Boundedness Of A Class Of Quasi-linear Parabolic Equations And Blow-up Analysis

In this paper, wo are concerned with the global existence and blow-up of soltuions for the following equation of the formThis equation cornes from a vairity of diffusion phenomena appeared widely in nature, such as filtration, phase tansition, plasma physics, biochemistry, etc. Comparing to linear equations and quasi-linear equations without degeneracy and singulartiy, such equations, to a certain extent, reflect even more exactly the physical reality. Recently, these diffusion-convection parabolic equations have been applied to describe some turbence, traffic flow, in particular, the model of sedimentation consolidation processes of flocculated suspensions.If m > 1, then the equation(*) degenerates at the point it = 0. If m < 1, then the equation (*) has the singularity at the point u = 0. We consider the problem in these two cases respectively.For the degenerate convection diffusion equation, namely, m > 1, a 7^ 0, we prove the global existence of solution of the equation(*) . Using DeGorgi Nash's technique, we get the uniform estimates of soltions for the approximate problems and the global solution is obtained by compactness.For the singular diffusion equation, namely, m < 1, a = 0, we prove the global existence and blow-up of solutions respectively by considering the eigenvalue problem of the elliptic equation.