The Analysis Of The Solutions To Three Partial Differential Equations

Partial differential equations can be divided into elliptical equations, parabolic equations and hyperbolic equations. This paper mainly studies the solutions for three partial differential problems. Chapte 2 deals with inhomogeneous fluctuate equation: which is the special case of hyperbolic equations. And the fomulas of the solution of power series are given for Cauchy problem.In chapter 3 we investigate the blow-up properties of the positive solutions to a nonlocal degenerate singular reaction-diffusion problems: |x|~mu_tdiv(|x|~α▽u)=f_Ωu~p(x, t)dx-ku~q with homogeneous Dirichlet boundary conditions, whereΩ=B(0,1),0≤α＜2,p＞q≥1. We obtain that the solution exists globally for sufficient small initial data while blows up in finite time for large enough initial ones as p=q＞1 and p＞q.And in Chapter 4 a nonlocal degenerate singular semilinear parabolic system is considerred. Since the local existence and finite time blow-up of classical positive solution have been obtained, we mainly investigate the asymptotic behaviour for blow-up solutions and the precise blow-up rate is established. And the result have the similar form as the case of the nonsingular parabolic equations with nonlocal source.