Blowup for biharmonic NLS

Abstract

We consider the Cauchy problem for the biharmonic (i. e. fourth-order) NLS with focusing nonlinearity given bywhere 0 ă σ ă 8 for d ď 4 and 0 ă σ ď 4{pd´4q for d ě 5; and µ P R is some parameter to include a possible lower-order dispersion. In the mass-supercritical case σ ą 4{d, we prove a general result on finite-time blowup for radial data in H 2 pR d q in any dimension d ě 2. Moreover, we derive a universal upper bound for the blowup rate for suitable 4{d ă σ ă 4{pd´4q. In the mass-critical case σ " 4{d, we prove a general blowup result in finite or infinite time for radial data in H 2 pR d q. As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems.In addition, we prove a radial symmetry result for ground states for the biharmonic NLS, which may be of some value for the related elliptic problem.

On the global well-posedness of focusing energy-critical inhomogeneous NLS

“…The condition (1.10) leads us to the inequality (|∇u| 2 −(1−η)g|u| p+1 ) dx < −C for some 0 ≤ η < 1 and C > 0 and hence to the finite time blowup through the localized virial identity (6.5) below. This argument also appears in some literatures (see [21,2,13]). In (2) the moment condition |x|ϕ ∈ L 2…”

“…For the construction of such function see Appendix B of [2]. Now let β r (s) = rβ( s r ) and b r (|x|) = |x| 0 β r (s) ds.…”

On the focusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space

“…[15], [3], [34], [35], [4] and references therein). It is closely related to ground states Q of (NL4S) which are solutions to the elliptic equation…”

“…Recently, Boulenger-Lenzmann in [4] proved a general result on finite-time blowup for the focusing generalized nonlinear fourth-order Schrödigner equation( i.e. (1.1) with µ = 1) with radial data in H 2 (R d ).…”

On a fourth‐order nonlinear Helmholtz equation

“…Despite being less studied than the classical (2NLS), an increasing attention has been given to (4NLS). We refer to the works of Pausader [35][36][37][38][39], Miao et al [33], Ruzhansky et al [40], Segata [41,42] concerning global well-posedness and scattering, to [6,8] for finite-time blow-up and to [4,5,34] for the stability of standing wave solutions. We also mention that (4NHE) also appears in the theory of water waves [9] and as a model to study travelling waves in suspension bridges [24,32] (see also [10,26]).…”