Global structure of solutions toward the rarefaction waves for the Cauchy problem of the scalar conservation law with nonlinear viscosity

Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the generalized Benjamin-Bona-Mahony-Burgers equation with dissipative term

“…and obtained the global stability of the rarefaction wave for the case 0 < p < 3/7. Recently, Yoshida [60] obtained this rarefaction stability for more general case 0 < p < 1/3 and its precise time-decay estimates. For the rarefaction problem of the Korteweg-de Vries equation…”

“…Because the proofs of Lemmas 2.1 and 2.2 are given in [15], [16], [29], [32], [35], [50], [53], [60], [63] and so on, we omit the proofs here.…”

Global asymptotic stability of the rarefaction waves to the Cauchy problem for the generalized Rosenau–Korteweg–de Vries–Burgers equation

“…), respectively. Because the proofs of Lemmas 2.1 and 2.2 are given in [12], [13], [23], [24], [27], [40], [43], [50], [53] and so on, we omit the proofs here.…”

Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the scalar diffusive dispersive conservation laws

“…it is investigated by Harabetian [14], Matsumura-Nishihara [27], [28], Matsumura-Yoshida [29], [30], and Yoshida [37], [38], [39], [40], [41], [43], that the asymptotic stability and time-decay estimates of various nonlinear waves, such as rarefaction wave, viscous shock wave, viscous contact wave (see also [22], [44]), and multiwave pattern which consists of the rarefaction waves and the viscous contact wave. For the rarefaction problem of the Korteweg-de Vries equation…”