Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain

TL;DR

This paper investigates the lifespan of solutions to certain semilinear evolution equations in a 2D exterior domain, establishing a double exponential upper bound for the lifespan when the nonlinearity exponent equals the critical value 2.

Contribution

It provides a new upper bound estimate for the lifespan of solutions to semilinear PDEs in exterior domains at the critical exponent, using novel test functions related to harmonic functions.

Findings

Lifespan of solutions is bounded above by a double exponential when p=2.

Solutions blow up in finite time under certain initial conditions.

The method involves harmonic function approximations and lifespan estimation techniques.

Abstract

In this paper we consider the initial-boundary value problem for the heat, damped wave, complex-Ginzburg-Landau and Schr"odinger equations with the power type nonlinearity $|u|^p$ with $p in (1,2]$ in a two-dimensional exterior domain. Remark that $2=1+2/N$ is well-known as the Fujita exponent. If $p>2$, then there exists a small global-in-time solution of the damped wave equation for some initial data small enough (see Ikehata'05), and if $p<2$, then global-in-time solutions cannot exist for any positive initial data (see Ogawa-Takeda'09 and Lai-Yin'17). The result is that for given initial data $(f,tau g)in H^1_0(Omega)times L^2(Omega)$ satisfying $(f+tau g)log |x|in L^1(Omega)$ with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to the problem is given as the following {it double exponential type} when $p=2$: [ lifespan(u) leq exp[exp(Cep^{-1})] . ] The crucial idea is to use test functions which approximates the harmonic function $log |x|$ satisfying Dirichlet boundary condition and the technique for derivation of lifespan estimate in Ikeda-Sobajima(arXiv:1710.06780).