This note studies the Monge-Ampère Keller–Segel equation in a periodic domain , a fully nonlinear modification of the Keller–Segel equation where the Monge-Ampère equation substitutes for the usual Poisson equation . The existence of global weak solutions is obtained for this modified equation. Moreover, we prove the regularity in for some .

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 studied coupled systems of the Fokker--Planck equation and the Navier--Stokes equation modeling the Hookean and the finitely extensible nonlinear elastic (FENE)-type polymeric flows. We proved the continuous embedding and compact embedding theorems in weighted spaces that naturally arise from related entropy estimates. These embedding estimates are shown to be sharp. For the Hookean polymeric system with a center-of-mass diffusion and a superlinear spring potential, we proved the existence of a global weak solution. Moreover, we were able to tackle the FENE model with $L^2$ initial data for the polymer density instead of the $L^\infty$ counterpart in the literature.

In this paper, we study the Li stability of perturbation of constant states for 2 x 2 systems of conservation laws. We introduce a nonlinear functional which is equivalent to the Li norm of the difference between the constant state and the approximate solutions consisting of elementary waves and is non-increasing in time for the limiting weak solution. This yields the Li stability of the constant states. The approximate solutions are constructed with the aid of wave tracing for the random choice method. Our functional reveals the aspects of nonlinear wave behavior, particularly the coupling of waves pertaining to different characteristic families, which affects the Li norm of the solutions.

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^∘}.