OPTIMAL DECAY RATES TO CONSERVATION LAWS WITH DIFFUSION-TYPE TERMS OF REGULARITY-GAIN AND REGULARITY-LOSS

Abstract

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ ℝ over the whole space ℝn for any spatial dimension n ≥ 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 ∈ ℝ, we not only obtain the Lp–Lq 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.