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We consider the Cauchy problem on nonlinear scalar conservation laws with adiffusion-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 sourceterm behaves as the usual diffusion term over the low frequency domain while itadmits on the high frequency part a feature of regularity-gain andregularity-loss for $s< 1$ and $s>1$, respectively. For all $s\in \R$, we notonly obtain the $L^p$-$L^q$ time-decay estimates on the linear solutionsemigroup but also establish the global existence and optimal time-decay ratesof small-amplitude classical solutions to the nonlinear Cauchy problem. In thecase of regularity-loss, the time-weighted energy method is introduced toovercome the weakly dissipative property of the equation. Moreover, thelarge-time behavior of solutions asymptotically tending to the heat diffusionwaves is also studied. The current results have general applications to severalconcrete models arising from physics.

Optimal Decay Rates to Conservation Laws with Diffusion-Type Terms of Regularity-gain and Regularity-loss