Perturbation Effects on the Decay of Discontinuous Solutions of Nonlinear First Order Wave Equations

Abstract

The decay of discontinuous solutions of quasi-linear hyperbolic equations of the form \[ u_t + g(u)u_x + \lambda h(u) = 0,\quad \lambda > 0,\quad g_u (u) > 0,\quad h_u (u) > 0\quad {\text{for}}\,u > 0\], where subscripts denote differentiation and $g(u), h(u)$ are nonnegative, is considered from a pedagogical point of view. For a given finite initial disturbance, zero outside a finite range in x, it is shown that if $h(u) = O(u^\alpha ),\alpha > 0$, for $0 < u \ll 1$, the initial disturbance decays (i) within a finite time and finite distance for $0 < \alpha < 1$ and is unique under certain conditions, (ii) within an infinite time, like $O(\exp - \lambda t)$, and in a finite distance for $\alpha = 1$, and (iii) within an infinite time and distance like $O(t^{ - 1/(\alpha - 1)} )$ for $1 < \alpha \leqq 3$, and $O(t^{ - 1/2} )$ for $\alpha \geqq 3$. The asymptotic speed of propagation of the discontinuity is given in each case together with its role in the decay process. A boundary value problem for stress wave propagation in a mildly nonlinear Maxwell rod with a finite nonlinear viscous damping is considered. The first term in an asymptotic solution, in the magnitude of the wave speed nonlinearity, is shown to be equivalent to that of the above equation with all its implications on decay now applying to the stress wave in the rod. The effect of a higher order perturbation on decay is studied by considering, by way of example, the initial value problem for \[ u_t + g(u)u_x = \varepsilon u_{xx} ,\quad \varepsilon \ll 1,\quad g_u (u) > 0, \] with $g(u)$ nonnegative. For initial data comparable to the above the effect on the asymptotic speed of propagation for t large is found to be $O(\varepsilon t^{ - 1/ 2} \log t)$ while it is $O(\varepsilon t^{ - 1} \log t)$ for the decay of the solution.