The Analysis Of The Solutions To Three Parabolic Equations

This paper studies the solutions to three parabolic equations. In Chapter 2 a nonlocal degenerate singular semilinear parabolic system is considered. There are degeneration and singularity, so the local existence and uniqueness of classical solution are established firstly. And we use the method of upper and lower solutions to investigate the factor which leads to blow up, and give some conditions under which the solution of the system exists globally or blows up in finite time. It is shows that for pq > 1 the solution exists globally for sufficiently small initial data while blows up in finite time for initial ones large enough.And in Chapter 3 we deal with a degenerate singular parabolic system with nonlocal source and homogeneous Dirichlet conditions in R^N space, whereΩ= B(0,1) and u₀,Ï…₀ is radical. Under appropriate hypotheses, the sufficient conditions on the global existence and finite time blow-up of posotive solution are given. Furthermore, we obtain the asymptotic behavior of blow-up solution.The last Chapter is dedicate to studying the nonhomogeneous heat conduction Cauchy problems, we give the the fomulas of the solution of power series by using iteration. This solutions can be applied in any heat conduction Cauchy equations with all dimentions.