In this talk we characterize the transonic stationary solutions of the hydrodynamic escape problem (HEP), which is significant on studying the evolution of atmospheres of planets. The characterization is established by the geometric singular perturbation method (GSP) on the wind solution of HEP. We will discuss the existence and non-existence of the adiabatic wind solution with or without heat ...

We prove the existence and uniqueness of solutions of weakly non-linear wave equations in the region x > 0; t > 0 under arbitrary initial and boundary conditions. We also establish the asymptotic validity of formal perturbation approximations to the solutions. The asymptotics of these equations require two slow scales, one spatial, one temporal. We formulate a multiple-scale perturbation proced...

Abstract. In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with n independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313–322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singul...

In this paper, in order to numerically solve for multiple positive solutions to a singularly perturbed Neumann boundary value problem in mathematical biology and other applications, a local minimax method is modified with new local mesh refinement and other strategies. Algorithm convergence and other related properties are verified. Motivated by the numerical algorithm and convinced by the nume...

We consider a nonlinear Dirichlet problem driven by the -Laplacian and with reaction consisting of singular term plus negative perturbation. Using regularization truncation comparison techniques, we show that has unique positive smooth solution.

In this paper, we study the existence and non-existence of normalized solutions to lower critical Choquard equation with a local perturbation $ \begin{equation*} \begin{cases} -\Delta u+\lambda u = \gamma (I_{\alpha}\ast|u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u+\mu |u|^{q-2}u, \quad {\rm{in}}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx c^2, \end{cases} \end{equation*} where \gamma, ...

Temperature dependence studies of electrochemical parameters provide insight into electron transfer processes. In cases where adding excess electrolyte causes experimental complications, e.g., colloidal systems, organic or biological samples, it is preferable to deal with the high resistivity medium. We validate use unsupported and weakly-supported solutions in thermoelectrochemical experiments...