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The time-decay rate toward the viscous shock wave for scalar viscous conservation law $$u_t+ f(u)_x =\mu u_{xx}$$ is obtained in this paper through an $L^p$ estimate and the area inequality in [1] provided that the initial perturbations are small, i.e., $||\Phi_0||_{H^2}≤ε,$ where $\Phi_0$ is the anti-derivative of the initial perturbation. It is noted that there is no additional weighted requirement on $\Phi_0,$ i.e., $\Phi_0(x)$ only belongs to $H^2 (R).$
}, issn = {2790-1939}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmaa/20662.html} }The time-decay rate toward the viscous shock wave for scalar viscous conservation law $$u_t+ f(u)_x =\mu u_{xx}$$ is obtained in this paper through an $L^p$ estimate and the area inequality in [1] provided that the initial perturbations are small, i.e., $||\Phi_0||_{H^2}≤ε,$ where $\Phi_0$ is the anti-derivative of the initial perturbation. It is noted that there is no additional weighted requirement on $\Phi_0,$ i.e., $\Phi_0(x)$ only belongs to $H^2 (R).$