Parabolic Maximum Principle Research Articles

Harnack inequality and maximum principle for degenerate Kolmogorov operators in divergence form

We prove the parabolic strong maximum principle and the Harnack inequality for classical solutions to degenerate second order partial differential equations of the formLu:=∑i,j=1m0∂xi(aij∂xju)+∑j=1m0bj∂xju+cu+∑i,j=1Nbijxj∂xiu−∂tu=f, where 1≤m0≤N, A0=(aij)i,j=1,…,m0 is a bounded, symmetric and uniformly positive matrix and the matrix B:=(bij)i,j=1,…,N has real constant entries. Moreover, we assume low regularity (i.e. Hölder continuity) on the coefficients aij, bj and c, for i,j=1,…,m0. We point out the proofs of our main results only rely on the classical theory developed for harmonic functions.

Pointwise eigenvector estimates by landscape functions: Some variations on the Filoche–Mayboroda–van den Berg bound

AbstractLandscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue—much in the spirit of earlier results by Donsker–Varadhan and Bañuelos–Carrol—as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including ‐Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments.

Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non‐smooth coefficients

AbstractWe prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C1‐boundary, the other stated for sets with finite perimeter.

Spreading speeds for time heterogeneous prey–predator systems with diffusion

We investigate the large time behavior for two components reaction–diffusion systems of prey–predator type in a time varying environment. Here we assume that these variations in time exhibit an averaging property, which will be called mean value in this work. This framework includes in particular time periodicity, almost periodicity and unique ergodicity. We describe the spreading behavior of the prey and the predator, wherein the two populations are able to co-invade the empty space. Our analysis is based the parabolic strong maximum principle for scalar equation and on the derivation of local pointwise estimates that are used to compare the solutions of the prey–predator problem with those of a KPP scalar equation on suitable spatio-temporal domains.

A quantitative strong parabolic maximum principle and application to a taxis-type migration–consumption model involving signal-dependent degenerate diffusion

The taxis-type migration–consumption model accounting for signal-dependent motilities, as given by u_{t} = \Delta (u\phi(v)) , v_{t} = \Delta v-uv , is considered for suitably smooth functions \phi\colon[0,\infty)\to\R which are such that \phi>0 on (0,\infty) , but that in addition \phi(0)=0 with \phi'(0)>0 . In order to appropriately cope with the diffusion degeneracies thereby included, this study separately examines the Neumann problem for the linear equation V_{t} = \Delta V + \nabla\cdot ( a(x,t)V) + b(x,t)V and establishes a statement on how pointwise positive lower bounds for nonnegative solutions depend on the supremum and the mass of the initial data, and on integrability features of a and b . This is thereafter used as a key tool in the derivation of a result on global existence of solutions to the equation above, smooth and classical for positive times, under the mere assumption that the suitably regular initial data be nonnegative in both components. Apart from that, these solutions are seen to stabilize toward some equilibrium, and as a qualitative effect genuinely due to degeneracy in diffusion, a criterion on initial smallness of the second component is identified as sufficient for this limit state to be spatially nonconstant.

HEAT EQUATIONS DEFINED BY SELF-SIMILAR MEASURES WITH OVERLAPS

We study the heat equation on a bounded open set [Formula: see text] supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. The structures of these measures are in general complicated and intractable. However, for a class of such measures that we call essentially of finite type, important information about the structure of the measures can be obtained. We make use of this information to set up a framework to study the associated heat equations in one dimension. We show that the heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study the propagation speed problem.

Pointwise monotonicity of heat kernels

In this paper we present an elementary proof of a pointwise radial monotonicity property of heat kernels that is shared by the Euclidean spaces, spheres and hyperbolic spaces. The main result was discovered by Cheeger and Yau in 1981 and rediscovered in special cases during the last few years. It deals with the monotonicity of the heat kernel from special points on revolution hypersurfaces. Our proof hinges on a non straightforward but elementary application of the parabolic maximum principle. As a consequence of the monotonicity property, we derive new inequalities involving classical special functions.

Harnack estimates for a kind of porous medium equation

In this paper, we derive several differential Harnack estimates (also known as Li–Yau–Hamilton type estimates) for nonnegative solutions and positive solutions of the porous medium equation ft=ffxx+fx2+f2, which is a nonlinear parabolic equation. Our derivations rely on an idea related to the parabolic maximum principle. We prove these estimates by different methods. As the applications of the estimates, Harnack inequalities are also obtained.

Blow-up analysis for parabolic p-Laplacian equations with a gradient source term

In this work, we deal with the blow-up solutions of the following parabolic p-Laplacian equations with a gradient source term: {(b(u))t=∇⋅(|∇u|p−2∇u)+f(x,u,|∇u|2,t)in Ω×(0,t∗),∂u∂n=0on ∂Ω×(0,t∗),u(x,0)=u0(x)≥0in Ω‾,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (b(u) )_{t} =\\nabla \\cdot ( \\vert \\nabla u \\vert ^{p-2}\\nabla u )+f(x,u, \\vert \\nabla u \\vert ^{2},t) &\\text{in } \\varOmega \\times (0,t^{*}), \\\\ \\frac{\\partial u}{\\partial n}=0 &\\text{on } \\partial \\varOmega \\times (0,t^{*}),\\\\ u(x,0)=u_{0}(x)\\geq 0 & \\text{in } \\overline{\\varOmega }, \\end{cases} $$\\end{document} where p>2, the spatial domain varOmega subset mathbb{R}^{N} (Ngeq 2) is bounded, and the boundary ∂Ω is smooth. Our research relies on the creation of some suitable auxiliary functions and the use of the differential inequality techniques and parabolic maximum principles. We give sufficient conditions to ensure that the solution blows up at a finite time t^{*}. The upper bounds of the blow-up time t^{*} and the upper estimates of the blow-up rate are also obtained.

Bi-Laplacians on graphs and networks

We study the differential operator $A=\frac{d^4}{dx^4}$ acting on a connected network $\mathcal{G}$ along with $\mathcal L^2$, the square of the discrete Laplacian acting on a connected discrete graph $\mathsf{G}$. For both operators we discuss well-posedness of the associated {linear} parabolic problems \[ \frac{\partial u}{\partial t}=-Au,\qquad\frac{df}{dt}=-\mathcal L^2 f, \] on $L^p(\mathcal{G})$ or $\ell^p(\mathsf{V})$, respectively, for $1\leq p\leq\infty$. In view of the well-known lack of parabolic maximum principle for all elliptic differential operators of order $2N$ for $N>1$, our most surprising finding is that, after some transient time, the parabolic equations driven by $-A$ may display Markovian features, depending on the imposed transmission conditions in the vertices. Analogous results seem to be unknown in the case of general domains and even bounded intervals. Our analysis is based on a detailed study of bi-harmonic functions complemented by simple combinatorial arguments. We elaborate on analogous issues for the discrete bi-Laplacian; a characterization of complete graphs in terms of the Markovian property of the semigroup generated by $-\mathcal L^2$ is also presented.

Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions

This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: \t\t\t{(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} (h(u) )_{t} =\\nabla\\cdot (|\\nabla u|^{p-2}\\nabla u )+f(u) &\\mbox{in } D\\times(0,t^{*}), \\\\ \\frac{\\partial u}{\\partial n}=g(u) &\\mbox{on } \\partial D\\times (0,t^{*}), \\\\ u(x,0)=u_{0}(x)\\geq0 & \\mbox{in } \\overline{D}. \\end{cases} $$\\end{document} Here p>2, the spatial region D in mathbb{R}^{N} (Ngeq2) is bounded, and ∂D is smooth. We set up conditions to ensure that the solution must be a global solution or blows up in some finite time. Moreover, we dedicate upper estimates of the global solution and the blow-up rate. An upper bound for the blow-up time is also specified. Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique.

Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations

In this work, we consider a one-dimensional Itô diffusion process Xt with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by the observation of the expectation of the process during a small time interval, and starting from any value X0 in a given subset of . With the same type of observation, and given the drift coefficient, we also show that the diffusion coefficient is uniquely determined. When both coefficients are unknown, we show that they are simultaneously uniquely determined by the observation of the expectation and variance of the process, during a small time interval, and starting again from any value X0 in a given subset of . To derive these results, we apply the Feynman-Kac theorem which leads to a linear parabolic equation with unknown coefficients in front of the first and second order terms. We then solve the corresponding inverse problem with PDE technics which are mainly based on the strong parabolic maximum principle.

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and $2s$ in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. H\"older and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic H\"older spaces. Our methods involve the \textit{parabolic language of semigroups} and the Cauchy Integral Theorem, which are original to define the fractional powers of $\partial_t-\Delta$. Though we mainly focus in the equation $(\partial_t-\Delta)^su=f$, applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.

The parabolic infinite-Laplace equation in Carnot groups

By employing a Carnot parabolic maximum principle, we show existence-uniqueness of viscosity solutions to a class of equations modeled on the parabolic infinite Laplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity.

Periodic-parabolic eigenvalue problems with a large parameter and degeneration

We consider a periodic-parabolic eigenvalue problem with a non-negative potential λm vanishing on a non-cylindrical domain Dm satisfying conditions similar to those for the parabolic maximum principle. We show that the limit as λ→∞ leads to a periodic-parabolic problem on Dm having a periodic-parabolic principal eigenvalue and eigenfunction which are unique in some sense. We substantially improve a result from [Du and Peng, Trans. Amer. Math. Soc. 364 (2012), p. 6039–6070]. At the same time we offer a different approach based on a periodic-parabolic initial boundary value problem. The results are motivated by an analysis of the asymptotic behaviour of positive solutions to semilinear logistic periodic-parabolic problems with temporal and spacial degeneracies.

Harnack estimate for the Endangered Species Equation

We prove a differential Harnack inequality for the Endangered Species Equation, a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle. As an application of this inequality, we will show that positive solutions to this equation must blowup in finite time. We also use this inequality to partially answer a question of Hamilton, 2011.

Estimates of heat kernels for non-local regular Dirichlet forms

In this paper we present new heat kernel upper bounds for a certain class of non-local regular Dirichlet forms on metric measure spaces, including fractal spaces. We use a new purely analytic method where one of the main tools is the parabolic maximum principle. We deduce an off-diagonal upper bound of the heat kernel from the on-diagonal one under the volume regularity hypothesis, restriction of the jump kernel and the survival hypothesis. As an application, we obtain two-sided estimates of heat kernels for non-local regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than 2 2 .

The parabolic p-Laplace equation in Carnot groups

By establishing a parabolic maximum principle, we show uniqueness of viscosity solutions to the parabolic p-Laplace equation and then examine the limit as t goes to infinity. Additionally, we explore the limit as p goes to infinity. 1. Background and motivation In (7), uniqueness of viscosity solutions to a class of fully nonlinear subelliptic equations in Carnot groups was established. The key tool used was the Carnot Group Maximum Principle, which is a sub-Riemannian analog of the Euclidean version of (10). In particular, the following theorem was proved:

Stable estimation of two coefficients in a nonlinear Fisher–KPP equation

We consider the inverse problem of determining two non-constant coefficients in a nonlinear parabolic equation of the Fisher–Kolmogorov–Petrovsky–Piskunov type. For the equation ut = DΔu + μ(x) u − γ(x)u2 in (0, T) × Ω, which corresponds to a classical model of population dynamics in a bounded heterogeneous environment, our results give a stability inequality between the couple of coefficients (μ, γ) and some observations of the solution u. These observations consist in measurements of u: in the whole domain Ω at two fixed times, in a subset ω⊂⊂Ω during a finite time interval and on the boundary of Ω at all times t ∈ (0, T). The proof relies on parabolic estimates together with the parabolic maximum principle and Hopf’s lemma which enable us to use a Carleman inequality. This work extends previous studies on the stable determination of non-constant coefficients in parabolic equations, as it deals with two coefficients and with a nonlinear term. A consequence of our results is the uniqueness of the couple of coefficients (μ, γ), given the observation of u. This uniqueness result was obtained in a previous paper but in the one-dimensional case only.

Lower inequalities of heat semigroups by using parabolic maximum principle

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly ρ-local Dirichle form on the abstract metric measure space. As an application, we obtain lower estimates for heat kernels on some Riemannian manifolds.