We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on R d , d = 5, 6, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on R d , d = 5, 6, in the defocusing case, and (ii) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space. Contents 2010 Mathematics Subject Classification. 35Q55. Key words and phrases. nonlinear Schrödinger equation; almost sure local well-posedness; almost sure global well-posedness; finite time blowup. c T γ φ 2 H s such that for each ω ∈ Ω T , there exists a unique solution, where S(t) = e it∆ and X 1 T is defined in Section 3 below. Almost sure local well-posedness with respect to the Wiener randomization has been studied in the context of the cubic NLS and the quintic NLS on R d [2, 3, 8] which are energy-critical in dimensions 4 and 3, respectively. Note that when d = 5, 6, the energycritical nonlinearity |u| 4 d−2 u is no longer algebraic, presenting a new difficulty in applying the argument in [2,3,8].

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order α > 0) of a space-time white noise. In particular, we show that the well-posedness theory breaks at α = 1 2 for SNLW and at α = 1 for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for 0 < α < 1 2 . We first revisit this argument and establish multilinear smoothing of order 1 4 on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of α. On the other hand, when α ≥ 1 2 , we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for α ≥ 1 2 . (ii) As for SNLH, we establish analogous results with a threshold given byThese examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values (α = 3 4 in the wave case and α = 2 in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schrödinger equation with the derivative quartic nonlinearity ∂ x (u 4 ) is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the d ≥ 2 and derivative polynomial type nonlinearity, for example |∇|(u m ) with (m − 1)d ≥ 4.

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Abstract. We consider ill-posedness of the Cauchy problem for the generalized Boussinesq and Kawahara equations. We prove norm inflation with general initial data, an improvement over the ill-posedness results by Geba et al., Nonlinear Anal. 95 (2014), 404-413 for the generalized Boussinesq equations and by Kato, Adv.