Singular backward self-similar solutions of a semilinear parabolic equation

Shota Sato,Eiji Yanagida

doi:10.3934/dcdss.2011.4.897

Abstract

We consider a parabolic partial differential equation with power nonlinearity. Our concern is the existence of a singular solution whose singularity becomes anomalous in finite time. First we study the structure of singular radial solutions for an equation derived by backward self-similar variables. Using this, we obtain a singular backward self-similar solution whose singularity becomes stronger or weaker than that of a singular steady state.

Highlights

For psg < p < p∗, we established the time-local existence, uniqueness and comparison principle for a solution of (F) with a moving singularity in some class
We consider singular solutions of the Cauchy problem for a semilinear parabolic equation: (F)ut = ∆u + up, x ∈ RN, where p > 1.Singular Steady State For N N ≥3 and p > psg := −
Aim 1 We find a solution of (BE) satisfying (1)

Summary

Introduction

For psg < p < p∗, we established the time-local existence, uniqueness and comparison principle for a solution of (F) with a moving singularity in some class. For some T > 0, there exists a solution of (F) with a moving singularity ξ(t) such that u(x, t) = L|x − ξ(t)|−m + o(|x − ξ(t)|−m) as x → ξ(t) for 0 < t < T.

Results

Conclusion