Properties Of Solutions To Two Nonlinear Parabolic Equations

This paper study properties of solutions to two nonlinear parabolic equa tions. Charpte 2 is dedicate to studying parabolic system we use the method of upper and lower solutions to investigate the factor which lead to blow up, and give some conditions under which solutions of the system existence globally or blow up in finite time. The following result as fomer works is obtained: large data make a solution blow up in finite time; however, small data make a solution existence globally. At the same time, it is proved that the domain is more large, the solution is more likely to blow up in finite time. For the two cases:pi > 1 and 0 < pi≤1, different method is introduced to study the problem. In the case of pi > 1, the result is expressed in the form of eigenvalue and eigenfunction of the domain. When 0 < Pi ≤ 1, the direct conclusion that large domain leads the solution blow up is obtained.For the fast diffusion equation :ut = △um + up (0 < m < 1), convex functional method is used in charpter 3 to obtain the upper estimate for the life span of the solution and a constructed upper solution is helpful to obtain the lower estimate. Accordingly, we give the estimate for the life span of the solution to fast diffusion equation. And the result have the same form as the case of slow diffusion equation and semilinear equation.