A Liouville theorem for vector-valued nonlinear heat equations and applications. (English) Zbl 0939.35086

Merle, Frank; Zaag, Hatem

Math. Ann. 316, No. 1, 103-137 (2000).

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.

Reviewer: Dian K.Palagachev (Bari)

MSC:

35K50	Systems of parabolic equations, boundary value problems (MSC2000)
35K55	Nonlinear parabolic equations
35A20	Analyticity in context of PDEs

Keywords:

vector-valued semilinear heat equation; blowing-up solutions