On the Strongly Damped Wave Equations with Critical Nonlinearities

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Received 6 August 2015; accepted 23 April 2016; published 26 April 2016

1. Introduction

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

(1)

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.

(2)

(3)

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

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

(4)

.

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,

. (5)

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

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

(6)

and

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

Hence, for each, we have that

(7)

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

(8)

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

(9)

(10)

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

(11)

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] ).

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.

(12)

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

(13)

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

(14)

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

(15)

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

(16)

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

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.

References

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