Blow-up of multi-componential solutions in heat equations with exponential boundary flux (Q680342)

The authors consider multi-componential solutions of heat equations with coupled nonlinear boundary flux \[ \begin{cases} (u_i)_t=\Delta u_i & (x,t)\in B_R\times(0,T),\\ \frac{\partial u_i}{\partial \nu}=\exp\{p_iu_i+q_{i+1}u_{i+1}\} & (x,t)\in \partial B_R\times(0,T),\\ u_i(x,0)=u_{i,0}(x),\quad i=1,\ldots,n,\;n\geq 2 & x\in B_R, \end{cases} \] where \(B_R\) is the \(N\)-dimensional ball of radius \(R,\) \(u_{n+1}=u_1,\) \(p_{n+1}=p_1,\) \(q_{n+1}=q_1,\) the exponents \(p_i,q_i\geq0\) are constant, \(u_{i,0}\) are positive, smooth and radially symmetric functions satisfying the compatibility conditions on the boundary. Under certain nonotonicity assumptions on the initial data with respect to the radial variable, the authors obtain necessary and sufficient conditions for simultaneous blow-up of at least two components of the solution.