Optimal Decay Rates to Conservation Laws with Diffusion-Type Terms of   Regularity-gain and Regularity-loss
  We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index $s\in \R$ over the whole space $\R^n$ for any spatial dimension $n\geq 1$. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for $s< 1$ and $s>1$, respectively. For all $s\in \R$, we not only obtain the $L^p$-$L^q$ time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics. 
Renjun Duan, Lizhi Ruan and Changjiang Zhu
math.AP