Radial Heat Diffusion from the Root of a Homogeneous Tree and the Combinatorics of Paths

Joel M. Cohen; Mauro Pagliacci; Massimo A. Picardello

Displaying similar documents to “Radial Heat Diffusion from the Root of a Homogeneous Tree and the Combinatorics of Paths”

Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity

María F. Natale, Domingo A. Tarzia (2006)

Bollettino dell'Unione Matematica Italiana

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We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity with a constant temperature or a heat flux condition of the type $-q_{0}/\sqrt{t}$ ( $q_{0}>0$ ) at the fixed face $x=0$ . We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for $t\geq t_{0}>0$ with $t_{0}$ an arbitrary positive time. These explicit solutions are obtained through the unique solution of an integral equation with the time...

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...

Weighted norm estimates for the maximal operator of the Laguerre functions heat diffusion semigroup

R. Macías, C. Segovia, J. L. Torrea (2006)

Studia Mathematica

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We obtain weighted $L^{p}$ boundedness, with weights of the type $y^{δ}$ , δ > -1, for the maximal operator of the heat semigroup associated to the Laguerre functions, ${ℒ_{k}^{α}}_{k}$ , when the parameter α is greater than -1. It is proved that when -1 < α < 0, the maximal operator is of strong type (p,p) if p > 1 and 2(1+δ)/(2+α) < p < 2(1+δ)/(-α), and if α ≥ 0 it is of strong type for 1 < p ≤ ∞ and 2(1+δ)/(2+α) < p. The behavior at the end points of the intervals where there is strong...

Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$

Grzegorz Łysik (2003)

Colloquium Mathematicae

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It is proved that the solution to the initial value problem $\partial_{t} u = \partial ²_{x} u + u ²$ , u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^{s}$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, Shang-Yuan Shiu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a family of nonlinear stochastic heat equations of the form $\partial_{t} u = ℒ u + σ (u) \dot{W}$ , where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$ , and $σ$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$ . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒ f = c f^{''}$ for some $c g t; 0$ , we prove that if $u_{0}$ is a finite measure of compact support, then the...

Blow up for a completely coupled Fujita type reaction-diffusion system

Noureddine Igbida, Mokhtar Kirane (2002)

Colloquium Mathematicae

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This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $u ₜ - Δ (a_{11} u) = h (t, x) {| v |}^{p}$ , $v ₜ - Δ (a_{21} u) - Δ (a_{22} v) = k (t, x) {| w |}^{q}$ , $w ₜ - Δ (a_{31} u) - Δ (a_{32} v) - Δ (a_{33} w) = l (t, x) {| u |}^{r}$ , for $x \in ℝ^{N}$ , t > 0, p > 0, q > 0, r > 0, $a_{i j} = a_{i j} (t, x, u, v)$ , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x \in ℝ^{N}$ , where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Nguyen Ngoc Khanh (2016)

Archivum Mathematicum

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In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M, g)$ for the following general heat equation $u_{t} = Δ_{V} u + a u log u + b u$ where $a$ is a constant and $b$ is a differentiable function defined on $M \times [0, \infty)$ . We suppose that the Bakry-Émery curvature and the $N$ -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently. ...

Uniform analytic-Gevrey regularity of solutions to a semilinear heat equation

Todor Gramchev, Grzegorz Łysik (2008)

Banach Center Publications

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We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation $\partial_{t} u - Δ u = u^{M}$ . The approach is based on suitable iterative fixed point methods in $L^{p}$ based Banach spaces with anisotropic Gevrey norms with respect to the time and the space variables. We also construct explicit solutions uniformly analytic in t ≥ 0 and x ∈ ℝⁿ for some conservative nonlinear terms with symmetries.

The Cauchy problem for a strongly degenerate quasilinear equation

F. Andreu, Vicent Caselles, J. M. Mazón (2005)

Journal of the European Mathematical Society

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We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation $u_{t} = div 𝐚 (u, D u)$ , where $𝐚 (z, ξ) = \nabla_{ξ} f (z, ξ)$ , and $f$ is a convex function of $ξ$ with linear growth as $∥ ξ ∥ \to \infty$ , satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.

The tree property at both $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$

Laura Fontanella, Sy David Friedman (2015)

Fundamenta Mathematicae

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We force from large cardinals a model of ZFC in which $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model $ℵ_{ω + 2}$ even satisfies the super tree property.

A note on the cubical dimension of new classes of binary trees

Kamal Kabyl, Abdelhafid Berrachedi, Éric Sopena (2015)

Czechoslovak Mathematical Journal

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The cubical dimension of a graph $G$ is the smallest dimension of a hypercube into which $G$ is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with $2^{n}$ vertices, $n \geq 1$ , is $n$ . The 2-rooted complete binary tree of depth $n$ is obtained from two copies of the complete binary tree of depth $n$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice...

Property C for ODE and Applications to an Inverse Problem for a Heat Equation

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $ℓ_{j} : = - d ² / d x ² + k ² q_{j} (x)$ , k = const > 0, j = 1,2, $0 < e s s i n f q_{j} (x) \leq e s s s u p q_{j} (x) < \infty$ . Suppose that (*) $\int_{0}^{1} p (x) u ₁ (x, k) u ₂ (x, k) d x = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_{j}$ solves the problem $ℓ_{j} u_{j} = 0$ , 0 ≤ x ≤ 1, $u_{j}^{'} (0, k) = 0$ , $u_{j} (0, k) = 1$ . It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

Second order elliptic operators with complex bounded measurable coefficients in $L^{p}$ , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$ , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^{p}$ , Sobolev, and some new Hardy spaces naturally associated to $L$ . First, we show...

Self-similar solutions in reaction-diffusion systems

Joanna Rencławowicz (2003)

Banach Center Publications

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In this paper we examine self-similar solutions to the system $u_{i t} - d_{i} Δ u_{i} = \prod_{k = 1}^{m} u_{k}^{p_{k}^{i}}$ , i = 1,…,m, $x \in ℝ^{N}$ , t > 0, $u_{i} (0, x) = u_{0 i} (x)$ , i = 1,…,m, $x \in ℝ^{N}$ , where m > 1 and $p_{k}^{i} > 0$ , to describe asymptotics near the blow up point.

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{- β}$ , with $β \in [0, \infty)$ . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$ . If $β \in [0, 1)$ , the hydrodynamic limit is given by the usual heat equation. If $β = 1$ , it is given by a parabolic equation involving an operator...

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

S. Thangavelu (2002)

Colloquium Mathematicae

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Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{δ, j} : δ \in K ̂ ₀, 1 \leq j \leq d_{δ}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_{t}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{δ} (i λ + ϱ)$ be the Kostant polynomials. We establish the following...

Observability inequalities and measurable sets

Jone Apraiz, Luis Escauriaza, Gengsheng Wang, C. Zhang (2014)

Journal of the European Mathematical Society

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This paper presents two observability inequalities for the heat equation over $Ω \times (0, T)$ . In the first one, the observation is from a subset of positive measure in $Ω \times (0, T)$ , while in the second, the observation is from a subset of positive surface measure on $\partial Ω \times (0, T)$ . It also proves the Lebeau-Robbiano spectral inequality when $Ω$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

Shadow trees of Mandelbrot sets

Virpi Kauko (2003)

Fundamenta Mathematicae

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The topology and combinatorial structure of the Mandelbrot set $ℳ^{d}$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in $ℳ^{d}$ . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, $Λ^{d}$ . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence...

Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen, Panki Kim, Renming Song (2010)

Journal of the European Mathematical Society

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We consider the fractional Laplacian $- {(- Δ)}^{α / 2}$ on an open subset in $ℝ^{d}$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in $C^{1, 1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $α$ -stable process killed upon leaving a $C^{1, 1}$ open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

Jan J. Dijkstra, Kirsten I. S. Valkenburg (2010)

Fundamenta Mathematicae

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One way to generalize complete Erdős space $_{c}$ is to consider uncountable products of zero-dimensional $G_{δ}$ -subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise...

Porous medium equation and fast diffusion equation as gradient systems

Samuel Littig, Jürgen Voigt (2015)

Czechoslovak Mathematical Journal

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We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot{u} - Δ u^{m} = f$ , with $m \in (0, \infty)$ , can be modeled as a gradient system in the Hilbert space $H^{- 1} (Ω)$ , and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $Ω \subseteq ℝ^{n}$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.