A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations

Abstract

In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)-type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS-type equations. This could answer the following open issue: consider, for instance, the classical one-dimensional cubic nonlinear Schrödinger equation u t + iu xx + i|u| 2 u + u = , ∈ L 2 (R). "How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity

Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations

“…Hence, thanks to the Young inequality, we obtain from (74) thatd dt ||θ R Z|| 2 L 2 + ||θ R Z|| 2 L 2 ||θ R Z|| 2 L 2 + ||θ R Y || 2 + ||Y || 2 L 2 .This leads, thanks to Gronwall's Lemma and in accordance with Lemma 5.6, to ||(Z(t), Y (t)||2 …”

“…proof of Lemma 5.4. To begin with, for arbitrary R > 0 we write R |ϑ| |u| 2 dx = [−R,R]∩{|ϑ|≤R} |ϑ| |u| 2 dx + [−R,R]∩{|ϑ|>R} |ϑ| |u| 2 dx + {|x|>R} |ϑ| |u| 2 dxwe obtain by applying the Hölder inequality thatR |ϑ| |u| 2 dx ≤ R ||u|| 2 L 2 ([−R,R]) + ||ϑ|| L 2 ({|ϑ|>R}) + ||ϑ|| L 2 ({|x|>R}) ||u|| 2 L 4 (R)by means of which and the continuous embedding ofH α 2 in L 4 (Lemma 2.3) we deduce that R |ϑ| |u| 2 dx ≤ R ||u|| 2 L 2 ([−R,R]) + C ||ϑ|| L 2 ({|ϑ|>R}) + ||ϑ|| L 2 ({|x|>R})||u|| Knowing that ||ϑ|| L 2 ({|ϑ|>R}) + ||ϑ|| L 2 ({|x|>R}) −→ 0 as R −→ +∞, we deduce that for every > 0 there exists R = R(ϑ, ) > 0 satisfying R |ϑ| |u| 2 dx ≤ ||u||2…”

“…The regularity of the global attractor A α for (1) − (2) will be discussed in Section 4. Finally, the section 5 we be dedicated to establish the finite fractal dimension of A α using the new idea recently introduced by the author in [2].…”

“…Then M is a compact subset of X and has a finite fractal dimension. Now, we state an important result (see [2]) that will enables us to apply Theorem 5.3.…”

Global attractor for a fractional nonlinear Klein-Gordon-Schrödinger system

“… → (Λ )2 ,  → Λ Λ and  → Λ are continuous from 2 into 2 . Hence, using (5.100) on the one hand and , (5.97)-(5.98) on the other hand one can easily obtain (5.99), which conclude the proof of Lemma 3.At this stage, we have obtained by Lebesgue's convergence theorem once again (5.94) for = and ( 0 , 0 , 0 ) = (− ) , we infer from (5.101) thatlim sup →+∞ ̃ ( ) ≤ [lim sup →+∞ ̃ ( (− ) ) + | ̃ ( (− ) )|] −2 + ̃ ( ).Thanks (− ) to and (− ) belong to  (⊂ 2 ), there exists = ( , , , , , ) such thatlim sup →+∞ ̃ ( ) ≤ −̃ ( ).…”