On Blow-up Criterion For The Nonlinear Schrodinger System

The Schrodinger equation is the basic equation in quantum mechanics.Its general form is i（?）tu+Δu+Vu=0,where V represents potential and its solution u is called wave function.In this paper,we study the blow-up problem of the nonlinear Schrodinger system:（?）The system describes how the quantum state of a physical system evolves with time.Nonlinear Schrodinger equations have attracted great attention of mathematicians,es-pecially because of its application in nonlinear optics.The potentials of the two equa-tions are V=（|u|²+|v|²）u and V=（|u|²+|v|²）v,respectively,and depend on the wave function u and v,which give the nonlinear terms.In the first chapter of this paper,we mainly introduce the research background and current situation of this Schrodinger system,and give the conservation laws of mass,momentum and energy that the system satisfies,and introduce the virial identity that the system satisfies.In the second chapter,the local virial identity and almost fi-nite propagation velocity are calculated.The third chapter mainly introduces the proof process of the solution explosion of the system.First,we use the thought of the counter proof method to assume that the system has a global solution and has always been bounded.Then,through the first derivative and the second derivative of the virial i-dentity,we finally derive the contradiction and prove the theorem.Chapter four is the summary and prospect of the article.The main results of this paper are as follows:when the dimension n≥ 2 and the energy of the equations E（u₀,v₀）<0,the solution will blow up in finite or infinite time.In particular,when n≥ 4,the system includes the energy supercritical（critical）cases.