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

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).