Asymptotic Behavior And Optimal Decay Rates Of Solutions To The P-system With Damping

This paper is concerned with the asymptotic behavior and the optimal decay rates for the solutions to the p-system with linear and nonlinear damping, and we obtain the following results :(â… ) : We consider the Cauchy problem of the p-system with nonlinear dampingwith the initial data(v(x,0),u(x,0)) = (v₀(x),u₀(x)), x∈R, (0.0.2)where p(v) is a decreasing smooth function, f(u) is a smooth function, and the initial data (v₀(x),u₀(x)) satisfies :(v₀(x),u₀(x))→(v_±,u_±), v_-≠v₊, x→±∞. (0.0.3)Under the following assumptions:(A₁) p(v)∈C³(R+), p'(v) < 0, for any v > 0;(A₂) v±> 0, u₊ = u_- = 0;(A₃) Let the nonlinear damping f(u) be a superposition of the linear and nonlinear damping parts, i.e., f(u) =αu + g₁(u),α> 0 is a constant, and g₁(u)∈C³(R),g₁(0)=g′₁(0)=g″₁(0) = 0.By using the a priori estimate and the energy estimates, we show that if the initial data is sufficiently small, the Cauchy problem (0.0.1)-(0.0.3) admits a unique global solution and the solution tends time-asymptotically to the corresponding nonlinear diffusion wave (v,u)(x,t) governed by the classical Darcy's law, where (v,u)(x,t) satisfiesFurthermore,L^p(2≤p≤+∞) convergence rates of the solutions are also obtained by using the Green function and energy estimates. (â…¡) : We consider the initial-boundary problem of the p-system with nonlinear dampingwith the initial data(v(x,0),u(x,0)) = (v₀(x),u₀(x))→(v₊,u₊), x→+∞, and v₊ > 0, (0.0.6)with the Dirichlet boundary conditionu|_x=0 = 0, (0.0.7)or the Neumann boundary conditionu_x|_x=0= 0, (0.0.8)whereα> 0 is a constant.Under the following assumptions:(B₁) p(v)∈C³(R⁺), p'(v) < 0, for all v > 0;(B₂) u₊ > 0, u₊ = 0;(B₃) the nonlinear function g(u)∈C²(R), and satisfies g(0) = g′(0) = 0.By using the a priori estimate and the energy estimates, we show that if the initial data is sufficiently small, the initial-boundary problem with the Dirichlet boundary condition (0.0.5)-(0.0.7) admits a unique global solution, and the solution tends time-asymptotically to the corresponding nonlinear diffusion wave (v^*, u^*)(x, t),where (v^*,u^*)(x,t) satisfiesFurthermore, the optimal L∞convergence rates of the solutions are also obtained by using the Green function.For the case of the Neumann boundary condition, we also get the existence and uniqueness of the smooth solution to the problem (0.0.5), (0.0.6) and (0.0.8). Furthermore, we consider the asymptotic behavior and the optimal L∞convergene rates for both the case of v₀(0) = v₊ and v₀(0)≠v₊.(â…¢): We consider the initial-boundary problem of the p-system with the linear dampingwith the initial data(u(x,0),u(x,0)) = (v₀(x),u₀(x))→(v₊,u₊), x→+∞, and u₊> 0, (0.0.11)with the Dirichlet boundary conditionu|_x=0 = 0, (0.0.12)whereα> 0 is a constant.We show that for a certain class of given large initial data, the initial-boundary problem (0.0.10)-(0.0.12) admits a unique global smooth solution and such a solution tends time-asymptotically to the nonlinear diffusion wave (v,u)(x,t), where (v,u)(x,t) satisfiesFurthermore, the optimal L² convergence rates of the solution are also obtained by using the Green function.