Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss-Weierstrass semi-groups

Franka Baaske, Hans-Jürgen Schmeißer, Hans Triebel

Abstract

We present a new proof of the caloric smoothing related to the fractional Gauss-Weierstrass semi-group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss-Weierstrass semi-groups

Abstract

We present a new proof of the caloric smoothing related to the fractional Gauss-Weierstrass semi-group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.
Paper Structure (7 sections, 8 theorems, 124 equations, 4 figures)

This paper contains 7 sections, 8 theorems, 124 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Definitions and basic ingredients
  4. Characterizations of some function spaces in terms of fractional Gauss--Weierstrass semi--groups
  5. Smoothing properties
  6. Nonlinear fractional heat equations
  7. Comments and special cases

Key Result

Proposition 2.3

Let $\varphi_0$ be as in 2.3 and $\varphi \in C^\infty ({\mathbb R}^n \setminus \{0 \} )$ with $\lvert\varphi (x)\lvert >0$ if $1/2 \le \lvert x\lvert\, \le 2$. (i) Let $1 \le p,q \le \infty$ and $0<s<\sigma$. Let and Then $($usual modification if $q= \infty)$ are equivalent norms in $B^s_{p,q} ({\mathbb R}^n)$. (ii) Let $1 \le p<\infty$, $1 \le q\le \infty$, $0<s<\sigma$ and $a>n$. Let and

Figures (4)

  • Figure 1: $\frac{1}{2}<\alpha\leq 1$
  • Figure 2: $1<\alpha\leq\frac{n+2}{4}$
  • Figure 3: $\frac{n+2}{4}<\alpha\leq \frac{n}{2}+1$
  • Figure 4: $\alpha>\frac{n}{2}+1$

Theorems & Definitions (27)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7
  • proof
  • ...and 17 more