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Solution of a nonlinear heat equation with arbitrarily given blow-up points. (English) Zbl 0785.35012

Summary: We consider the equation \[ u(0,x)=\varphi(x),\;u_ t=u_{xx}+| u|^{p-1}u\text{ on }[0,T)\times I,\;u=0\text{ on }[0,T)\times\partial I, \tag{1} \] where \(I\subset\mathbb{R}\), \(u\) is scalar- valued and \(p>1\). It has been proven that if \(u(t)\) blows up at time \(T\), the blow-up points are finite in number and located in \(\overset\circ I\).
Our aim is to prove that this result is optimal. That is, for any given points \(x_ 1,\ldots,x_ k\) in \(\overset\circ I\), there is a solution \(u\) such that its blow-up points are exactly \(x_ 1,\dots,x_ k\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs
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