Shock reflection for the damped $P$-system

The global existence and the asymptotic behavior of the weak entropy solution, the piecewise smooth solution with one shock discontinuity, on a strip domain is investigated in the present paper. We show that, for small smooth initial data and boundary value with only one small jump at $\left ( x, t \right ) = \left ( 0, 0 \right )$, the piecewise smooth solution with one shock discontinuity exists globally in time. The shock discontinuity begins from $\left ( x, t \right ) = \left ( 0, 0 \right )$, moves forward and reflects in a finite time at the boundary $x = 1$ to form a 1-shock, which goes backward and reflects at $x = 0$ also in a finite time to create a new 2-shock. The shock strength decays exponentially and never disappears in finite time. As $t \to \infty$, this solution converges to a constant state determined by the initial and the boundary conditions.