LIFESPAN ESTIMATE FOR A POROUS MEDIUM EQUATION WITH GRADIENT TERMS UNDER NONLINEAR BOUNDARY CONDITIONS

In the present paper, we investigate the following porous media equations with gradient terms: g(u))(t) = Delta u(m) + f(x, u, vertical bar del u vertical bar(2), t), x is an element of D, t is an element of (0, T*), partial derivative u/partial derivative v = h(u), x is an element of partial derivative D, t is an element of(0, T*), u(x, 0) = u(0)(x), x is an element of(D) over bar. Here m > 1, D is a convex bounded domain in R-n (n >= 2) with smooth boundary partial derivative D. We establish some suitable conditions to ensure that the solutions of the above problem blow up in a finite blow-up time T*. Furthermore, with the help of some first-order differential inequalities and some embedding theorems in Sobolev space, the upper and lower bounds of T* are also given.