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

In this paper, we will show that under some smallness conditions, the planar diffusion wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar v\left( {\frac{{x_1 }} {{\sqrt {1 + t} }}} \right)$$\end{document} is stable for a quasilinear wave equation with nonlinear damping: vtt − Δf(v) + vt + g(vt) = 0, x = (x1, x2, ⋯, xn) ∈ ℝn, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar v\left( {\frac{{x_1 }} {{\sqrt {1 + t} }}} \right)$$\end{document} is the unique similar solution to the one dimensional nonlinear heat equation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v_t - f(v)_{x_1 x_1 } = 0,f'(v) > 0$\end{document}, v(±∞, t) = v±, v+ ≠ v−. We also obtain the L∞ time decay rate which reads \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\| {v - \bar v} \right\|_{L^\infty } = O(1)(1 + t) - \tfrac{r} {4} $\end{document}, where r = min{3, n}. To get the main result, the energy method and a new inequality have been used.