The Study For The Solution Of Nonlinear Dispersion Equation With The Initial Boundary Value

In this paper, we consider the initial boundary value problem of semilinear Schrodinger equation on half-line. Blow up result will be established, assuming the power of the non-linear term satisfying 1< p< 2. The proof is based on a contradiction argument, by constructing a special test function. The conclusion are divided into 1< p< 2 and p= 2 to prove and we obtain the upper bound of the lifespan in the case 1< p< 2. Further-more, we establish blow up theorems for critical semilinear wave equations with focusing nonlinear term in 3-D bounded domains, for both Dirichlet and dissipative boundary conditions. The proof is based on the concavity method.This paper consists of three chapters.The first chapter is the introduction. It mainly introduces the research background, situation and research significance of Schrodinger equation, the wave equation and the necessity and its important meaning of the existence of the test function, giving the preliminaries and main content of this paper.The second chapter apply the test function method φ(t, x)= Cxφ1l(x)T-α(1-t/T)+k-and the contradiction argument to prove the blow up and lifespan estimate for one-dimensional semilinear Schrodinger equation with initial boundary value. Blow up result will be established, assuming the power of the nonlinear term satisfying 1< p< 2. Furthermore, we obtain the upper bound of the lifespan T(ε)< Cεp-2/p-1 in the case 1< p<2.The third chapter introduce the concavity method to prove the blow up theorems for critical semilinear wave equations with focusing nonlinear term in 3-D bounded domains, for both Dirichlet and dissipative boundary conditions.