ASYMPTOTIC BEHAVIOR OF SOLUTION TO NONLINEAR DAMPED p-SYSTEM WITH BOUNDARY EFFECT

with a specially selected initial data v̄0(x). The optimal convergence rates ‖∂ x (v− v̄, u− ū)(t)‖L2 = O(1)(t 2k+3 4 , t 2k+5 4 ), k = 0, 1, are also obtained, as the initial perturbation is in L(R+) ∩ H(R+). If the initial perturbation is in the weighted space L(R+) ∩ H(R+) with the best choice of γ = 1 4 , some new and much better decay rates are further obtained: ‖∂ x(v− v̄)(t)‖L2 = O(1)(1 + t) 2k+3 4 − γ 2 , k = 0, 1. The proof is based on the technical weighted energy method combining with the Green function method. However, when β < 0 and |β| > α |u+| q−1 , then the solution will blow up at a finite time. Finally, numerical simulations are carried out to confirm the theoretical results by using the central-upwind scheme. In particular, the interest phenomenon of coexistence of the global solution v(x, t) and the blow-up solution u(x, t) is observed and numerically demonstrated.