Semilinear reaction--diffusion systems of several components
The paper deals with the reaction-diffusion system of the form \[ u_t = \Delta u + f(u) \tag{1} \] with an initial condition and the boundary condition of Dirichlet form. Here \[ u(x,t) = (u_1(x,t), \dots , u_n(x,t)) \] is the unknown vector function, \(x \in \Omega \subset \mathbb{R}^N\), \(t>0\) and \(\Omega\) is a bounded domain with smooth boundary. It is supposed that the functions \(u_j(x,t)\) (which can be thought as the temperatures of \(n\) substances which constitute a combustible mixture) are coupled according to the formula \[ f(u) = \left( \prod_{j=1}^n u_j^{p_{1j}}, \prod_{j=1}^n u_j^{p_{2j}}, \dots , \prod_{j=1}^n u_j^{p_{nj}} \right) \] The case \(n=1\) was studied by many authors (see the references in the paper) and the case \(n=2\) was studied in [\textit{F. Dickstein} and \textit{M. Escobedo}, Nonlinear Anal., Theory Methods Appl. 45, 825-837 (2001; Zbl 0986.35044)]. Assuming \(n \geq 1\) the authors give the blow-up conditions of (1) in terms of the matrix \(P = (p_{ij})\).
scientific article; zbMATH DE number 1884234
scientific article