# American Institute of Mathematical Sciences

## Journals

JIMO
We propose a new smoothing technique based on discretization to solve semi-infinite variational inequalities. The proposed algorithm is tested by both linear and nonlinear problems and proven to be efficient.
keywords: Variational inequalities smoothing. semi-infinite
JIMO

This paper develops a multi-period product pricing and service investment model to discuss the optimal decisions of the participants in a supplier-dominant supply chain under uncertainty. The supply chain consists of a risk-neutral supplier and two risk-averse manufacturers, of which one manufacturer can provide real-time customer service based on the Internet of Things (IoT). In each period of the Stackelberg game, the supplier decides its wholesale price to maximize the profit while the manufacturers make pricing and service investment decisions to maximize their respective utility. Using the backward induction, we first investigate the effects of risk-averse coefficients and price sensitive coefficients on the optimal decisions of the manufacturers. We find that the decisions of one manufacturer are inversely proportional to both risk-averse coefficients and its own price sensitive coefficient, while proportional to the price sensitive coefficient of its rival. Then, we derive the first-best wholesale price of the supplier and analyze how relevant factors affect the results. A numerical example is conducted to verify our conclusions and demonstrate the advantages of the IoT technology in long-term competition. Finally, we summarize the main contributions of this paper and put forward some advices for further study.

keywords: Stackelberg game IoT pricing service investment risk
DCDS-B
In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.
keywords: free boundary a priori estimates mixed interpolation inequality. physical vacuum Compressible Euler equations
CPAA
In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
keywords: Cauchy problem Derivative nonlinear Schrödinger equations local well-posedness.
DCDS-B
We are concerned with the Cauchy problem for a viscous shallow water system with a third-order surface-tension term. The global existence and uniqueness of the solution in the space of Besov type is shown for the initial data close to a constant equilibrium state away from the vacuum by using the Friedrich's regularization and compactness arguments.
keywords: Shallow water equation with surface tension global-in-time solution. homogeneous Besov space Littlewood-Paley decomposition