This paper studies the asymptotic behavior of the solution to the Cauchy problem of a semilinear wave equation with damping $v_{tt}+v_t+f(Dv)=\mathrm{\Delta }v$, $x\in {\mathbb{R}}^n$, under some smallness conditions. By applying elementary energy method, we prove the solution of the above equation tends to the planar diffusion wave $\overline{v}(\frac{x_1}{\sqrt{1+t}})$ time-asymptotically, where $\overline{v}(\frac{x_1}{\sqrt{1+t}})$ is a self-similar solution of the one dimensional equation ${\overline{v}}_t+C_0{\overline{v}}_{x_1}^2={\overline{v}}_{x_1{x}_1},\overline{v}(\pm \infty ,t)=v_{\pm },v_{+}\ne v_{-}$, with $C_0=\frac{1}{2}\frac{{\partial }^{2}f(\xi )}{\partial {\xi }_1^2}{|}_{\xi =0}$. In addition, this paper gives the $L^{\infty }$ time decay rate, namely, ${\Vert v-\overline{v}\Vert }_{L^{\infty }}=O(1){\epsilon }_2{(1+t)}^{-\frac{\gamma }{4}}$, where $\gamma =\min \{3,n\}$.

中文翻译：

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带阻尼半线性波动方程的扩散波渐近稳定性

本文研究了具有阻尼的半线性波动方程柯西问题解的渐近行为 $v_{吨吨}+v_{吨}+F(Dv)=\mathrm{\Delta }v$, $X\in {\mathbb{电阻}}^n$，在一些小的条件下。应用初等能量法，证明上述方程的解趋于平面扩散波$\overline{v}(\frac{X_1}{\sqrt{1+吨}})$ 时间渐近，其中 $\overline{v}(\frac{X_1}{\sqrt{1+吨}})$ 是一维方程的自相似解 ${\overline{v}}_{吨}+C_0{\overline{v}}_{X_1}^2={\overline{v}}_{X_1{X}_1},\overline{v}(\pm \infty ,吨)=v_{\pm },v_{+}\ne v_{-}$， 和 $C_0=\frac{1}{2}\frac{{\partial }^{2}F(\xi )}{\partial {\xi }_1^2}{|}_{\xi =0}$. 此外，本文给出了${升}^{\infty }$ 时间衰减率，即 ${\Vert v-\overline{v}\Vert }_{{升}^{\infty }}=哦(1){\epsilon }_2{(1+吨)}^{-\frac{\gamma }{4}}$， 在哪里 $\gamma =\mathrm{分钟}\{3,n\}$.