A Liouville theorem for vector-valued nonlinear heat equations and applications
The paper deals with the blowing-up solutions to the Dirichlet problem for vector-valued semilinear heat equation \[ u_t=\Delta u+F(|u|) \quad \text{in} \Omega \times [0,T), \qquad u(x,t)=0 \quad \text{on} \partial \Omega\times [0,T), \qquad u(x,0)=u_0(x),\;x\in \Omega, \] where \(\Omega\) is a bounded and convex domain of \({\mathbb R}^N\) or \(\Omega={\mathbb R}^N,\) \(F\in C^1({\mathbb R},{\mathbb R})\) satisfying \(F(|u|)\sim |u|^{p-1}\) as \(|u|\to\infty,\) \(p>1\) and \((N-2)p<N+2.\) By the aid of scalar-case-techniques, the authors extend the parabolic Liouville theorem to the vectorial setting.
scientific article; zbMATH DE number 1406573