Different asymptotic behavior of global solutions for a parabolic system with nonlinear gradient terms

In this paper we study the global existence and the different asymptotic behavior of mild solutions for the nonlinear parabolic system: ∂ t u = Δ u + a | ∇ v | p , ∂ t v = Δ v + b | ∇ u | q , t > 0 , x ∈ R N , where a , b ∈ R , N ⩾ 1 , 1 < p ⩽ q < 2 and p q > q N + 1 + N + 2 N + 1 . We prove, in particular, that if the initial values behave as u ( 0 , x ) ∼ ω 1 ( x / | x | ) | x | − α , v ( 0 , x ) ∼ ω 2 ( x / | x | ) | x | − β as | x | → ∞ , 0 < α , β < N , β + 2 − q q < α , α + 2 − p p < β and under suitable conditions on ω 1 , ω 2 , then the resulting solutions are global. Furthermore, although the scaling invariance properties of these initial values and the system are different, we prove that some of the solutions are asymptotic to self-similar solutions of appropriate asymptotic systems which depend on the values of α and β. The asymptotic behavior estimates are given in the W 1 , ∞ ( R N ) × W 1 , ∞ ( R N ) -norm and are stable under some small perturbations. The results of this paper complete those of Al-Elaiw and Tayachi (2010) [1] known only for β + 2 − q q = α and α + 2 − p p = β .