Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

“…Section 2 is devoted to present the notations, more precisely, the Sobolev spaces, the Bourgain spaces, the Riemann-Liouville fractional integral operator and the Duhamel boundary forcing operator associated of (4NLS), which are paramount to prove the main result of the article. In the Section 3, we will give an overview of the main estimates proved by the authors in [16]. With these two sections in hand, we are able to prove Theorem 1.1, in several steps, in the Section 4.…”

“…Considering an different domain instead of the torus T, the authors, in [16], considered the cubic fourth order Schrödinger equation on the right half-line…”

“…for γ, λ ∈ R. When γλ < 0 system (2) is so-called focusing otherwise, that is, γλ > 0, is called defocusing. In [16], Capistrano-Filho et al consider γ = −1 and suitable choices of f (t) and g(t) in the equation (2), precisely, by assuming…”

“…Therefore, this work gives an answer for the Problem A, in a star graph structure, when the boundary conditions (4), (5) or (6) are considered. This problem was left as an open problem in [16]. Before to enunciate the principal result of this work, we will denote the classical Sobolev space on the star graph G by…”

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In a recent article [16], the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [16] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure G. Precisely, consider G composed by N edges parameterized by half-lines (0, +∞) attached with a common vertex ν. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the boundary forcing operator approach.

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