We deal with $m$-component vector-valued solutions to the Cauchy problem for linear both homogeneous and nonhomogeneous weakly coupled second order parabolic system in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$. We assume that coefficients of the system are real and depending only on $t$, $n\geq 1$ and $T<\infty$. The homogeneous system is considered with initial data in $[L^p({\mathbb R}^n)]^m$, $1\leq p \leq \infty $. For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to $[L^p({\mathbb R}^{n+1}_T)]^m\cap [C^α\big (\overline{{\mathbb R}^{n+1}_T} \big )]^m $, $α\in (0, 1)$. Explicit formulas for the sharp coefficients in pointwise estimates for solutions of these problems and their directional derivative are obtained.
We obtain pointwise estimates for solutions of semilinear parabolic equations with a potential on connected domains both of $\mathbb R^n$ and of general Riemannian manifolds.
In this paper, we prove boundary pointwise $C^{k,α}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in obstacle-type problems.
We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results …
We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-Δu\leq v^λ\] \[0\leq v_t-Δv\leq u^σ\] in $Ω\times (0,1)$, where $λ$ and $σ$ are nonnegative constants and $Ω$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $λ$ and $σ$ such that solutions of this system satisfy pointwise bounds in compact subsets of $Ω$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.
We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic $p$-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic pro…
Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Hölder continuity of the coefficient functions and inhomogeneous term implies the Hölder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order "parabolic" equations of the form $$\partial_{t}u(x,t)-\sum_{i,j=1}^{m_1} a_{ij}(x,t)X_iX_ju(x,t)=f(x,t)$$ where …
In this paper we give both an historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developements of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introduci…
We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second order equations with real constant coefficients in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$ and $T<\infty$. The homogeneous equation is considered with initial data in $L^p({\mathbb R}^n)$, $1\leq p \leq \infty $. For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to $f\in L^p({\mathbb R}^{n+1}_T)\cap C^α\big (\bar{{\mathbb R}^{n+1}_T} \big ) $ , $p>n+2$ and $α\in (0, 1)$. Ex…
We investigate nonnegative solutions $u(x,t)$ and $v(x,t)$ of the nonlinear system of inequalities \[0\leq(\partial_t -Δ)^αu\leq v^λ\] \[ 0\leq (\partial_t -Δ)^βv\leq u^σ\] in $\mathbb{R}^n \times\mathbb{R}$, $n\geq 1$, satisfying the initial conditions \[ u=v=0\quad\text{ in }\mathbb{R}^n \times(-\infty,0) \] where $λ,σ,α$, and $β$ are positive constants. Specifically, using the definition of the fractional heat operator $(\partial_t-Δ)^α$ given in \cite{T}, we obtain, when they exist, optimal pointwise upper bounds on $\mathbb{R}^n \times(0,\infty)$ for nonnegative solutions $u$ and $v$ of t…
We study the behavior as $t\to 0^+$ of nonnegative functions \begin{equation}\label{0.1} u\in C^{2,1} (\mathbb{R}^n\times (0,1)) \cap L^λ(\mathbb{R}^n\times (0,1)),\quad n\ge 1, \end{equation} satisfying the parabolic Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq u_t-Δu\leq(Φ^{α/n}*u^λ)u^σ\quad \text{ in }B_1 (0)\times (0,1) \end{equation} where $α\in(0,n+2)$, $λ>0$, and $σ\geq0$ are constants, $Φ$ is the heat kernel, and $*$ is the convolution operation in $\mathbb{R}^n\times (0,1)$. We provide optimal conditions on $α,λ$, and $σ$ such that nonnegative solutions $u$ satis…
We investigate pointwise upper bounds for nonnegative solutions $u(x,t)$ of the nonlinear initial value problem \begin{equation}\label{0.1} 0\leq(\partial_t-Δ)^αu\leq u^λ\quad\text{ in }\mathbb{R}^n \times\mathbb{R},\,n\geq1, \end{equation} \begin{equation}\label{0.2} u=0\quad\text{in }\mathbb{R}^n\times(-\infty,0) \end{equation} where $λ$ and $α$ are positive constants. To do this we first give a definition---tailored for our study of this problem---of fractional powers of the heat operator $(\partial_t-Δ)^α:Y\to X$ where $X$ and $Y$ are linear spaces whose elements are real valued functions …
We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension $3$ and higher and for coefficients having a finite range of dependence, we prove a pointwise version of the two-scale expansion. A similar expansion is derived for elliptic equations in divergence form. The result is surprising, since it was not expected to be true without further symmetry assumptions on the law of the coefficients.
We consider the pointwise in space Lp-type regularity for elliptic and parabolic equations of order m in Rn. We provide pointwise Schauder estimates for the general range of Lp exponents, extending previous results from p > n/m to 1 < p < n/m.
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
We prove optimal pointwise Schauder estimates in the spatial variables for solutions of linear parabolic integro-differential equations. Optimal Hölder estimates in space-time for those spatial derivatives are also obtained.
We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.
We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and its applications to strongly coupled systems does not require higher $L^p$ integrability of derivatives.
This paper establishes sharp local regularity estimates for viscosity solutions of fully nonlinear parabolic free boundary problems with singular absorption terms. The main difficulties are due to the blow-up of the source along the free boundary and the lack of a variational structure. The proof combines the power of the Ishii-Lions method with intrinsically parabolic oscillation estimates. The results are new, even for second-order linear operators in nondivergence form.
Inspired by a recent pointwise differential inequality for positive bounded solutions of the fourth-order Hénon equation $Δ^2 u = |x|^a u^p$ in ${\mathbb R}^n$ with $a \geqslant 0$, $p > 1$, $n \geqslant 5$ due to Fazly, Wei, and Xu [ Anal. PDE., 8(2015) 1541--1563], first for some positive constants $α$ and $β$ we establish the following pointwise inequality \[ Δu \geqslant αu^{-\frac{q-1}2} + βu^{-1} |\nabla u|^2 \] in ${\mathbb R}^n$ with $n \geqslant 3$ for positive $C^4$-solutions of the fourth-order equation \[ Δ^2u=-u^{-q} \quad \text{ in } \mathbb R^n \] where $q > 1$. Next, we prove a…
We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a system of two parabolic equations, we establish that for data which are, in a suitable sense, smaller than the diffusion parameter $τ$ in the equation for the chemoattractant, we obtain global solutions, and for some data larger than $τ$ , a finite time blowup. In this way, we check that our size condition for the global existence is sharp for large $τ$ , up to a logarithmic factor.
We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form \begin{equation}\label{Meq}\tag{Eq} u_t- F(D^2u, Du, X, t) = f(X,t) \quad \mbox{in} \quad Q_1, \end{equation} where $F$ is elliptic with respect to the Hessian argument and $f \in L^{p,q}(Q_1)$. The quantity $κ(n, p, q):=\frac{n}{p}+\frac{2}{q}$ determines to which regularity regime a solution of \eqref{Meq} belongs. We prove that when $1< κ(n,p,q) < 2-ε_F$, solutions are parabolic-Hölder continuous for a sharp, quantitative exponent $0< α(n,p,q) < 1$. Precisely at the critical border…
We prove global gradient estimates for parabolic $p$-Laplace type equations with measure data, whose model is $$u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = μ\quad \textrm{in} \ Ω\times (0,T) \subset \mathbb{R}^n \times \mathbb{R},$$ where $μ$ is a signed Radon measure with finite total mass. We consider the singular case $$\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$$ and give possibly minimal conditions on the nonlinearity and the boundary of $Ω$, which guarantee the regularity results for such measure data problems.
Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）
We are interested in the three-dimensional quasilinear wave equations with null condition. Global existence and pointwise decay for this model have been proved in the celebrated works of Klainerman \cite{Klainerman86} and Christodoulou \cite{Christodoulou86} for small smooth initial data. In this work, we illustrate the precise pointwise asymptotic behavior of the solutions for initial data posed on a hyperboloid and show that the decay rate $v^{-1}u^{-1}$ is optimal for a generic set of initial data.
We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+Vu^q=f$, where $L$ is an elliptic operator in divergence form, $q\in\mathbb{R}\setminus \{0\}$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $Ω$ in $\mathbb{R}^{n}$, or in a weighted Riemannian manifold.
In this article, we obtain hessian estimates for Kolmogorov-Fokker-Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in Harmonic Analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative hessian estimates for the Kolmogorov-Fokker-Planck operators.
We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic $p$-Laplace equation $u_t-Δ_p u=μ$ with $p\in (1,2)$. The case when $p\in \big(2-\frac{1}{n+1},2\big)$ were studied in [15]. In this paper, we extend the results in [15] to the open case when $p\in \big(\frac{2n}{n+1},2-\frac{1}{n+1}\big]$ if $n\geq 2$ and $p\in(\frac{5}{4}, \frac{3}{2}]$ if $n=1$. More specifically, in a more singular range of $p$ as above, we establish pointwise gradient estimates via linear parabolic Riesz…
We study the boundary regularity properties and derive a priori pointwise supremum estimates of weak solutions and their derivatives in terms of suitable weighted $L^2$-norms for a class of degenerate parabolic equations that satisfy homogeneous Dirichlet boundary conditions on certain portions of the boundary. Such equations arise in population genetics in the study of models for the evolution of gene frequencies. Among the applications of our results is the description of the structure of the transition probabilities and of the hitting distributions of the underlying gene frequencies process…
