Stability of planar diffusion wave for the quasilinear wave equation with nonlinear damping

Abstract

In this paper, we will show that under some smallness conditions, the planar diffusion wave $\bar v\left( {\frac{{x_1 }} {{\sqrt {1 + t} }}} \right)$ is stable for a quasilinear wave equation with nonlinear damping: v _tt − Δf(v) + v _t + g(v _t ) = 0, x = (x ₁, x ₂, ⋯, x _n ) ∈ ℝⁿ, where $\bar v\left( {\frac{{x_1 }} {{\sqrt {1 + t} }}} \right)$ is the unique similar solution to the one dimensional nonlinear heat equation: $v_t - f(v)_{x_1 x_1 } = 0,f'(v) > 0$, v(±∞, t) = v _±, v ₊ ≠ v ₋. We also obtain the L ^∞ time decay rate which reads $\left\| {v - \bar v} \right\|_{L^\infty } = O(1)(1 + t) - \tfrac{r} {4}$, where r = min{3, n}. To get the main result, the energy method and a new inequality have been used.