Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity

Abstract

We consider the following semirelativistic nonlinear Schrödinger equation (SNLS): { i ∂ t u ± ( m 2 − Δ ) 1 / 2 u = λ | u | p , a m p ; ( t , x ) ∈ [ 0 , T ) × R d , u ( 0 , x ) = u 0 ( x ) , a m p ; x ∈ R d , \begin{equation} \left \{ \begin {array}{ll} i\partial _t u \pm (m^2-\Delta )^{1/2} u = \lambda |u|^{p}, & (t,x)\in [0,T)\times \mathbb {R}^d, \\ u(0,x)=u_0(x), & x \in \mathbb {R}^d, \end{array} \right . \notag \end{equation} where m ≥ 0 m\geq 0 , λ ∈ C ∖ { 0 } \lambda \in \mathbb {C} \setminus \{ 0\} , d ∈ N d\in \mathbb {N} , T > 0 T>0 , and ∂ t = ∂ / ∂ t \partial _t=\partial /\partial t . Here ( m 2 − Δ ) 1 / 2 := F − 1 ( m 2 + | ξ | 2 ) 1 / 2 F (m^2-\Delta )^{1/2}:=\mathcal {F}^{-1} (m^2+|\xi |^2 )^{1/2} \mathcal {F} , where F \mathcal {F} denotes the Fourier transform. Fujiwara and Ozawa proved the nonexistence of global weak solutions to SNLS for some initial data in the case of d = 1 d=1 , m = 0 m=0 , and 1 > p ≤ 2 1>p\leq 2 by a test function method. In this paper, we extend their result to a more general setting: for example, m ≥ 0 m\geq 0 , d ∈ N d\in \mathbb {N} , or p > 1 p>1 . Moreover, we obtain the upper estimates of weak solutions to SNLS. The key to the proof is to choose an appropriate test function.

Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case

“…We recall that in three dimensional case, p = p 3,1 = 3 is a critical value in view of the result in [18]. However, the result in [18] treats non-gauge invariant nonlinearities having constant sign, for which the test function method works. The question of the existence of local and global solutions for n ≥ 3 and p ≥ 1 + 2/(n − 2) seems, at the best of our knowledge, still open.…”

A special form of solution to half-wave equations

“…We need to check the compatibility of the equations ( 13) and ( 14). Applying (11), we have H(2uHu) = (Hu) 2 − u 2 , which is equivalent to…”

Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

“…This equation has rich mathematical problems, and it has recently gained much attention; to mention a few papers treating power-type nonlinear terms, see [2], [5], [10] for local/global wellposedness, [18] for finite-time blow up, [24], [2] for stability/instability of ground states, [5] for illposedness for low-regularity data, and [10] for the proof of various a priori estimates. The present paper also treats the power-type nonlinear term of the form λ|u| p−1 u (which is algebraic if p odd) or λ|u| p (which is algebraic if p even) (λ ∈ C \ {0}, p > 1), and discusses three problems left unexplored in the existing literature.…”