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Large time behaviour for the heat equation on $\Z,$ moments and decay rates

Luciano Abadias, Jorge González-Camus, Pedro J. Miana, Juan C. Pozo

Abstract

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh $\Z$ on $\ell^p$ spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and $\ell^p$ decay results for the fundamental solution. We use that estimates to get rates on the $\ell^p$ decay and large time behaviour of solutions. For the $\ell^2$ case, we get optimal decay by use of Fourier techniques.

Large time behaviour for the heat equation on $\Z,$ moments and decay rates

Abstract

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh on spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and decay results for the fundamental solution. We use that estimates to get rates on the decay and large time behaviour of solutions. For the case, we get optimal decay by use of Fourier techniques.
Paper Structure (10 sections, 8 theorems, 142 equations, 1 figure, 1 table)

This paper contains 10 sections, 8 theorems, 142 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and known results
  3. Moments of Bessel functions
  4. Pointwise and $\ell^p$ estimates of the fundamental solution
  5. $\ell^p$-$\ell^q$ theory
  6. Large time behaviour of solutions
  7. Graphics, open questions, and final comments
  8. The sequence of polynomials $(p_k)_{k\ge 0}$
  9. Decayment of finite differences of arbitrary order
  10. Self similar form for the heat semigroup

Key Result

Lemma 3.1

Let $(p_k)_{k\ge 0}$ be the sequence of polynomials $(p_k)_{k\ge 0}$ given by $p_0(t):=1$ and for $k\ge 1$ and $t\in \mathbb{{C}}$. Then the polynomial $p_k$ has positive integer coefficients, its degree is $k$, and $p_k(t)> 0$ for $t> 0$. For $k\ge 1$, we write then $a_{k,1}=1$, and for $2\le n\le k$; in particular $a_{k,2}= 4^{k-1}-1$, $a_{k, k}=(2k-1)!!$.

Figures (1)

  • Figure 1:

Theorems & Definitions (22)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Remark 3.5
  • Remark 3.6
  • Lemma 4.1
  • ...and 12 more