Unstable blow-up patterns
The author of this interesting paper studies the blowing-up properties of solutions to the following initial-boundary value problem: \[ u_t= \Delta u+ u^p,\;t> 0,\;x\in \Omega,\;u(x, 0)= \overline u(x),\;u(x, t)= \phi(x, t),\;x\in \partial\Omega,\tag{1} \] where \(\Omega\) is an open set in \(\mathbb{R}^N\), \(p> 1\) and \(\overline u\), \(\phi\) are positive continuous and bounded functions, coinciding on \(\partial\Omega\times \{t= 0\}\). The main goal of the paper is to construct for each \(a\in \Omega\) a solution which blows up at \(a\), at a finite time \(T\) (i.e., \(\lim_{n\to \infty} u(x_n, t_n)= +\infty\) for a sequence \((x_n, t_n)\to (a, t^-)\)) according to specific asymptotic behaviour. More precisely, the following result is proved: Let \(m\geq 2\) be an integer and \(a\in \Omega\). Then for every boundary function \(\phi> 0\), and for every choice of the coefficients \(\{c_\alpha: |\alpha|= 2m\}\) such that \[ \sum_{|\alpha|= 2m} c_\alpha {x^\alpha\over \alpha!}\geq c|x|^{2m}\quad \forall x\in \mathbb{R}^N \] for some \(c= \text{const}> 0\), there exists an initial function \(\overline u\in C(\Omega)\) such that the corresponding solution of (1) blows up at a finite time \(T\) at the single point \(a\).
scientific article; zbMATH DE number 811756