相关论文: Properties of Generalized Degenerate Parabolic Sys…

We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$ u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u) $$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution…

This paper studies a type of degenerate parabolic problem with nonlocal term \begin{equation*} \begin{cases} u_t=u^p(u_{xx}+u-\bar{u}) & 0<t<T_{{\max}},\ 0<x<a, u_x(0,t)=u_x(a,t)=0 & 0<t<T_{{\max}}, u(x,0)=u_0(x) & 0<x<a, \end{cases}…

Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely H\"older continuous…

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=\Omega\times (0,T)$, where $\Omega\subset…

This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}={\rm div}(|\nabla u|^{p-2}\nabla u)$ in $(0,\infty)\times\mathbb{R}^d$, $u(0,x)=u_0(x),$…

We consider the following parabolic approximation for hyperbolic system of conservation laws in 1-D with non-singular viscosity matrix $B(u)$ and $A(u)$ strictly hyperbolic,…

Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-\Delta_{p} u=\mu & \text{in}\…

We analyze a family of degenerate parabolic equations with linear growth Lagrangian having the form $u_t=\div (\varphi(u)\psi(\nabla u/u))$. Here $|\psi|\le 1$ and saturates at infinity. We present a simple and natural set of assumptions on…

Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular…

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt…

It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are H\"{o}lder continuous and satisfy the interior Harnack inequality. We show that even in the…

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely, $$ u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \ 0<s<1. $$ The problem is posed in $\{x\in\ren, t\in…

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…

We consider the non-degenerate second-order parabolic partial differential equations of non-divergence form with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to…

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and…

This paper is concerned with parabolic gradient systems of the form \[ u_t=-\nabla V (u) + \mathcal{D} u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, $\mathcal{D}$ denotes a fixed…

In this paper, we study the following chemotaxis--haptotaxis system with (generalized) logistic source $$ \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w),…

We establish partial regularity for vector-valued solutions to parabolic systems where the coefficients are possibly discontinuous with respect to (x,t). More precisely, we assume a VMO-condition with respect to the (x,t) and continuity…

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…

In this paper we study the initial boundary value problem for the system $\Delta v= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla…