Global solutions for bi-harmonic inhomogeneous non-linear Schrödinger equations in Lebesgue spaces

Abstract

Abstract This paper studies the inhomogeneous bi-harmonic nonlinear Schrödinger equation i u ˙ + Δ 2 u ± | x | − τ | u | p − 1 u = 0 , $$ \mbox{i}\dot{u} +\Delta ^{2} u\pm |x|^{-\tau}|u|^{p-1}u=0, $$ where u : = u ( t , x ) : R × R N → C $u:=u(t,x):\mathbb{R}\times \mathbb{R}^{N}\to \mathbb{C}$ , τ > 0 $\tau >0$ and p > 1 $p>1$ . One proves the existence of global solutions with datum in Lebesgue space L ϱ ( R N ) $L^{\varrho}(\mathbb{R}^{N})$ , for a certain 1 < ϱ < 2 $1<\varrho <2$ . This work complements the known global well-posedness result in the mass-sub-critical regime, namely 1 < p < 1 + 8 − 2 τ N $1< p<1+\frac{8-2\tau}{N}$ and with finite mass. This work aims to develop a local theory in non-Hilbert Lebesgue spaces. Moreover, one complements the local solution to a global one under some suitable assumptions on the parameters. To this end, one uses a Strichartz-type estimate based on L ϱ $L^{\varrho}$ . Moreover, one breaks down the data into two parts: the first is in L 2 $L^{2}$ , and the second has a small Strichrtz norm under the free bi-harmonic Schrödinger propagator. Then, one uses a standard fix point argument coupled with Strichartz estimates. First, one resolves the above problem in L 2 $L^{2}$ and then takes the considered equation with a perturbed source term. The local solution is complemented by a global one with induction. Since the free associated kernel e i ⋅ Δ 2 $e^{\mbox{i}\cdot \Delta ^{2}}$ is unitary in L 2 $L^{2}$ , the Schrödinger equations seem more adapted to be mathematically studied in functional spaces based on L 2 $L^{2}$ , such as the Sobolev spaces W s , 2 ( R N ) $W^{s,2}(\mathbb{R}^{N})$ . The main novelty here is investigating the above inhomogeneous bi-harmonic nonlinear Schrödinger problem in some Lebesgue spaces different from L 2 $L^{2}$ . One essential challenge is overcoming the lack of conservation laws, namely the mass and the energy, representing standard tools in the Schrödinger context. The second technical difficulty is the presence of an inhomogeneous source term, namely the singular decaying term | x | − τ $|x|^{-\tau}$ , which causes some serious complications. In order to deal with the inhomogeneous term, one uses Lorentz spaces with the property | x | − τ ∈ L N τ , ∞ $|x|^{-\tau}\in L^{\frac {N}{\tau},\infty}$ .

Keywords

• Fourth-order Schrödinger equation
• Nonlinear equations
• Inhomogeneous
• Global well-posedness
• Strichartz estimate