The Blow Up Problems On Parabolic System

The thesis includes four parts. The first one is existence and uniqueness on parabolic system, the second one is the continuous dependence on coupling functions, the third one is the blow up problems on parabolic system.Ⅰ. Consider coupled equations with n equations: Conclusion:The classical solutions of equations are existent and unique.Ⅱ. Consider coupled equations with2equations: and nonlinear equations with the same initial values Conclusion:The solutions of coupled equations are dependence on the parameters of coupling functions, and the explicit estimate is given: where, a1≥0, a2≥0,1/2≤p,p2≤1,A1,A2are positive constants relating to p1, p2, T.Ⅲ. First consider the equation with source item which is piecewise continuous: where, Conclusion:The complete blow up time of solutions is continuous dependence on the initial values.Second consider coupled equations: where, p>0, q>0, m>1, n>1.Conclusion:If (m-1)(n-1)> pq, or (m-1)(n-1)<pq and initial values uo(x)、v0(x) take enough small, then the global solutions are exist;If(m-1)(n-1)<pq and initial values uo(x)、v0(x) take enough large, then the solutions are blow up in finite time.