Saddle points and instability of nonlinear hyperbolic equations

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

“…Indeed, using the potential well method due to Payne-Sattinger [36], we extend the local solution with data in some stable sets to a global one. Here, we are reduced to prove that the fractional elliptic problem…”

“…Section seven deals with existence and stability of ground states. The last section is about global well-posedness of the Schrödinger problem (1.1) in the focusing case using potential well method [36]. In the appendix, some compact Sobolev injection is proved.…”

Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy

“…Some further sharpening of this result is possible in the mass case using the ideas of [17] for bounded domains. We shall describe this situation more precisely below.…”

“…For example, there are many related results in bounded domains [3,5,17,22], as well as for abstract problems [8,9,22].…”

Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity

“…It's well known that problem (1.2) has been studied by many authors. A powerful technique for treating problem (1.2) is the so called "potential well method", which was established by Sattinger [22], Payne and Sattinger [21], and then improved by Liu and Zhao [17] by introducing a family of potential wells. Recently, there are some interesting results about the global existence and blow-up of solutions for problem (1.2) in [3], in which Chen and Tian proved global existence, blow-up at +∞, the behavior of vacuum isolation and asymptotic behavior of solutions with initial energy J(u 0 ) ≤ d. For other related works, we refer the readers to [2,11,6,15] and the references therein.…”