedness of and the fact that as , and assumption (5.4) follows from the estimate from below in (3.18).
The first two authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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About the article
Published Online: 2017-12-13
This project was partially supported by the PRA “Problemi di evoluzione: studio qualitativo e comportamento asintotico” of the University of Pisa.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 902–927, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0181.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0