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Classification of singularities for blowing up solutions in higher dimensions. (English) Zbl 0803.35015

Summary: We consider the Cauchy problem \[ u_ t - \Delta u = u^ p, \quad x \in \mathbb{R}^ N,\;t>0, \;N \geq 1, \qquad u(x,0) = u_ 0(x), \quad x \in \mathbb{R}^ N, \tag{P} \] 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 (P), \(u(x,t)\), blows up in a finite time, by which we mean that \(\limsup_{t \uparrow T} (\sup_{x \in \mathbb{R}^ N} u(x,t)) = + \infty\). In this paper we 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.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
35K15 Initial value problems for second-order parabolic equations
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