Decay of solutions to a fourth-order nonlinear Schrödinger equation
-
Tarek Saanouni
Abstract
Let
A Appendix
As an application of the main result, we give a short proof of scattering. Precisely we prove the next result.
Let
and there exist
For any time slab I, take the space
endowed with the complete norm
Let us state an intermediate result.
For any time slab I and any time
Proof.
Using the Strichartz estimate, we have
Letting
we get
It remains to estimate the quantity
Write
Using the Hölder inequality, we obtain
Letting
Similarly,
The proof is achieved with previous computations. ∎
Finally, we are ready to prove scattering. By the previous lemma and Theorem 1.1, we have
where
By the computations done in the proof of Lemma A.2, we have
So, applying the Strichartz estimate, we get, for
Taking
Scattering is proved.
References
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Articles in the same Issue
- Frontmatter
- Periodic solutions of nonautonomous second-order differential equations with a p-Laplacian
- Double integral inequalities of Hermite–Hadamard type for h-convex functions on linear spaces
- Characterization of univalent harmonic mappings with integer or half-integer coefficients
- Weighted composition operators between weak spaces of vector-valued analytic functions
- Decay of solutions to a fourth-order nonlinear Schrödinger equation
Articles in the same Issue
- Frontmatter
- Periodic solutions of nonautonomous second-order differential equations with a p-Laplacian
- Double integral inequalities of Hermite–Hadamard type for h-convex functions on linear spaces
- Characterization of univalent harmonic mappings with integer or half-integer coefficients
- Weighted composition operators between weak spaces of vector-valued analytic functions
- Decay of solutions to a fourth-order nonlinear Schrödinger equation
