Sentence examples for the sharp thresholds from inspiring English sources

In particular, the sharp thresholds of global existence and blow-up for nonlinear Schrödinger equations are pursued strongly (see [1, 2, 19, 23 25] and the references therein).

We identify the sharp threshold for K4-percolation on the Erdos-Renyi graph G(n,p).

The sharp threshold condition (R_{0}) of our model is obtained.

The VFH algorithm also displayed the tendency of creating oscillations in environments with multiple narrow openings due to the sharp thresholding near the entrances.

However, the estimation of the "sharp" threshold is one of the main contributions from Onatski (2010), which as we have mentioned, consistently separates convergent and divergent eigenvalues.

The sharp threshold of global existence and blow-up, the instability of standing wave of (1.1) with (lambda_{1}=0) and a harmonic potential have been investigated in [19].

Zhang [14] investigates the sharp threshold of blow-up and global existence for equation (1.1) with (H^{1} -sub-critical nonlinearity (i.e. (1} -sub-criticalhe variational argument.

We remark that the sharp threshold obtained in [14] is not the energy criteria due to the threshold is not fully determined by the (dot{H}^{1} -norm of the corresponding ground state solutions.

3, we firstly show the existence of blow-up solutions to (1.1) with (lambda_{1}=1), (lambda_{2}=-1), (0< p_{1}the sharp threshold mass (Vert uVert _{L^{2}}) of global existence and blow-up.

For (1.1), we find that the ground state solution u to (1.4) exactly describes the sharp threshold mass of global existence and blow-up, the dynamical properties of blow-up solutions, including (L^{2} -concentration, L^{2} -concentration blimitingates.

Hence, in this paper, we first show the existence of blow-up solutions and obtain the sharp threshold mass (Vert uVert _{L^{2}}) of global existence and blow-up for (1.1), where u is a ground state solution of the elliptic equation -Delta u+ u-bigl(I_{alpha }ast vert uvert ^{p}bigr)vert uvert ^{p-2}u=0.