Existence and asymptotic behavior of global smooth solution for [formula omitted]-System with damping and boundary effect

Abstract

In this paper, we study the existence and asymptotic behavior of global smooth solution for p -System with damping on the strip [ 0 , 1 ] × ( 0 , ∞ ) v t − u x = 0 , u t + p ( v ) x = − α u , with the null-Dirichlet boundary condition, u ( 0 , t ) = u ( 1 , t ) = 0 . First, we prove the existence of the global smooth solution under the assumption that only the C 0 -norm of the first derivatives of the initial data is sufficiently small, while the C 0 -norm of the initial data is not necessarily small. Second, when the initial data is in C 2 and the C 0 -norm of the first derivatives of the initial data is large enough, then the first derivatives of the corresponding C 2 smooth solution must blow up in finite time. Finally, we consider the asymptotic behavior of the global smooth solution. Precisely, we show that the solution of this problem tends to the corresponding nonlinear diffusive wave governed by the classical Darcy’ s law exponentially fast as time goes to infinity in the sense of H 1 -norm, provided the initial total mass is the same and thus justifies the validity of Darcy’s law in large time.