A note on the inhomogeneous fourth-order Schrodinger equation

This paper studies the non-linear bihannonic Schodinger equation i(u)over dot + Delta(2)u +/- F (x, vertical bar u vertical bar)u = 0, where F(x, vertical bar u vertical bar) is an element of {vertical bar x vertical bar(-2b) vertical bar u vertical bar(2(q-1)) , vertical bar x vertical bar(-b)vertical bar u vertical bar(p-2)(I-alpha (*) vertical bar.vertical bar-b vertical bar u vertical bar(p))}, where b > 0 and the source terms are inter-critical. First one develops a local theory in L-2 and in the energy space H-2, by use of a sharp Gagliardo-Nirenberg type inequality. Then, one considers the global theory. Indeed, a sharp dichotomy of global versus non-global existence of solutions is obtained by use of the existence of ground states. Moreover, the strong instability of standing waves is proved. This note is a natural extension of Saanouni (Commun Pure Appl Anal 19(10): 5033-5057, 2020) to the inhomogeneous regime and gives some essential tools for the scattering of the focusing global solutions proved by Saanouni (Calc Var 60(113), 2021).