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Moving gradient singularity for the evolutionary $p$-Laplace equation

Erik Lindgren, Jin Takahashi

Abstract

We consider the evolutionary $p$-Laplace equation in $\mathbb{R}^n$. For $p>n$, we construct a solution $u$ with a moving gradient singularity in the sense that $|\nabla u(x,t)|\to \infty$ for each $t$ as $x\toξ(t)$, where $ξ:[0,\infty)\to\mathbb{R}^n$ is a given curve.

Moving gradient singularity for the evolutionary $p$-Laplace equation

Abstract

We consider the evolutionary -Laplace equation in . For , we construct a solution with a moving gradient singularity in the sense that for each as , where is a given curve.

Paper Structure

This paper contains 5 sections, 1 theorem, 53 equations.

Table of Contents

  1. Introduction
  2. Comparison functions
  3. Supersolution
  4. Subsolution
  5. Existence

Key Result

Theorem 1.1

Let $n\geq2$, $p>n$ and $k>k'>0$. Fix $\lambda$ and $\lambda'$ such that Then, there exist $0<C_\xi<1$ and $A>1$ such that the following statement holds: Assume that $\xi \in C^1([0,\infty); \mathbb{R}^n)$ and $u_0\in C(\mathbb{R}^n) \cap C^1(\mathbb{R}^n\setminus\{\xi(0)\})$ satisfy respectively. Then, eq:main admits a nonnegative weak solution $u$ satisfying (i), (ii) and (iii).

Theorems & Definitions (2)

  • Theorem 1.1
  • proof : Proof of Theorem \ref{['th:main']}