Research And Simulation On The Blow-up Of Solutions For Hyperbolic Nonlinear Systems

Many phenomena in natural science can be regarded as nonlinear systems,and the problems of nonlinear systems can be described by nonlinear equations.For nonlinear problems,control theory on manifold is often used to deal with nonlinear problems.An analytical method of manifold control theory with differential geometry as its main mathematical tool can be used to study local and global solutions of hyperbolic equations in nonlinear systems.Hyperbolic equation is a kind of representative equation in nonlinear system.The local wellposedness and blow-up results of solutions are important research contents of hyperbolic equations.In this paper,the blow-up results and the upper bound lifespan estimates of solutions for hyperbolic equations in Minkowski space and de Sitter space-time are investigated.This article mainly carries on the research from the following five aspects.(1)The variable coefficient wave equations with combined nonlinear terms in the external domain are considered.It also discusses the two cases of presence and absence of damping terms.When the damping term is present,by introducing damping in the multiplicative absorption equation,the upper bound lifespan estimate for the solution is obtained by using the test function method and the iterative method,where the result n ≥2 is better than n ≥1.Also for the case of no damping term,the upper bound lifespan estimate of the solution is obtained by using Kato’s Lemma.Finally,Matlab is used to simulate numerically the variation of the wave.(2)The quasilinear wave equations with damping and negative mass terms are considered in the external domain.By using the test function method and the iterative method,the blow-up results and upper bound lifespan estimates of solutions for the initial-boundary value problems with power nonlinear term and derivative nonlinear term are obtained,respectively.At the same time,the wave equation with combined nonlinear terms in Minkowski space-time is considered,the blow-up results and upper bound lifespan estimates are obtained.Finally,the numerical simulation is carried out by Matlab.(3)The weakly coupled wave equations with damping,negative mass and divergence nonlinear terms in Minkowski space-time are considered.Discuss in two scenarios: subcritical case and critical case.In the subcritical case,the test function method and the iterative method are used.In the critical case,by using the auxiliary function method and the iterative method.Finally,the numerical simulation is carried out in the case n =2.(4)The weakly coupled wave equations with damping and mass terms in de Sitter spacetime are considered.The cases of exponential growth,polynomial growth and logarithmic growth are considered,respectively.The blow-up results and the upper bound lifespan estimate of solutions are derived by iterative method.Finally,the changes of waves in the de sitter space are numerically simulated.(5)The Tricomi equation with memory term in the external domain are considered.Upper bound lifespan estimates of solutions for the equations with the power nonlinear term and combined nonlinear term are obtained by using the test function method and the iterative method,respectively.In addition,the upper bound lifespan estimates of solutions for the coupled Tricomi equation with the initial value problem are derived.