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

Abstract

Abstract This paper deals with the large time behavior of non-negative solutions for the porous medium equation with a nonlinear gradient source u t = Δ u m + | ∇ u l | q , ( x , t ) ∈ Ω × ( 0 , ∞ ) , 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 ) . MSC:35K55, 35K65, 35B40.