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Local existence and nonexistence of solutions to the Hardy parabolic equation with general nonlinearity

Yo Tsusaka

Abstract

In this paper, we consider the Cauchy problem for the Hardy parabolic equation with general nonlinearity and establish the local existence and nonexistence results. Our results provide the optimal integrability conditions on initial function for the existence of a local-in-time nonnegative solution. The proof of the existence result is based on the supersolution method.

Local existence and nonexistence of solutions to the Hardy parabolic equation with general nonlinearity

Abstract

In this paper, we consider the Cauchy problem for the Hardy parabolic equation with general nonlinearity and establish the local existence and nonexistence results. Our results provide the optimal integrability conditions on initial function for the existence of a local-in-time nonnegative solution. The proof of the existence result is based on the supersolution method.
Paper Structure (4 sections, 15 theorems, 122 equations, 1 figure)

This paper contains 4 sections, 15 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence results
  4. Nonexistence result

Key Result

Proposition 1.1

(Weissler W80) Let $N\ge1$ and $f(u)=|u|^{p-1}u$$(p>1)$.

Figures (1)

  • Figure 1: Existence and nonexistence area of a local-in-time solution in Theorem \ref{['Exist-thm']} and Theorem \ref{['Nonexist-thm']} . Existence (i), (ii) and (iii) correspond to the results of Theorem \ref{['Exist-thm']} (i), (ii) and (iii), respectively.

Theorems & Definitions (30)

  • Proposition 1.1
  • Proposition 1.2
  • Remark 1
  • Definition 1.1
  • Theorem 1.3
  • Remark 2
  • Theorem 1.4
  • Remark 3
  • Lemma 2.1: MS21
  • Proposition 2.2
  • ...and 20 more