Existence and Nonexistence of Global Solutions of Higher Order Parabolic Equations with Slow Decay Initial Data

In this paper we consider the Cauchy problem ∂tu+(−∆)mu=|u|p, x∈RN, t>0, (1) u(x,0)=u0(x), x∈RN, (2) wheremisanintegergreaterorequalto1,p>1,u0 ∈X≡L1(RN)∩L∞(RN). From the general theory of evolution equations, there exists a unique bounded solution of (1)–(2) u(x,t)=u(x,t;u0), defined on a maximal time interval [0,T∗), where T∗ = T∗(u0)∈(0,∞]. If T∗(u0)=∞, we say that u(x,t;u0) is a global solution of (1)–(2). On the other hand, if T ∗(u0) < ∞, we say that the corresponding solution does not exist globally or that it blows up in finite time. Recent results by Egorov et al. [3] show that, if 1<pp∗(m)=1+2m/N,thenT∗(u0)<∞foranyu0 ∈L1loc(RN)\{0}suchthat RN u0(x)dx 0.