We characterize the global and nonglobal solutions of the Timoshenko equation in a bounded
domain. We consider nonlinear dissipation and a nonlinear source term. We prove blowup of
solutions as well as convergence to the zero and nonzero equilibria, and we give rates of decay to the
zero equilibrium. In particular, we prove instability of the ground state. We show existence of global
solutions without a uniform bound in time for the equation with nonlinear damping. We define and
use a potential well and positive invariant sets.
with initial conditions
and with one set of the following boundary conditions:
where is a bounded domain with sufficiently smooth boundary, is the norm in ,
When the source term , there is a considerable set of works studying several properties of equation (1.1), see for instance, the early papers by Ball [1, 2], Haraux and Zuazua , and the books by Hale , Haraux , and references therein. For a destabilizing source term, , in the works of Payne and Sattinger , Georgiev and Todorova , and Ikehata , qualitative properties of (1.1) are studied, when . To understand the dynamics of second-order equations in time, similar to (1.1), active research is reported in Alves and Cavalcanti , Barbu et al. , Cavalcanti et al. [11–16], Rammaha  Rammaha and Sakuntasathien , and Todorova and Vitillaro , Vitillaro . For the Timoshenko equation, with , Bainov and Minchev  gave sufficient conditions for the nonexistence of smooth solutions of (1.1), with negative initial energy, and gave an upper bound of the maximal time of existence. For positive and sufficiently small initial energy, blowup and globality properties are characterized in Esquivel-Avila . For the Kirchhoff equation, that is, (1.1) with , the nonexistence of global solutions is studied in . In [24, 25], we characterized properties such as blowup and asymptotic behavior of solutions, for (1.1) with and . To the knowledge of the author, such problems are still open for the Timoshenko equation (1.1). Here, we want to give some results about the dynamics of problem (1.1). To do that we will generalize the concept of the depth of the potential well in such manner that our results of the dynamics be as sharp as the ones in [24, 25]. Furthermore, for particular cases, our definition of depth of the potential well will coincide with the one introduced in .
2. Preliminaries and Framework
We begin this section with an existence, uniqueness, and continuation theorem for (1.1). The proof is similar to the ones in [7, 8], where semilinear wave equations are studied.
Theorem 2.1. Assume that and if . For every initial data , where is defined either by , or , there exists a unique (local) weak solution of problem (1.1), that is,
a.e. in and for every , such that
Here, denotes the corresponding semigroup on , generated by problem (1.1), and is the inner product in . The following energy equation holds:
Here, is the initial energy, and denotes the norm in the space. If the maximal time of existence , then as , in the norm of :
In that case, from (2.3)–(2.6), as .
Now, we define, respectively, the stable (potential well) and unstable sets:
Here, denotes the set of with that property, and the depth of the potential well is defined as follows:
We assume that , and since , then , and . Also note that if , then , , and we have the following characterization of the depth of the potential well (2.10)-(2.11):
which is the definition given in , where a nondissipative nonlinear wave equation is studied.
Consider any , , and if , then
where , is any constant in the Sobolev-Poincaré’s inequality
If denotes any nonzero equilibria of equation (1.1),
then, by (2.1) in Theorem 2.1 with , we get that belongs to the Nehari manifold, , that is,
Consequently, . Furthermore, from (2.17) which is an equality when , we conclude that the Nehari manifold can be represented by the line: , in the plane with axes and , beginning at the point: . We also note that
From these facts it follows that the depth of the potential well (2.10) is characterized by
Hence, any equilibrium is such that . Moreover, like in , the set of extremals of (2.21) is characterized by set of equilibria with least energy, that is the ground state
Observe that is a tangent line to the curve defined by the equality in (2.17) with , at the point , which holds if . On the other hand, we notice that
and is equal to zero if and only if . Hence, if , then
Therefore, next results about the stable and unstable sets follow.
Lemma 2.2. The following properties of and hold: (i) is a neighborhood of .(ii) (closure in ), in particular .(iii), where
(iv). (v). (vi), .
Lemma 2.3. One has that
for any such that , in particular if , and
for any , such that , in particular if .
A set is positive invariant, with respect to problem (1.1), if the corresponding generated semigroup on is such that
Lemma 2.4. Let denote any solution of (1.1), given by Theorem 2.1. Then, the sets
are positive invariant.
Proof. First, we show that is positive invariant. In order to do that, we take . Then, by (2.4), , for any . Now, if is not positive invariant, there exists some , such that , with . Then, by (2.21), . But this is impossible because . The proof of the positive invariance of is quite similar. Indeed, if this is not true there exists some , such that . From (ii) of Lemma 2.2 , and this implies the same contradiction as before.
Next result gives an interpretation of sets and and follows from Lemma 2.2.
Lemma 2.5. The sets and have the properties
The following result is a direct consequence of (vi) in Lemma 2.2 and Lemma 2.4.
Lemma 2.6. For every solution of (1.1), only one of the following holds: (i)there exists some such that , and remains there for every ,(ii)there exists some such that , and remains there for every ,(iii) for every .
Hence, we notice that the sets and play an important role in the dynamics of (1.1). Moreover, we will prove that any solution eventually contained in converges to the zero equilibrium. If enters in , either blowups in a finite time or it is global but without a uniform bound in for every , in the case that , in (1.6). Also, we will prove that any solution with , for every , is bounded and converges to the set of nonzero equilibria .
We will need the following inequalities to show blowup and convergence to the zero equilibrium, respectively, in the dissipative case.
Lemma 2.7. Let be a nonnegative function such that
with and . Then, there exists some such that .
Proof. Define , then
Hence, , which is only possible if .
Lemma 2.8. Let be a nonnegative function such that
with and . Then, for , if
Proof. Consider , and notice that . Then, we integrate and obtain the first inequality. Now, let , and the second one follows.
3. Timoshenko Equation
Due to our assumptions on and , we restrict our analysis to dimensions . Indeed, since , and , if , then our analysis considers, whenever . We also notice that in any case we do not consider the interval . Moreover, whenever . We begin with a characterization of blowup when and .
Theorem 3.1. Let be a solution of problem (1.1), and suppose that . A necessary and sufficient condition for nonglobality, blowup by Theorem 2.1, is that and there exists such that .
Proof. Sufficiency By Lemma 2.4, for all . Now, we consider the function defined, along the solution, by
and notice that because of energy equation (2.3),
where, now . Notice that from (2.29) in Lemma 2.3,
We will need some estimates. First, we notice that from energy equation in terms of and (3.3),
where , , is the constant in the continuous embedding , , and will be chosen later. Consider a positive number to be chosen later, from (3.2)-(3.3), we obtain
If , we choose , and from (3.5) we get
If , then we notice that from (3.2)-(3.3),
Hence and from (3.5), we have the estimate
In this case, we choose the number so that the coefficient of in (3.8) be equal to zero, then
We note that , and we get
Therefore, from (3.6) and (3.10),
Now, we define the function, along the solution, by
where and will be choosen later. We intend to apply Lemma 2.7 to functional (3.13). First, we calculate the derivative, along solutions, with respect to . Let us start with the second term of (3.13). From (3.2)–(3.4) and (3.11), one has
where , and is sufficiently small. Consequently, if is sufficiently small,
where . From (3.15) and choosing small enough, we get
Utilizing two times (3.3), we get
where is the imbedding constant of , , and . Hence and from (3.15), we obtain the inequality in order to apply Lemma 2.7. Therefore, the maximal time of existence is finite: . Necessity Suppose that . Define the function, along the solution, by
where , , and is the imbedding constant of . Hence, by Gronwall inequality, it follows that is bounded in for any finite time. A contradiction. Proceeding again by contradiction suppose that, for all , . Then, by Lemma 2.6, we have either for all , or for all . In the first case, from (2.28) in Lemma 2.3
that is, is bounded in . This is not possible. In the second case,
where . Hence, by the Hölder inequality,
for , where . From Theorem 2.1,
hence, by Sobolev-Poincaré's inequality (2.16), for every , there exists some , such that
for every . This implies the first inequality of (2.29) in Lemma 2.3, replacing by . Now, we consider the function (3.13)
defined for , where , is sufficiently small, and here
and repeat the sufficiency part of the proof. Then, by Lemma 2.7, blowups as , . Moreover, for ,
hence and from (3.26), (3.27), and since ,
But this contradicts (3.23), since . The proof is complete.
Remark 3.2. From the last result, if and , any solution of problem (1.1), , is global if and only if either (i) there exists such that or (ii) , for every . On the other hand, if and , then any solution is global if and only if one of the following holds: (i), (ii), or (iii) and there exists such that .
We next prove a characterization of convergence to the zero equilibrium, and we give rates of decay.
Theorem 3.3. Let be a solution of problem (1.1) with . Suppose that and that , if . A necessary and sufficient condition for , strongly in as , is that there exists such that . In this case, if denotes either the energy
or the norm of the solution in
One has the rates of decay, for ,
and, for linear dissipation, ,
where is sufficiently large, and , are constants depending only on initial conditions.
Proof. Necessity By (ii) in Lemma 2.2, , and since the equilibrium , strong closures in , then, by Lemma 2.6, the solution must eventually enter in .Sufficiency By energy equation and (2.27) in Lemma 2.3, the solution must be global and uniformly bounded in the norm of , that is , for any . Hence, there exists a sequence of times, , such that if then , weakly in and, since the embedding is compact, . Also, notice that the energy is such that
Consequently, from the energy equation and the continuous embedding ,
in particular, for any sequence of times such that as ,
where , for . By Fatou Lemma,
for a.e. , and by the weak convergence to ,
where we choose such that , for some . It can be shown that the semigroup generated by problem (1.1) is continuous in with the weak topology, and then that the weak limit set is positive invariant, see Ball . Consequently must be an equilibrium of (1.1). Furthermore, by the lower-semicontinuity of the norm in , one has
Then, by (2.19) and (2.21), , and
Strong convergence follows if we get the rates of decay in our statement. Here, we will adapt the technique used in Haraux and Zuazua , to (1.1). That technique is based on the construction of suitable Liapunov functions defined along solutions and the application of Lemma 2.8. One of them is the energy, and we will need one more, defined by
where is a constant to be chosen later. We next prove that is equivalent to both, the energy and the norm of the solution, in the sense of (2.30) and (2.28) below. First we note that from (2.27) in Lemma 2.3,
where . Also, notice that from (3.43),
where is a constant that depends on the continuous embedding . Hence and from (3.44), if is sufficiently small, then
We will need the following estimate:
where we applied (3.44) in the third step and Young inequality in last step, and the constants , depend on the continuous embedding , and also depends on . It follows that, by (3.42) and since is compact, for any , there exists some such that for any
where is the corresponding embedding constant and we used (3.44) in the last step. Since we will apply Lemma 2.8, we need to calculate the time derivative of (3.43) and we begin with
which holds for any , and where we used (3.47), (3.48) and definition of . We notice that for any small , and by Young inequality and energy equation
where , depend on the continuous embedding , and depends on . Then, for and sufficiently small, (3.49) and (3.50), imply
for any , where . Consequently, for sufficiently small and any
where and is the constant in (3.45); also we used (3.44), the fact that the energy is decreasing and (3.46). Then, from (3.52) and Lemma 2.8, we obtain the desired rates of decay for . The result now follows by (3.46) and (3.44), and the proof is complete.
Remark 3.4. By (2.35), the ground state is: . Then, in any -neighborhood of that subset of nonzero equilibria, one can choose initial conditions either in or in . Hence, by Theorem 3.1 and (3.3), the ground state is unstable in the sense of Liapunov when the dissipation term is either linear or nonlinear.
Next we will study the behavior of solutions such that for all . First, we prove that those solutions are uniformly bounded in time. To that end we will study the cases: and separately, First, we consider the case .
Theorem 3.5. Let be a solution of problem (1.1). Assume that , and . Also, assume that if . If for all , then the solution is global and uniformly bounded in , for all .
Proof. Suppose that is not global, then by Theorem 2.1 blowups and by Theorem 3.1, for some . Hence, for all . A contradiction. Next, we will prove that is uniformly bounded for all . Let , where is the constant given below. Then, we obtain
where is the imbedding constant of , and . We define , the positive part of . We claim that, along solutions of (1.1), the time derivative satisfies . Indeed, if this is no the case, there exists some such that
By a standard comparison result for ordinary differential equations, (3.53) and (3.54) imply that as . Consequently, for any constant , there exists some , such that for
This is (2.29) in Lemma 2.3, replacing by . If we now define, for , the function
we can repeat the sufficiency part of the proof of Theorem 3.1 and show that the solution blowups in a finite time, consequently is nonglobal. A contradiction. Then, , for all , and some constant . Next, we will prove that uniform boundedness of implies uniform boundedness of in , for all . To that end, we consider the functions , where now and is defined below. From the second line in (3.53),
where and . Hence, for ,
and consequently, from definition of , for ,
Notice that if , for some , we obtain from (3.59) that . A contradiction. Then, for all ,
Now, we define , and like in (3.57)
where and . Hence,
Hence and from (3.60), is uniformly bounded in time. We integrate the second line of (3.53) in terms of and, by the energy equation, we obtain
Hence and since is uniformly bounded in time,
where is a constant, and
Next, we will show that there exists a constant , such that
for any . To this end, we calculate
If , we integrate and obtain, for , that
By Gronwall inequality and (3.65),
where , depend on the continuous embeddings , and . If , we use Galiardo-Niremberg's inequality,
where , and . Notice that if , and if , because . Then, from
we integrate and apply Gronwall inequality, for ,
where . Notice that because by hypothesis , then we use the Hölder inequality, and from (3.65) we get
Then (3.67) holds for any , under our assumptions on . Consequently, (3.65) and (3.67) imply that
and the proof is complete.
Next, we consider the case . Due to our assumptions on , we restrict our analysis to . Since , our analysis considers, at most, dimensions , whenever .
Theorem 3.6. Let be a solution of problem (1.1). Assume that , and with if . Also assume that for . If