In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condition for the generalized Rosenau–Korteweg–de Vries–Burgers equation. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity. We can further obtain the same global asymptotic stability of the rarefaction wave to the generalized Rosenau–Benjamin–Bona–Mahony–Burgers equation with a third-order dispersive term as the former one.
Let $E$ be the attractor of an iterated function system ${\lbrace \phi_i (x) = \rho R_i x + a_i \rbrace}^N_{i=1}$ on $\mathbb{R}^d$, where $0 \lt \rho \lt 1$, $a_i \in \mathbb{R}^d$ and $R_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that ${\lbrace \phi_i \rbrace}^N_{i=1}$ satisfies the open set condition, but not the strong separation condition. We show that $E$ can not be generated by any iterated function system of similitudes satisfying the strong separation condition. This gives a partial answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.
In this paper we study a system of partial differential equations which models lithium-ion batteries. The system describes the conservation of Lithium and conservation of charges in the solid and electrolyte phases, together with the conservation of energy. The mathematical challenge is due to the fact that the reaction terms in the system involve the hyperbolic sine function along with possible degeneracy in one of the high-order terms. We obtain a local existence assertion for the initial boundary problem for the system. In particular, a lower bound for the blow-up time can be derived from our result. We hope that our analysis can lead to a deeper understanding of battery life.