Tuoc Phan

Gradient Estimates and Global Existence of Smooth Solutions to a Cross-Diffusion System

We investigate the global time existence of smooth solutions for the Shigesada--Kawasaki--Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global

Local Gradient Estimates for Degenerate Elliptic Equations

W^{1,p}

Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation

-estimates of Calderon--Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing the Caffarelli--Peral perturbation technique together with a new two-parameter scaling argument.

Properties of Generalized Forchheimer Flows in Porous Media

Abstract This paper is focused on the local interior W 1 , ∞

Small solutions of nonlinear Schrödinger equations near first excited states

{W^{1,\infty}}

Asymptotic stability of solitary waves in generalized Gross–Neveu model

-regularity for weak solutions of degenerate elliptic equations of the form div ⁡ [ 𝐚 ⁢ ( x , u , ∇ ⁡ u ) ] + b ⁢ ( x , u , ∇ ⁡ u ) = 0

On small energy stabilization in the NLKG with a trapping potential

{\operatorname{div}[\mathbf{a}(x,u,\nabla u)]+b(x,u,\nabla u)=0}

Interior gradient estimates for quasilinear elliptic equations

, which include those of p-Laplacian type. We derive an explicit estimate of the local L ∞

Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

{L^{\infty}}

Weighted-

-norm for the solution’s gradient in terms of its local L p