In this paper we study the asymptotic profile (as ) of the solution to the Cauchy problem for the linear plate equation
when is a decreasing function, assuming initial data in the energy space and verifying a moment condition. For sufficiently small data, we find the critical exponent for global solutions to the corresponding problem with power nonlinearity
In order to do that, we assume small data in the energy space and, possibly, in . In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities.
In this paper, we study the asymptotic profile of the solutions to the Cauchy problem for the damped plate equation
where is a decreasing function.
Fourth-order evolution partial differential equations as in (1) arise in problems of solid mechanics as, for example, in the theory of thin plates and beams. Also, in particular formulations of problems related with the Navier–Stokes equations (see [35]) appear elliptic equations of fourth-order. Models to study the vibrations of thin plates () given by the full von Kármán system have been studied by several authors, in particular, see [6,24,33].
The equation in (1) represents a plate equation with a time-dependent decreasing coefficient, under the action of a damping term. In space dimension , it may be considered as a linearized model of the Woinovsky-Krieger equation with damping
which represents the vibration of an extensible beam proposed in [39]. The nonlinear term represents the change in the tension of the beam due to its extensibility [1].
The energy for problem (1) is given by
and it dissipates as , as a consequence of
We discuss the possible asymptotic profiles for the solution to (1) as , under the moment condition
We define the diffusive-type kernels
where
The definition of diffusive-type kernels is motivated by the fact that and are the fundamental solutions to two diffusive problems: the kernel solves the fourth order diffusive equation
and the kernel solves the heat equation with time-dependent coefficient :
Here is the classical Gauss-Weierstrass kernel. We also define
Here denotes the Fourier transform of a function φ in , denote the inverse of the Fourier transformation, and we write . If , we always intend .
The asymptotic profile of the solution to (1) as may be described by one of the three kernels in (4), (5) or (6), according to the behavior of for sufficiently large t. In this paper, we assume that has a polynomial speed
but this assumption may be easily relaxed to consider more general functions with a prescribed behavior for sufficiently large t. As a consequence,
whereas if . We remark that if .
We will distinguish the asymptotic profile of the solution to (1) according to three cases: , , and the limit case .
Let. Assume, together with (
3
), and letbe as in (
7
). Letbe the solution to (
1
). Then we may distinguish three cases.
Ifthen the asymptotic profile of u is described by (
4
) in, namely,
Ifthen the asymptotic profile of u is described by (
5
) in, namely,
In the limit casethe asymptotic profile of u is described by (
6
) in, namely,
By the scaling properties of the Fourier transform, one gets
for any . The right-hand sides of the two equalities are finite. In view of this information, thanks to the reverse triangle inequality, estimates (9) and (11) in Theorem 1 describe the asymptotic profile of the solution to (1) when . The case is more complicated since the asymptotic profile of the solution to (1) is not scale-invariant so we do not have a norm equality, but still
In view of this latter, estimate (13) in Theorem 1 describes the asymptotic profile of the solution to (1) when .
If the moment condition in (3) is dropped, that is,
then estimates (9), (11) and (13) are still valid, in the sense that
but they no longer determinate the asymptotic profile of u. Assuming additional regularity on , (for instance, , for a sufficiently large ), it is possible to describe the “secondary” asymptotic profiles of u, when . By “secondary asymptotic profile”, we mean that
if and (8) holds, where the components of the first-order moment are given by (see, for instance, [9])
Similarly if (10) or (12) holds. On the other hand, the assumption has a different influence on the semilinear problem (21). Indeed, the secondary asymptotic profiles, which are relevant for the linear problem, do not come into play to describe the asymptotic profile of the solution to the semilinear problem (21), as we show in Section 7.
The asymptotic profile for the damped evolution equation with constant coefficients
has been studied for a general by G. Karch [23] (see also [26,29,40]), who proved that it is given by , with . The fact that the asymptotic profile for damped evolution equations as in (14) is given by diffusive-type kernels, under a moment condition on the initial data, like (3), is a consequence of the diffusion phenomenon. The diffusion phenomenon means that the solution to (14) may be approximated by the solution to the corresponding heat-type equation with initial data . Namely, by . This latter phenomenon holds, more in general, for evolution equations with effective damping (according to the classification introduced by the authors in [12]) and possibly with time-dependent coefficients:
In particular, in the case of constant coefficients
the damping is effective, in the sense that the diffusion phenomenon holds for (16), if . In [10] the authors showed that for a double diffusion phenomenon holds, when the norm of the solution to (16) is considered, with , inspired by the estimates obtained in [28]. That is, one has to consider two different heat-type equations to approximate the solution to (16), according to the space dimension and to the space considered. On the other hand, when , the asymptotic profile to (16) changes completely, since the fundamental solution may be approximated by considering the convolution between a diffusive-type kernel and the fundamental solution of an evolution operator [22]. The presence of the oscillations has a strong influence on estimates, in particular on estimates (see [14,34]). In the case of the wave equation with viscoelastic or strong damping (), the profile has been investigated in [20].
The diffusion phenomenon for (1) has been investigated in [8], in the case of constant coefficient , and in [41], in space dimension , with . Further details on the diffusion phenomenon for (1) are provided by Lemma 3 in Section 4.
As a consequence of the diffusion phenomenon, we may derive optimal long time decay estimates for the solution to (1) and its energy, under different assumptions for the initial data.
Letandbe as in (
7
). Assume, for some. Letbe the solution to (
1
). Then we have the following decay estimates
The optimality of decay estimates (17) and (19) in Theorem 2 is guaranteed by Lemma 3 in Section 4. For the sake of brevity, we will not discuss the optimality of estimate (18), but we expect this estimate to be optimal, as well.
Estimates (17)–(18)–(19) may be written in a compact form as
As a consequence of (17) and (19), we get
By using Theorem 2, one may estimate the energy for the solution to (1) given in (2), according to the different scenarios. Due to
we obtain
Thanks to the estimates obtained for the solution to the linear Cauchy problem (1), we may investigate the critical exponent of small data global solutions for the following problem with power nonlinearity:
Indeed, the diffusion phenomenon for evolution equations with effective damping as in (16) makes the study of global small data solutions for the associated problem with power nonlinearity (or even , when ) analogous to the study for the heat-type equation [11,13,17,32]. The critical exponent for the classical damped wave ( in (14)) with power nonlinearity was first determined by A. Matsumura [27] in space dimension , and by G. Todorova and B. Yordanov [36] in space dimension . The critical exponent for global small data solutions to semilinear evolution equations with effective time-dependent damping has been widely investigated in these years [3,4,7,16,25,30,31,37].
In the study of global small data solutions for (21) some new interesting effects appear. There is competition between two critical exponents, one related to the fourth-order, constant coefficient term , which becomes dominant when , and the other related to the second-order, time-dependent term , which is the Fujita exponent modified by the vanishing speed of . Moreover, since the equation still has the regularity of the plate, we are able to obtain a global existence result, assuming only small data in the energy space, in any space dimension , for sufficiently slowly decaying speed of (Theorem 5).
Let , with . By the test function method (see [15,18]), we know that global-in-time solutions to (21) cannot exist if
in space dimension , provided that we make a suitable sign assumption on the data (analogous to the moment condition (3)), namely,
We show that small data global solutions exist for , in low space dimension, when . The exponent is optimal, since in the subcritical case and critical case, the nonexistence result holds. We don’t discuss the case of high space dimension, since it requires more sophisticated estimates (for instance, weighted energy estimates, as in [36]) to deal with power nonlinearities . We also don’t discuss the case of , since it requires a different approach to prove Theorem 2.
We mention that the local-in-time existence of the solution to the semilinear problem (21) may be derived by standard arguments in the energy space, by using the energy estimates obtained for the linear problem (see, for instance, [5]). In particular, assuming initial data in , as in Theorem 5, local-in-time energy solutions exist for any if and for any if .
Let,, and. Assume thatif,iforif. Then there existssuch that for any initial datathere is a (unique) global-in-time energy solutionto (
21
). Moreover, it satisfies the following long time decay estimates
Estimates (23) are the same as (20), with . Thanks to Theorem 3, we may also describe the asymptotic profile of the solution to (21), for sufficiently small data in the energy space and in , in space dimension .
Let,, and. Assume thatif,iforif. Assume initial data as in (
22
), withas provided by Theorem
3
.
Then the (unique) global-in-time energy solutionto (
21
) has its asymptotic profile described by (
4
) or, respectively, (
5
), or (
6
), in, if (
8
) or, respectively, (
10
), or (
12
) holds, provided that, where now M is defined byMore precisely, (
9
) or, respectively, (
11
), or (
13
) holds.
If we remove the assumption that the initial data are in and we only assume that they are in the energy space, then the critical exponent is modified into
For the classical damped wave equation, this phenomenon has been investigated in [21].
In particular, global solutions cannot exist, in any space dimension, if , provided that we assume
for some , and for sufficiently large (see [13]). We may prove an existence result of global solution for small data in the energy space, in space dimension if , and in space dimension if . In the limit case , the result holds in any space dimension .
Let, and. Assume thatif, orif. Moreover, assume thatif. Then there existssuch that for any initial datathere is a (unique) global-in-time energy solutionto (
21
). Moreover, it satisfies the following long time decay estimates
We mention that in Theorem 5 a peculiar property of the solution to the linear problem (1) is heavily used. Even if the asymptotic profile of its solution is described by , for , the regularity of the solution is still determined by the higher order terms of the equation, i.e., by . This dichotomy, which is typical for the plate equation in which two different powers of the Laplace operator appears ( and ), remains valid also in the constant coefficients case (, i.e. ), and allows us to obtain an existence result valid in any space dimension in Theorem 5, for a sufficiently slowly decaying speed of . We also mention that the critical exponent may be included in the global existence result in Theorem 5, following the ideas in [19].
Discussion of a model with a different power nonlinearity
Our interest for studying the critical exponent for a power nonlinearity is related to the variety of the mathematical properties which appears when the decay rate obtained for the linear problem is employed to find the critical exponent for the nonlinear problem. However, problems with different nonlinearities may have an interest from a physical background. The following problem
is related to Falk’s model for the thermomechanical phase transitions in shape memory alloys, in space dimension , that is,
In this model, shape memory alloys modify their shapes according to the temperature. At low temperature, an alloy behaves like a plastic body, whereas at high temperature the behaviour is pseudoelastic (see [2, chapter 5]).
We have the following result, analogous to Theorems 3 and 4.
Let, for some, and assume thatiforif. Then there existssuch that for any initial datathere is a (unique) global-in-time energy solutionto (
28
). Moreover, it satisfies the following long time decay estimatesand its asymptotic profile described by (
4
) or, respectively, (
5
), or (
6
), in, if (
8
) or, respectively, (
10
), or (
12
) holds, provided that, where now M is defined byMore precisely, (
9
) or, respectively, (
11
), or (
13
) holds.
Representation of the fundamental solution of the problem
We perform the Fourier transform of (1) with respect to the x variable to obtain
Applying the change of variable
problem (32) reads as
where
In the following, we will separately study the solution at low and high frequencies, since the asymptotic behaviour of the solution is determined at low frequencies, while its regularity is determined at high frequencies. To divide the two zones, we fix a constant K such that
and we denote
For any , we denote
and we separately estimate the fundamental solution to the problem for our micro-energy , in and in . Namely,
The choice of the bound is motivated by the following property.
For any , it holds
In particular,
as a consequence of as , and due to .
To simplify our analysis, we define the “equivalent” micro-energies in and in :
and, consequently, the fundamental solutions and , such that
We first derive an estimate and a representation formula for .
The fundamental solution verifies the estimatewhereFor, we defineand we setandThen, for all, for any,can be represented in the formwhereis uniformly bounded. Moreover, for any, there existsand it verifies
We remark that
uniformly in , as a consequence of Remark 2. By using the trivial property
we may estimate
so that we obtain , with d as in (36).
To prove our statement, we may assume that , where is as in (35), since the set is compact. Indeed, for any , the function behaves as the function , due to the fact that the difference is uniformly bounded with respect to .
For any and , the fundamental solution solves the system
where
We define
where and are as in (37) and (38). We remark that , are uniformly bounded in , due to , see (35).
Then solves the following problem:
Now we diagonalize the principal part of the system above by defining , where
are constant, unitary matrices, so that solves the problem
Let us write in the form
where e is as in (39). Now solves the problem
where
Here and in the following by we denote the maximum norm of the entries of a matrix. Then, we may estimate
Therefore, recalling that h is nonnegative, is uniformly bounded. Following as in [12], one may verify that
and that
Indeed, setting
we find that
Formally,
Due the fact that is not bounded, we do not have the boundedness of itself, so we cannot use this representation. However,
where we used , , and so on. This leads to obtain
and (46).
We now have that
is uniformly bounded in and
This concludes the proof. □
Now we consider the fundamental solution at high frequencies.
There exists a positive constant, depending on K, such that the fundamental solutionverifies the estimate
For any and , the fundamental solution solves the system
where
We diagonalize the principal part of the system (47). We set , with P given by (44), so that
and
We may write , where
and is the solution to:
where , since we used the property . It is clear that is unitary, so it remains to prove that
for some , depending on K. However,
so the proof follows by recalling that , and noticing that , for sufficiently large s, due to
for sufficiently large s. □
By using Lemmas 1 and 2, one may easily prove Theorem 2.
Estimates (17), (18) and (19) trivially hold at short time, since problem (1) is well-posed in , namely,
Therefore, we shall only prove the decay estimates for . Let u be the solution to (1). By Plancherel’s theorem,
and we may now use Lemma 1 to estimate the first term, and Lemma 2 to estimate the second one, recalling (see (33)) that
For any , we define , its Hölder conjugate, and by
Using Lemma 1 with , thanks to
we obtain
for . To obtain a sharp estimate for , we shall refine the estimate in , following the ideas in [38]. Setting , this latter solves the first-order equation with inhomogeneous term
that is,
Using the estimate for in and integrating by parts, we derive
where
For a sufficiently large t and for , we may estimate , so that
The presence of the additional term leads to the additional decay rate for with respect to the decay rate for :
for . Indeed, if , then
whereas, if , then
The proof of Theorem 2 follows thanks to the exponential decay
where is as in Lemma 2. □
The asymptotic profiles given in (4) and (5) are, respectively, the fundamental solutions to the Cauchy problems for diffusive equations given by
and
Also, (6) is the fundamental solution to the following Cauchy problem for a diffusive equation:
Since we are interested in the fundamental solution of the Cauchy problems (49), (50) and (51), we will prove Theorem 1 as a corollary of the following results.
Letbe the solution to (
1
). Then we have the following estimatewhere v is the solution to (
51
) with initial data.
We postpone the proof of Lemma 3 to the end of the section.
The proof of Theorem 1 is based on Lemma 3 with , and on the following result.
Assume that, and letThen, for everyand, it holdswhere the action ofmay be defined by.
The statement of Lemma 4 can be found in [23, Lemma 3.2], but, for the ease of reading, we provide an alternative, more direct proof.
For the sake of brevity, we fix , so that we shall prove the following estimate:
By interpolation, it is sufficient to prove the statement for and for . We first consider and we define
Then g is continuous and bounded, since . Due to the continuity of g,
since . Then, we may estimate
It is clear that
On the other hand, the integral may be estimated by
For a sufficiently large t, it holds . Therefore, for a fixed h and for any , for sufficiently large t, we proved that
and this concludes the proof of the statement for .
Now let . We may write
Then
Let us fix . Due to the fact that , there exists such that
On the other hand, since is self-similar, we have
for any fixed . By Lebesgue’s dominated convergence theorem, we derive that
for a sufficiently large t. As a consequence, we derive
for sufficiently large t. This concludes the proof. □
we use Plancherel and Riemann–Lebesgue theorem to show that the solution v to (51) verifies the estimate
where the last estimate is a consequence of assumption (8);
Then we conclude the proof of (9) by triangle inequality.
On the other hand, if (10) holds, we may proceed as before and show that the solution v to (51) verifies the estimate
By the change of variables , applying Lemma 4, with , and , we derive
and this concludes the proof of (11).
If we assume (12), then we first use the estimate
then we apply again Lemma 4, with and , to estimate
This concludes the proof of (13). □
At high frequencies, we have an exponential decay (in time) for the diffusive-type kernels and for the solution to (1). Hence we may localize our analysis to low frequencies.
The solution for the Cauchy problem
for is given by
On the other hand, for a fixed , we may write
for any . Hence
where . Thanks to and we conclude
Let v be the solution to (51) with initial data . For any , with sufficiently large , we may write
where
The solution for the system (45) at is given by
and
Therefore
with
We may now estimate
and
Moreover,
and, using (40), we get
The last inequality is a consequence of
Therefore,
and we conclude that
This concludes the proof of Lemma 3. □
To prove Theorems 3 and 5, we rely on Duhamel’s principle. However, since the equation in (1) is not invariant by time-translations, we shall extend the estimates in Theorem 2 to the following family of parameter-dependent Cauchy problems (see [16]):
Let,, andbe as in (
7
). Assume, for some. Letbe the solution to (
53
). Then we have the following decay estimatesif, andif.
We remark that
It is sufficient to follow the proof of Theorem 2. We may assume and and use Lemma 1, recalling that
but we now distinguish two cases. If , then we obtain
for and r given by (48). On the other hand, if , we estimate
To obtain a sharp estimate for , we shall refine the estimate in . Setting , we now have
Using the estimate for in and integrating by parts, we derive
where
Proceeding as in the proof of Theorem 2, for a sufficiently large and for , we get
The presence of the additional term leads to the additional decay rate for with respect to the decay rate for . Indeed:
where we used (56) if .
The proof follows once again thanks to the exponential decay at high frequencies (Lemma 2). □
To derive the existence and uniqueness of the global solution to (21), for any , we employ a classical contraction argument in the space
equipped with the norm induced by the estimates for the linear problem (1) which we obtained in Theorem 2. That is,
if we prove Theorem 3, and
if we prove Theorem 5.
In particular, thanks to Theorem 2, the solution to (1), with initial data as in (22) or, respectively, as in (25), is in and
uniformly with respect to T.
We define the operator F such that, for any ,
where is the fundamental solution to (53). Then we prove the estimates
where does not depend on T. By standard arguments (Duhamel’s principle, contraction argument), estimates (60)–(61) lead to the existence of a unique global-in-time solution to (21), for sufficiently small .
Let . Taking into account of the norm in (58), we may estimate
for any . Thanks to the Gagliardo–Nirenberg inequality, we obtain
for any if , and for any if .
We first assume and we prove (60). For , thanks to Lemma 5 with and , we may estimate
thanks to (62) (we used the assumption that if ). Due to the fact that
for , thanks to the assumption of , the latter integral may be estimated by
Here and in the following, we use the well-known estimate
This concludes the proof of (60). In order to prove (61), we use the inequality
and the Hölder inequality
to derive
Now let . We only prove (60), being the proof of (61) analogous. Thanks to (56), we may now estimate
where we used (62). Due to the fact that
for , thanks to the assumption of , recalling (63), the latter integral may be estimated by
This concludes the proof. □
We only prove (60), being the proof of (61) analogous.
Let and . Taking into account of the norm in (57), we may estimate
for any . Thanks to the Gagliardo–Nirenberg inequality, we obtain
for any .
We first assume . We employ Lemma 5 with for and Lemma 5 with for , so that, for , we may estimate
thanks to (64), and to the property for and for . Due to the fact that
for , thanks to the assumption of , the two integrals may be estimated by
This concludes the proof for . Now let . We employ again Lemma 5 with for and Lemma 5 with for . Moreover:
Therefore, we get
where we used (64). Due to the fact that
for , thanks to the assumption of , the both two integrals may be estimated by
This concludes the proof. □
Letbe the solution to (
53
). Then we have the following estimatesfor, andfor some sufficiently small, if.
We wrote (67) including an explicit polynomial decay gain , since later we plan to use property (56) to prove our results.
Applying the change of variable , as in (33), problem (53) reads as
where
For a fixed , we may write
for any . Hence
Thanks to and we conclude
Hence we may write
with
We may now estimate
and
Therefore,
and we conclude that
if , and
for a sufficiently small , if . □
The proof of this theorem is very similar to the arguments employed by G. Karch in [23]. Taking into account of Theorem 1, it remains to estimate only the nonlinear contribution to the solution for (21). Let . We first estimate:
where we denote by the solution operator to (53), that is,
and we define
Let us assume . Thanks to Lemma 6 with , using (64) and that , we may estimate
We proceed similarly for , but now, for a sufficiently small δ, we obtain:
where we assumed that δ was sufficiently small to verify
Now it remains to estimate
For we do not use cancelations. First, let . Then
We proceed similarly for . In this case, we have
Now we have to estimate
First, let . For any small , let us define
Now we may estimate
and, for any fixed , the latter integral tends to 0, as . Indeed , since
Therefore, since monotonously tends to as , we get that
Now, we cannot use the self-similar profile, as in [23], to deal with , since our is the convolution of two different diffusive-type kernels. However, both and are self-similar, that is
for any . Therefore, setting
we obtain
Now, as a consequence of and in , we have that
where in the last inequality we used the assumption . As a consequence of the continuity, for any , there exists such that
so that
Taking the limit as , we conclude the proof. We proceed similarly for , but now using
and we conclude the proof. □
The case of a null moment condition
The result in Theorem 4 may appear confusing, since we didn’t exclude the chance that the moment of the initial data is zero, that is,
The above condition implies that vanishes at . As a consequence, the solution v to problems (49)–(50)–(51) has the property
for any . As a matter of fact, we possibly have that the solution v to problems (49)–(50)–(51) is identically zero, in the special case , for any . This suggests that the properties for the solution u to problems (1) and (21) may be proved independently of the diffusion phenomenon, when the null moment condition holds.
Indeed, if and , then we may replace the statement of Lemma 3 (with and ) by the following.
Letandbe as in (
7
). Assume, be such thatLetbe the solution to (
1
). Then we have the following estimate
In view of Lemma 3, it is sufficient to prove that
where v is the solution to (51). However, due to
where is as in (6), we may apply Lemma 4 with
as we did in the proof of Theorem 1, and obtain (69). □
By the same reasoning, estimates (9) or, respectively, (11), or (13), remain valid also for the solution to the nonlinear problem (21), when , that is,
in (24). Moreover, for nontrivial data, the above quantity is nonzero. That is, we may determinate the asymptotic profile of the solution to (21) using in (6), even when the moment condition for the initial data in (3) is not verified.
To prove Theorem 6, we shall now deal with a nonlinearity , instead of .
We follow the proof of Theorem 3, but now the Duhamel’s operator is
where is the fundamental solution to (53). Again we shall prove (60) and (61), where
equipped with the norm in (58), that is,
For any , we may estimate
for any . By Gagliardo–Nirenberg inequality, we also obtain
As a consequence,
We first assume . We employ Lemma 5 with for and Lemma 5 with for , so that, for , we may estimate
thanks to the property for and for . Due to the fact that
for , thanks to the assumption of , the two integrals may be estimated by
This concludes the proof for . Now let . We employ again Lemma 5 with for and Lemma 5 with for . Moreover:
Therefore, we get
Due to the fact that
for , thanks to the assumption of , the both two integrals may be estimated by
In order to prove (61), we need more computation than in Theorem 3. We first write
then we use the inequality
valid for , and the Hölder inequality to get
and similarly for the norm. We then prove (61), as we did in the proof of Theorem 5.
To prove the asymptotic profile, we shall estimate the nonlinear contribution to the solution for (28). We follow the proof of Theorem 4, but now . Let . Using , we easily obtain
On the other hand,
We notice that , due to
and this implies that the integral in the right-hand side in
tends to 0, as . The rest of the proof is as in the proof of Theorem 4. □
Footnotes
Acknowledgements
The discussions on this paper began during a two weeks research stay of the first author at the Department of Mathematics and Computer Science of University of São Paulo, FFCLRP, in August 2018. The stay of the first author was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), grant 2017/19497-3. The first author has been supported by the “National Group for Mathematical Analysis, Probability and their Applications” (GNAMPA – INdAM). The second author has been partially supported by FAPESP grant 2017/19497-3.
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