Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian

: We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.


Introduction, function spaces and auxiliary results
Here, r Δ , 0, 1 r ( ) ( ) − ∈ is the fractional Laplacian. The fractional Laplacian of the function w is a singular integral operator defined by Similar problems were studied, we refer for example to the pioneer works of MacCamy et al. [1,2] and the books of Zuazua [3] and other authors [4][5][6][7][8][9][10][11] and references therein, for a complete analysis and review on this topic.
Motivated by the aforementioned works, we complete the study of weak solutions for problem (1.1) in the setting of fractional Laplacian by potential well theory and Galerkin approximations. More precisely, we shall prove the existence of global solutions for problem (1.1). Furthermore, we show anew decay estimates of global solutions.
It is very important to note that our model involving fractional Laplacian is surely well studied in recent years. This type of problem arises much more in many different applications, such as for example image processing, finance, population dynamics, fluid dynamics, minimal surfaces and game theory and especially in physics.
The rest of the paper is organized as follows. In Section 1, we introduce our problem and recall some necessary definitions and properties of the fractional Sobolev spaces. In Section 3, we study the global existence of weak solutions for problem (1.1). In Section 4, we show the decay rate of global solutions of (1.1).
Some necessary definitions and properties regarding the fractional Sobolev spaces are stated here, see [12] for further details.
We define the fractional-order Sobolev space by 0a . e . i nΩ, is a closed linear subspace of W Ω r,2 1 ( ), and its norm is given by The potential well We define We define then the stable set as follows: where the mountain pass level d is defined by We introduce the so-called "Nehari manifold" then potential depth d is characterized by We will prove the invariance of the set .
Consequently, for all w ∈ we have w ∈ , where w W w R Ω : .
The proof of Lemma 2.2 is completed. □ Denote by μ μ μ 1 2 3 < < <⋯ the distinct eigenvalues and e k the eigenfunction corresponding to μ k of the elliptic eigenvalue problem More precisely, the following weak formulation of ( ). Furthermore, we have the following property: , there exists a sequence w n n 0, Definition 2.5. Let w t ( ) be a weak solution of problem (1.1). We define the maximal existence time T of w t ( ) as follows: For problem (1.1) and δ where K is the best imbedding constant of the embedding W Ω r 0 and d δ 0 Proof.
The proof is completed.
The proof is now completed. □ Let us define the following family of potential wells for all δ 0, 1 The following result is a consequence of Lemma 2.6.
where δ¯is such that r δ δ r δ 1 The following Lemmas are a consequence of Lemmas 2.7 and 2.10.
, and that δ δ 2 1 > are the two solutions of the equation Then w δ ( ) does not change sign for δ δ δ ,

Existence of global solutions
As in [13], we are now ready to apply the Galerkin method by constructing finite-dimensional Galerkin approximations for (1.1) and then present a priori estimates, which allow us to pass to the limit to obtain the desired weak solution w of (1.1). Indeed, w verifies the conditions of initial data and belongs to the family of potential wells.
, we say that w is a strong global solution of problem (1.1).
We introduce the energy of solution at time t as and for all t ∈ + * .
Proof. By (2.16) there exists a sequence u C Ω  According to the standard ordinary differential equation theory, problem admits a solution g jm of class for each n. Multiplying problem (1.1) by g j n′ , summing for j, we have provided that n is sufficiently large. Thus, for all δ δ δ , , w ., 0 n δ ( ) ∈ and n is sufficiently large. Next we t claim ∀ ∈ + * that w t ., n δ ( ) ∈ for sufficiently large n. Suppose that w t ., n ( ) is not contained in δ , and let T be the smallest time t for which w t ., n ( ) is not contained in δ . Then, w T ., n δ ( ) ∈ ∂ by the continuity of w t ., n ( ). Hence, either A w T d δ ., n ( ) ( ) = or w T ., 0 δ n ( ( )) = . Therefore, for n large enough, we have which contradicts the fact that w t d δ ., Hence, there exist ξ , w and a subsequence of w n n ( ) , such that as n → ∞ and 1 p p Integrating with respect to τ from 0 to t, we have w t u w τ u τ w τ u τ w u ξ u τ ., , 1 2 ., , d ., , d , , d .
, with t fixed, and integration with respect to t, we conclude that w is a global solution of the problem. Finally, w t ., n δ ( ) ∈ for any δ δ δ , , for any n and for t ∈ + * , so that w t ., δ ( ) ∈ for any δ δ δ , 1 2 ( ) ∈ and t ∈ + * . □ and for all t ∈ + * .
Proof. If w