Blow-up Of Solutions And Existence Of Solitary Wave Solutions For Hyperbolic(Equations) Systems

Many problems in natural science study nonlinear systems.The stability of system is an important branch.Many nonlinear systems are described by partial differential equations.Therefore,the blow-up of solutions and the existence of peakon solutions of such equations are hot topics in recent years.From the perspective of system stability,the following two types of problems are studied in this paper.First,we use the test function method to verify the blow-up behavior of solutions for several hyperbolic partial differential equations（systems）.Secondly,the existence and simulation of peakon solutions for generalized rotating two-component Camassa-Holm system are established.The main contents include the following five parts.（1）In the second chapter,we consider the wave equations with critical exponent and the wave equations with scale invariant damping in exterior domain in high dimensions.The blow-up results and upper bound lifespan estimates of solutions are obtained by using the test function method.The innovation is that we give the asymptotic behavior of the test function by using maximum principles.The method used in our paper is different from the existing literature^[93],we avoid using the modified Bessel function to construct the test function.Meanwhile,we also provide an alternative proof which is different from the references^[58,59].（2）In the third chapter,we study the wave equations with scale invariant damping in exterior domain in one space dimension.And we use two different methods to obtain the blow-up result and upper bound lifespan estimate of the solution.The first method is to construct the functional which is related to the solution,then the blow-up result and upper bound lifespan estimate of solution are obtained by solving the ordinary differential inequality.In the second method,the blow-up result and lifespan estimate of the solution are derived by using the test function method.The upper bound lifespan estimates that are obtained by two different methods are consistent.Finally,according to the idea of finite difference method,we give the simulation of the solution.（3）In the fourth chapter,we discuss the wave equations with combined nonlinear terms and space dependent damping.The blow-up results of the solution are obtained by test function method,and the corresponding upper bound lifespan estimates of the solution are derived.We conclude that the range of blow-up of the solution produced by the combined nonlinear terms is wider than that produced by the power nonlinear term and derivative nonlinear term.（4）In the fifth chapter,we introduce the coupled Tricomi equations with derivative nonlinear terms.For the Cauchy problem with small initial values,the blow-up result and upper bound lifespan estimates of the solution are obtained by using the test function method and iterative method,respectively.The innovation is that the lifespan estimates of solutions in sub-critical and critical cases are related to Glassey conjecture.（5）In the sixth chapter,we prove the existence of peakon solutions for the generalized rotating two-component Camassa-Holm system.By using the properties of Dirac function,we obtain that this system admits the following peakon solutions,（?）,where δ=x-ct,β,r,p,q,c care constants.Meanwhile,the above system at most admits two peakon solutions（when p,q≠0）.At last,we use Matlab to give the simulation of the solution when p≠0,q=0;p=0,q≠0;p,q≠0.