Classification of singularities for blowing up solutions in higher dimensions

Abstract:

Consider the Cauchy problem (P) \[ \left \{ {\begin {array}{*{20}{c}} {{u_t} - \Delta u = {u^p}} \hfill & {{\text {when}}\;x \in {\mathbb {R}^N},} \hfill & {t > 0,N \geq 1,} \hfill \\ {u(x,0) = {u_0}(x)} \hfill & {{\text {when}}\;x \in {\mathbb {R}^N},} \hfill & {} \hfill \\ \end {array} } \right .\] where $p > 1$, and ${u_0}(x)$ is a continuous, nonnegative and bounded function. It is known that, under fairly general assumptions on ${u_0}(x)$, the unique solution of $({\text {P}})$, $u(x,t)$, blows up in a finite time, by which we mean that \[ \lim \sup \limits _{t \uparrow T} \left ( {\sup \limits _{x \in {\mathbb {R}^N}} \;u(x,t)} \right ) = + \infty .\] In this paper we shall assume that $u(x,t)$ blows up at $x = 0$, $t = T < + \infty$ , and derive the possible asymptotic behaviours of $u(x,t)$ as $(x,t) \to (0,T)$, under general assumptions on the blow-up rate.