GLOBAL AND BLOW-UP SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS WITH A GRADIENT TERM

In this paper, we are concerned with the following nonlinear parabolic equation with a gradient term and Neumann boundary condition: {(b(u))(t) = del . (a(u)del u) + f(x, u,vertical bar del u vertical bar(2), t) in D x (0, T), partial derivative u/partial derivative n = 0 on partial derivative D x (0,T), u(x,0) = u(0)(x)>0 in (D) over bar, where D subset of R-N (N >= 2) is a bounded domain of R-N with smooth boundary partial derivative D. The upper and lower solution technique is adopted in investigations. The sufficient conditions for the existence of global positive solution and an upper estimate of global solution are given. Moreover, under some appropriate assumptions on the functions a, b, and f, we prove the existence of blow-up positive solution. An upper bound of "blow-up time" is also presented.