Study On The Properties Of Solutions To Tricomi Equations(System)

Tricomi equation is a kind of important partial differential equation,which has significant mathematical theory and physical application,and has attracted extensive attention and explo-ration of mathematical researchers.In this thesis,we mainly study some properties of solutions to the Tricomi equation(system),and further study the long-term behavior of solutions to a sys-tem of Tricomi equations with mixed nonlinear terms.Specifically,we consider the following topics:Firstly,the fundamental solutions of the Tricomi operator are systematically studied,based on known results.In particular,in the one-dimensional case,the explicit expression of the solutions to the inhomogeneous Tricomi equation are given.Secondly,the relationship between the semilinear generalized Tricomi equation and the semilinear damped wave equation is studied.For semilinear generalized Tricomi equation with nonlinear term of |u|~p and |u_t|~p,one can always find some suitable transformation,such that it will be changed to semilinear wave equation with variable coefficient(time dependent)damp-ing.And hence study method can be borrowed from each other.Finally,we study the long time behavior of solutions to the Cauchy problem of the follow-ing generalized Tricomi system with small initial data and mixed nonlinear terms(?)Here,m≥0 is a real constant,n∈?denotes the spatial dimension,and ε>0 is a parameter representing the smallness of the initial data.Blow-up and lifespan estimate are established by using solutions of two ordinary differential equations to construct two appropriate test functions.The key point is to connect the estimates of the two nonlinear terms by analyzing the relationship between the two test functions delicately.The topic in this thesis is helpful to understand the long time behavior of mixed type partial differential equations,for both boundary problem and Cauchy problem.