Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $ω_{\varepsilon}=(x_{0}-\varepsilon, x_{0}+\varepsilon )$, in the limit $\varepsilon\rightarrow 0$, where $x_{0}\in (0,1)$. It is known that depending on arithmetic properties of $x_{0}$, there may exist a minimal time $T_{0}$ of pointwise control at $x_{0}$ of the heat equation. Besides, for any $\varepsilon$ fixed, the heat equation is controllable with control set $ω_{\varepsilon}$ in any time $T>0$. We relate these two phenomena. We show that the observability constant on $ω_\varepsilon$ does not converge to $0$ as $\varepsilon\rightarrow 0$ at the same speed when $T>T_{0}$ (in which case it is comparable to $\varepsilon^{1/2}$) or $T<T_{0}$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^{2}$ null-controls on $ω_{\varepsilon}$ in the limit $\varepsilon \rightarrow 0$.
This paper is concerned with the kinetic model of Othmer-Dunbar-Alt for bacterial motion. Following a previous work, we apply the dispersion and Strichartz estimates to prove global existence under several borderline growth assumptions on the turning kernel. In particular we study the kinetic model with internal variables taking into account the complex molecular network inside the cell.
On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash inequalities. We are able to treat more general pointwise estimates,…
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 apply Davies' method for obtaining pointwise lower bounds on the heat kernels of higher-order differential operators to obtain pointwise lower bounds in the presence of a polynomialy bounded potential.
Let $u(t,x)$ be a solution of the heat equation in $\mathbb{R}^n$. Then, each $k-$th derivative also solves the heat equation and satisfies a maximum principle, the largest $k-$th derivative of $u(t,x)$ cannot be larger than the largest $k-$th derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction $-Δϕ_k = λ_k ϕ_k$ with Dirichlet conditions on sm…
We find optimal integrability conditions on the initial data $f$ for the existence of solutions $e^{-tΔ_λ}f(x)$ and $e^{-t\sqrt{Δ_λ}}f(x)$ of the heat and Poisson initial data problems for the Bessel operator $Δ_λ$ in $\mathbb{R}^{+}$. We also characterize the most general class of weights $v$ for which the solutions converge a.e. to $f$ for every $f\in L^{p}(v)$, with $1\le p<\infty$. Finally, we show that for such weights and $1<p<\infty$ the local maximal operators are bounded from $L^{p}(v)$ to $L^{p}(u)$, for some weight $u$.
We obtain pointwise lower bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in $\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close to the boundary. We make no smoothness assumptions on our operator coefficients which we assume only to be bounded and measurable.
In this paper the authors present a proof of a pointwise radial monotonicity property of heat kernels that is shared by the euclidean spaces, spheres and hyperbolic spaces. The main result deals with monotonicity from special points on revolution hypersurfaces from which the aforementioned are deduced. The proof relies on a non straightforward but elementary application of the parabolic maximum principle.
We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of $Δu -u_t$ at the origin is Dini (in $L^p$ average), then the origin is a Lebesgue point of continuity (still in $L^p$ average) for $D^2 u$ and $\dd_t u$. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right hand side. In particular, we prove that the solution to the obstacle problem has, at regular points of the free boundary, a Taylor expansion up to order two in space and one in time (in the $L^p$ average). Moreo…
In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays wit…
Let $p:C\to R$ be a subharmonic, nonharmonic polynomial and $τ\in R$ a parameter. Define $\bar Z_{τp} = \partial_{\bar z} + τp_{\bar z} = e^{-τp} p_{\bar z} e^{τp}$, a closed, densely defined operator on $L^2(C)$. If $\Box_{τp} = \bar Z_{τp}\bar Z^*_{τp}$ and $\tilde\Box_{τp} = \bar Z^*_{τp}\bar Z_{τp}$, we solve the heat equations $\partial_s u + \Box_{τp} u=0$, $u(0,z)=f(z)$ and $\partial_s \tilde u + \tilde\Box_{τp} \tilde u=0$, $\tilde u(0,z) = \tilde f(z)$. We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We…
We consider the the pointwise estimates and the blow-up rate estimates for the zero Dirchilet problem of the semilinear heat equation with a gradient term.
Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities
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…
By working with the periodic resolvent kernel and Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations.With our linearized estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, we obtain $L^p$- behavior($p \geq 1$) of a nonlinear solution to a perturbation equation of a reaction-diffusion equation with respect to initial data in $L^1 \cap H^1$ recovering and slightly sharpening results obtained by Schneider using weighted en…
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.
Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $L^p$. Derivation of the coefficients is based on solving certain optimization problems with respect to a vector parameter inside of an integral over the unit sphere.
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…
The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case
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 …
The purpose of this article is to establish regularity and pointwise upper bounds for the (relative) fundamental solution of the heat equation associated to the weighted dbar-operator in $L^2(C^n)$ for a certain class of weights. The weights depend on a parameter, and we find pointwise bounds for heat kernel, as well as its derivatives in time, space, and the parameter. We also prove cancellation conditions for the heat semigroup. We reduce the $n$-dimensional case to the one-dimensional case, and the estimates in one-dimensional case are achieved by Duhamel's principle and commutator properti…
In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise c…
Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.
Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier--Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, % under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection…
Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients and a possibly unbounded potential term.
We obtain pointwise upper bounds on the derivatives of the heat kernel on Damek-Ricci spaces. Applying these estimates we prove the $L^p$-boundedness of Littlewood-Paley-Stein operators.
We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation $\partial_t u +(-Δ)^s u=0$, $0<s<1$, posed in the whole space $\mathbb{R}^N$ with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions, which is given in one direction by the representation formula, in the other one by the initial trace. We review many of the typical properties of the solutions, in p…
In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $\mathbb{R}^d$ and a complete Riemannian manifold $\mathrm{M}$. On one hand, in $\mathbb{R}^d$, we prove that any solution $u=u(t,x)$ to $u_t(t,x)-\mathrm{L}_α^κ u(t,x)=0$, where $\mathrm{L}_α^κ$ is a nonlocal operator of order $α$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|\leq C(1+|x|)^{α-ε}$ for any $(t,x)\in (0,1]\times \mathbb{R}^d$ and $ε\in(0,α)$. We also obtain pointwise estimates for $\partial_t^kp_α(t,x;y)$, where $p_α(t,x;y)$ is the f…
We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order $2$ on $\M$. Here $\M$ is $\R^d$ or a complete noncompact manifold with Ricci curvature bounded from below by a constant. An implication is a sharp solvability condition for the Cauchy problem of the backward heat equation, which is a well known ill-posed problem. Another implication is a sharp criteria for time analyticity of solutions down to the initial time. The other pe…
We study pointwise and $L^p$ gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness results on $L^p$ spaces for the heat operator of the Hodge Laplacian on differential forms.
We give pointwise upper estimate for the gradient of the heat kernel on some fractal-like cable systems including the Vicsek and the Sierpiński cable systems. Applications to $L^p$-boundedness of quasi-Riesz transforms are derived.
In this paper, we recover the boundary null controllability for the degenerate heat equation by analyzing the asymptotic behavior of an eligible family of state-control pairs $((u_{\varepsilon}, h_{\varepsilon}))_{\varepsilon >0}$ solving corresponding singularly perturbed internal null controllability problems. As in other situations studied in the literature, our approach relies on Carleman estimates and meticulous weak convergence results. However, for the degenerate parabolic case, some specific trace operator inequalities must be obtained, in order to justify correctly the passage to the …
In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of $RCD^*(K,N)$-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in $RCD^*(K,N)$-spaces. We use then these results to initiate the study of Weyl's law in the $RCD$ setting
We build the iDLA cluster using drifted random walks, and study the limiting shapes they exhibit, with the help of sandpile models. For constant drift, the normalised cluster converges to a canonical shape S, which can be termed a true heat ball, in that it gives rise to a mean value property for caloric functions. The existence and boundedness of such a shape answers a natural yet open question in PDE theory.
We consider pointwise linear elliptic equations of the form $\mathrm{L}_x u_x = η_x$ on a smooth compact manifold where the operators $\mathrm{L}_x$ are in divergence form with real, bounded, measurable coefficients that vary in the space variable $x$. We establish $\mathrm{L}^{2}$-continuity of the solutions at $x$ whenever the coefficients of $\mathrm{L}_x$ are $\mathrm{L}^{\infty}$-continuous at $x$ and the initial datum is $\mathrm{L}^{2}$-continuous at $x$. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a …
Long time existence and uniqueness of solutions to the Yang-Mills heat equation have been proven over a compact 3-manifold with boundary for initial data of finite energy. In the present paper we improve on previous estimates by using a Neumann domination technique that allows us to get much better pointwise bounds on the magnetic field. As in the earlier work, we focus on Dirichlet, Neumann and Marini boundary conditions. In addition, we show that the Wilson Loop functions, gauge invariantly regularized, converge as the parabolic time goes to infinity.
