Large time behavior of solutions for the porous medium equation with a nonlinear gradient source

This paper deals with the large time behavior of non-negative solutions for the porous medium equation with a nonlinear gradient source u(t) = Delta u(m) + vertical bar del u(l)vertical bar(q), (x,t) is an element of Omega x (0, infinity), where l >= m > 1 and 1 <= q < 2. When l(q) = m, we prove that the global solution converges to the separate variable solution t(-)1/m-1 f(x). While m < l(q) <= m + 1, we note that the global solution converges to the separate variable solution t(-)1/m-1 f(0)(x). Moreover, when l(q) > m +1, we show that the global solution also converges to the separate variable solution t(-)1/m-1 f(0)(x) for the small initial data u(0)(x), and we find that the solution u(x, t) blows up in finite time for the large initial data u(0)(x).