A regularity result for a quasilinear equation and its consequences for blowing-up solutions of semilinear heat equations
Positive, smooth blow-up solutions \(u(x,t)\) \((u \geq 0, u \in C^ \infty (\Omega \times (0,T))\), \(\lim_{t \to T} \| u(x,t) \|_{L^ \infty (\Omega)} = \infty)\) of the initial-boundary value problem \[ u_ t - \Delta u = f(u) \quad \text{in} \quad \Omega \times (0,T), \quad u = 0 \quad \text{on} \quad \partial \Omega \times (0,T), \quad u(x,0) = u_ 0(x) \] are considered. Here \(\Omega\) is a smooth, bounded domain in \(\mathbb{R}^ n\), and \(f(u)=e^ u\) or \(f(u) = (u + \lambda)^ p\), \(\lambda \geq 0\), \(p>1\). Let \(v=e^{-u}\) (or \(v=(u+\lambda)^{1-p})\). Then \(v\) satisfies the equation \[ v_ t - \Delta v = - 1 - {| \nabla v |^ 2 \over v} \quad \text{or} \quad v_ t - \Delta v = - (p-1) - {p \over p - 1} {| \nabla v |^ 2 \over v}. \] Evidently, the blow-up set of \(u\) corresponds to the extinction set of \(v\). In the symmetric case \((\Omega\) is a ball and \(u_ 0\) is a radial function, \((u_ 0)_ r \leq 0)\) \(x=0\) is the only blow-up point for \(u(x,t)\). In this case the author proves: Theorem 1. Assume that for some \(a>0\), \(u_ t \geq \delta\) in the cylinder \(B(x_ 0,a) \times (T-a,T)\). If \(f(u) = (u + \lambda)^{1-p}\), \(p>5\), then \(v \in C^{1 + \alpha, \alpha} (\Omega \times (0,T))\). In the nonsymmetrical case \((n=1\) and \(f(u) = e^ u)\) the following theorem is proved: Theorem 2. Assume that \(u_ t \geq \delta\) in a neighborhood of the blow- up set. Then \(v \in C^{\alpha, \alpha/2} (\Omega \times (0,T))\).
scientific article; zbMATH DE number 97344
scientific article