Abstract:
We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrodinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability.

Abstract:
It is proved that blow-up solutions to N coupled Schrodinger equations $$ ivarphi_{jt} + varphi_{jxx} + mu_j|varphi_j|^{p-2}varphi_j +sum_{k eq j,;k=1}^Neta_{kj}|varphi_k|^{p_k}|varphi_j|^{p_j-2} varphi_j=0 $$ exist only under the condition that the initial data have strictly negative energy.

Abstract:
We study the nonlinear Schrodinger equations with a linear potential. A change of variables makes it possible to deduce results concerning finite time blow up and scattering theory from the case with no potential.

Abstract:
In this paper, we establish two new types of invariant sets for the coupled nonlinear Schrodinger system on $\mathbb{R}^n$, and derive two sharp thresholds of blow-up and global existence for its solution. Some analogous results for the nonlinear Schrodinger system posed on the hyperbolic space $\mathbb{H}^n$ and on the standard 2-sphere $\mathbb{S}^2$ are also presented. Our arguments and constructions are improvements of some previous works on this direction. At the end, we give some heuristic analysis about the strong instability of the solitary waves.

Abstract:
We consider the energy critical Schrodinger map to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map in the scale invariant norm which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. The concentration rate is given by $$\lambda(t)=\kappa(u)\frac{T-t}{|\log (T-t)|^2}(1+o(1))$$ for some $\kappa(u)>0$. The detailed proofs of the results will appear in a companion paper.

Abstract:
This paper addresses the blow-up problem associated with the parameter estimation part of an adaptive controller. A partial solution to the problem has been devised by the introduction of a variable forgetting factor. However, this does not eliminate the blow-up possibility. This is shown by simulation experiments on two different models.

Abstract:
In this paper we study blow-up phenomena in general coupled nonlinear Schrodinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several applications of our results to heteronuclear multispecies Bose-Einstein condensates and to degenerate boson-fermion mixtures.

Abstract:
This dissertation is to discuss the initial-boundary value problem under the third nonlinear boundary condition for a kind of quasi-linear parabolic equations. To apply the maximum value theory and convex method, it is proved that the blowing up of solution in the definite time. And for application, this paper research into a mathematics model of fluids in porous medium. And have got the blowing up behavior for the problem in limited time.

Abstract:
We consider the nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{p-1}u$ in dimension $N\geq 3$ in the $L^2$ super critical range $1+\frac{4}{N}

Abstract:
We consider equations of nonlinear Schrodinger type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic confinement in all spatial directions drives the solution of our model to zero for large time. In the case without external potential we prove that the solution may not go to zero for large time due to (non-trivial) scattering.