On the Strongly Damped Wave Equations with Critical Nonlinearities

We study the strongly damped wave equations with critical nonlinearities. By choosing suitable state spaces, we prove sectorial property of the operator matrix together with its adjoint operator, investigate the associated interpolation and extrapolation spaces, analysis the criticality of the nonlinearity with critical growth, and study the higher spatial regularity of the Y-regular solution by bootstrapping.

• Negative Laplacian
• Wave Equation
• Strong Damping
• Sectorial Operator
• Fractional Power
• Global Attractor

1. Introduction

This paper deals with a class of wave equations with strong damping

Here is a bounded domain with boundary, and is the coefficient of strong damping. Let, then the negative Laplacian, denoted by A, is a positive definite and self-adjoint operator defined in X with compact inverse. For each, there define and as the fractional power of A and its domain endowed with the graph norm respectively. Evidently, in this setting, , , , and for all, we have.

Introduce the energy space as our work space, and let, , , then Equation (1) turns to be an abstract Cauchy problem

and we can treat it in the framework of semigroup of operators.

Recall that, the operator matrix itself is not closed in Y, and consequently its negative is not a generator of any -semigroups except. But its closure, which is still denoted by, is a sectorial operator whenever, and its negative generates an analytic and exponential decaying semigroup (see [1] - [3] for references).

By using the notation of e-regular solution introduced in [4] [5] together with interpolation and extrapolation spaces, and under the Lipschitz condition,

.

Carvalho-Cholewa in [1] and lately Carvalho-Cholewa-Dlotko in [2] studied the local existence and regularity of the e-regular (or Y-regular in this paper) solution of Equation (1). Under the dissipative condition,

Carvalho-Cholewa in [6] investigate the global existence of e-regular solutions in the subcritical case , together with the existence and regularity of the universal attractors. As for the critical case, there are few references except. According to the general theory of the e-regular solutions, in this case, the related nonlinear map is critical (i.e., and can only take the value), consequently for a e-regular solution arising in the energy space, boundedness of the Y-norm on its maximal existence interval could not guarantee the global existence (see [1] [2] ).

Here we are concerned with the higher regularity and global existence of the Y-regular solution of Equation (1). By introducing a new state space () weak than Y somewhat, we will reveal that, the operator matrix is also sectorial, together with its dual operators. Moreover, all the interpolation and extrapolation spaces () can be expressed by the Cartesian products. And consequently, for and, the corresponding nonlinearity turns to be subcritical. Using these properties, we will prove by bootstrapping that every -regular solution of (2) with the initial value taken in Y is a strong one exactly. Moreover, this solution exists on the whole interval, or its Y-norm blows up in finite time. Results obtained here, which can be viewed as useful supplements to the references listed above, tell us that in a semilinear parabolic equation, substitution of phase spaces may change the criticality of the nonlinear perturbation attached to it. In other words, criticality is not absolute for the parabolic systems in many concrete situations.

2. Main Results and Proofs

Lemma 2.1 Suppose that X and Y are two Banach spaces, A is a sectorial operators defined in X, and B is a linear operator densely defined in Y. Suppose also there is a homeomorphism satisfying, then B is also sectorial together with and (see [7] , §5.2).

Lemma 2.2 The operator matrix is sectorial in the new space, and. Moreover, the domain equipped with the graph norm is equivalent to the product space (cf. [8] ).

For the Hilbert space and the operator introduced above, consider the interpolation-

extrapolation Hilbert scale, where if, if, and is the realization of A in the space. For the real and complex interpolation methods, please refer to

[9] , Ch.1, and for the extrapolation method, see [10] , Ch. V for references. Recall that, for every, is also a sectorial operator in, and is, is for all (cf. [10] , § 5.1.3).

Define the realization of in as follows:

It is easy to check that, for all, in the sense of equivalent norms. Furthermore, we have

Lemma 2.3 is sectorial in the state space with the same spectrum as has.

Proof: This lemma can be easily verified by Lemma 2.1, together with the fact that the following operator

is an isomorphism between and, satisfying. ,

Consider another operator matrix defined below,

Evidently, is closed in the space with domain. And for all and, we have

This tell us that, is contained in, the adjoint operator of. In order to show the equality, it suffices to check that, which is a consequence of the following lemma.

Lemma 2.4 is sectorial in with the spectrum.

Proof of this lemma is much similar to that of Lemma 2.3, and here we omit it.

Denote, which is isomorphic to the product space according to the graph norm.

Now we can give some representations for the interpolation and extrapolation spaces attached to. For each, we have

and

Thus by the dual principle (refer to [10] , Ch. V, thm. 1.5.12), we obtain

Hence, for each, we have that

in the sense of isomorphism.

Let us study the nonlinear operator in the case and in new state spaces.

Theorem 2.5 Take, then under the assumption (4), for each, is bounded and locally Lipschitz. More precisely, verifies

Proof: Firstly using the embedding, we can easily deduce that if, and for all if. Notice that for all. Hence for the number s satisfying, by invoking (4), we find that the Nemytskij operator of f, denoted also by f verifies

This inequality, together with the definition of and (7) leads to the desired inequality (8).

If, then we have the following embedding

And simple calculations show that in case, for all and, inequalities

and

hold simultaneously. Thus for the number r verifying the restriction in (10), the other number satisfies the restriction in (9). Hence by invoking (9), (10) and (4), we obtain

which means that inequality (8) still holds in the case. This complete the proof. ,

Theorem 2.6 Let, then under the assumption (4), for all, the operator satisfies

Similar to Thm. 2.5, core of the proof for this theorem is to check the validity of the following inequality

under condition (4). Here we omit the whole process.

Remark 2.7 In the new state spaces, the nonlinearity turns to be a subcritical map (please compare to [1] [2] ).

Now we can investigate higher regularity and global existence of solutions of the abstract Cauchy problem (2) + (3) for the critical growth exponent in the case. In view of [4] and [1] [2] ,

we know that for the initial point, there exists a unique e-regular (or in other words Y-regular) solution defined on an interval for some, s.t.

for some, and Equtaion (2) is satisfied in the space. If lies in the space, then thanks to (8), there exists another interval, on which there is a unique -regular solution satisfying

for all together with Equation (2) satisfied in the space (see [7] , Ch. 6 or [11] , Ch. 3 for references).

Take, then by the uniqueness and regularity mentioned above, we can easily find that an -regular solution is equal to a Y-regular one on the common existing interval if they have the same initial value.

Denote by and respectively the maximal intervals of existing as a Y-regular solution and as an -regular one with. In the following paragraph, we will prove that. Evidently since. For the inverse inequality, it suffices to show that for arbitrary (cf. [12] ). This can be done by bootstrapping.

Taking any, and using (13) and (6), we obtain

Regard and as the initial time and space respectively, then by invoking the local existence and uniqueness of the -regular solution, we can find a time, such that

Here the time depends on the norm due to the subcriticality of (8). Notice that is uniformly continuous in on any bounded interval thanks to (13) and

(14), therefore it can be extended to the whole interval as an -regular solution. And similar to (14), for any, we have that

The above inclusion is valid for all due to the arbitrariness of. Thus using the procedure performed above, we can deduce that, as an -regular solution,

for all.

Select so that, and repeat the above step k times, we finally obtain

for all, and. Thus, for any, we can conclude that, which leads to the desired conclusion.

Theorem 2.8 Every Y-regular solution of the problem (2) + (3) with is exactly the strong one on its maximal interval of existence. More precisely, verifies all the following properties

・ for all,

・ Equation (2) holds in for all, and

・ either, i.e. blows up in finite time, or, i.e. exists globally.

Proof: Choose so that, then the inclusion (15) and the imbedding jointly produce 1). Moreover, thanks to (11), if we regard () as the initial space, and use the existence and uniqueness of the -regular solution, we can derive 2). Suppose that condition

holds, then as an -regular solution, can be extended onto the whole interval since is subcritical and. Therefore, and exists globally as a Y-regular solution (it is a global strong solution indeed). This results means that (iii) holds. ,

Remark 2.9 From Thm. 2.8(i), one can conclude that the first component function of a Y-regular solution belongs to for all, and satisfies Equation

(1) in the strong sense on its maximal existing interval definitely. In [6] , the authors showed that, is the strong solution under the extra conditions and. And in [2] , the authors proved that is the classical one whenever. In this sense, Thm 2.8 is a useful supplement to the above two results.

Remark 2.10 Under the assumptions (4) and (5), the following estimate is valid for (see [6] [13] ):

where

is the energy functional attached to (2). Thus for every, condition (2.11) holds, and consequently, is globally defined.

3. Further Discussions

By introducing some new state spaces, we investigate the higher regularity and global existence of the weak solution of the wave Equation (1) for the critical growth exponent in the case. Results obtained here show that criticality of the nonlinearity attached to a semilinear parabolic system is not absolutely. It depends on the state spaces selected in many concrete situations. On the other hand, we have to admitted that, methods used here are inadequate for, since criticality of does not change anymore (), regardless of the space we selected. In this case, condition (2.11) does not guarantee the global existence of the Y-regular solution any more. In [14] , the authors proved that, under

hypotheses (4) and (5), every Y-regular solution arising in Y can be extended onto the whole interval as a -regular solution () or a piece-wise e-regular solution in other words (see

[12] for references). More precisely, verifies

1) for every,

2), and

3) there is a sequence of singular times with, s.t. on each (),is a Y-regular solution, and for each.

Thus, we can also consider the existence and regularity of the universal attractors.

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