In this paper, we investigate the qualitative properties of positive solutions for a two-coupled elliptic system in the punctured space. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity.

We derive an exact solution for Stokes flow in an in a channel with permeable walls. We assume that at the channel walls, the normal component of the fluid velocity is described by Darcy's law and the tangential component of the fluid velocity is described by the no slip condition. The pressure exterior to the channel is assumed to be constant. Although this problem has been well studied, typical studies assume that the permeability of the wall is small relative to other non-dimensional parameters; this work relaxes this assumption and explores a regime in parameter space that has not yet been well studied. A consequence of this relaxation is that transverse velocity is no longer necessarily small when compared with the axial velocity. We use our result to explore how existing asymptotic theories break down in the limit of large permeability.

We establish an optimal error estimate for a random particle blob method for the Keller-Segel equation in Rd (d  2). With a blob size " = h

Let G=(V,E) be a locally finite connected weighted graph, and Ω be an unbounded subset of V. Using Rothe's method, we study the existence of solutions for the semilinear heat equation ∂_t u+|u|^{p−1}⋅u=Δu (p≥1) and the parabolic variational inequality ∫_{Ω^∘} ∂_t u⋅(v−u)dμ ≥ ∫_{Ω^∘} (Δu+f)⋅(v−u)dμ for any v∈H, where H={u∈W^{1,2}(V):u=0 on V∖Ω^∘}.

In this article, we investigate the global stability of the wave patterns with the superposition of viscous contact wave and rarefaction wave for the one-dimensional compressible Navier-Stokes equations with a free boundary. It is shown that for the ideal polytropic gas, the superposition of the viscous contact wave with rarefaction wave is nonlinearly stable for the free boundary problem under the large initial perturbations for any > 1 with being the adiabatic exponent provided that the wave strength is suitably small.