相关论文: H\"older estimates of weak solutions to chemotaxis…

In this note we consider degenerate chemotaxis systems with porous media type diffusion and a source term satisfying the Hadamard growth condition. We prove the H\"older regularity for bounded solutions to parabolic-parabolic as well as for…

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized…

This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a…

This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption:…

Motivated by models for biofilm growth, we consider Cauchy problems for quasilinear reaction diffusion equations where the diffusion coefficient has a porous medium type degeneracy as well as a singularity. We prove results on the…

We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla (-\Delta)^{-1/2}u).$ For definiteness, the problem is posed…

We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-\Delta)^s(|u|^{m-1}u) = 0…

This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller-Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter…

We study the boundedness and convergence to equilibrium of weak solutions to reaction-diffusion systems with nonlinear diffusion. The nonlinear diffusion is of porous medium type and the nonlinear reaction terms are assumed to grow…

This paper deals with a parabolic-elliptic chemotaxis system with nonlinear diffusion. It was proved that there exists a solution of a Cahn-Hilliard system as an approximation of a nonlinear diffusion equation by applying an abstract theory…

We investigated existence of global weak solutions for a system of chemotaxis type with nonlinear degenerate diffusion, arising in modelling Multiple Sclerosis disease. The model consists of three equations describing the evolution of…

We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If…

We show interior H\"older continuity for a class of quasi-linear degenerate reaction-diffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term…

In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis in $\mathbb{R}$. Under some suitable structural assumption on the pressure function, we first…

We study the weak solvability of a quasilinear reaction-diffusion system nonlinearly coupled with an linear elliptic system posed in a domain with distributed microscopic balls in $2D$. The size of these balls are governed by an ODE with…

In this paper we investigate regularity aspects for solutions of the nonlinear parabolic equation $$ u_t= \Delta u^m, \quad m > 1 $$ usually called the porous medium equation. More precisely, we provide sharp regularity estimates for…

This paper proposes a novel reaction-diffusion system approximation tailored for singular diffusion problems, typified by the fast diffusion equation. While such approximation methods have been successfully applied to degenerate parabolic…

We consider the chemotaxis-fluid system \begin{align}\label{star}\tag{$\diamondsuit$} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n^m-\nabla\!\cdot(n\nabla c),\ &x\in\Omega,& t>0,\\…

In this paper, we study a nutrient-taxis model with porous medium slow diffusion \begin{align*} \left\{ \begin{aligned} &u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)+\xi uv-\rho u, \\ &v_t-\Delta v=-vu+\mu v(1-v), \end{aligned}\right.…

In this paper, we utilize the De Giorgi iteration to quantitatively analyze the upper bound of solutions for Keller-Segel type systems. The refined upper bound estimate presented here has broad applications in determining large time…