Global attractors for the -Laplacian equations with nonregular data

Let Ω be a smooth bounded domain in R N , ( N ≥ 3 ) . We consider the long time behavior of solutions to the p -Laplacian equation u t − Δ p u + f ( u ) = g where 2 ≤ p < N , − Δ p = − div ( | ∇ u | p − 2 ∇ u ) is the p -Laplace operator. Assume that g and the initial condition u 0 lie in L 1 ( Ω ) , f : R → R is of class C 1 and satisfies proper growth conditions. Firstly, we prove the uniqueness of the entropy solution and establish some regularity results. Then we show the existence of a global attractor A in L r − 1 ( Ω ) ∩ W 0 1 , s ( Ω ) with s < max { N ( p − 1 ) N − 1 , p ( r − 1 ) r } . To obtain the results, a decomposition method and a bootstrap technique are used.