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 ut=Δum+|∇ul|q, (x,t)∈Ω×(0,∞), where l≥m>1 and 1≤q<2. When lq=m, we prove that the global solution converges to the separate variable solution t−1m−1f(x). While m<lq≤m+1, we note that the global solution converges to the separate variable solution t−1m−1f0(x). Moreover, when lq>m+1, we show that the global solution also converges to the separate variable solution t−1m−1f0(x) for the small initial data u0(x), and we find that the solution u(x,t) blows up in finite time for the large initial data u0(x).