Interactions among multi-nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux
The following nonlinear diffusion equations with nonlinear absorption and coupled nonlinear boundary flux are studied: \[ \begin{alignedat}{3} u_t &= \Delta e^{mu} - a_1e^{\alpha_1u}, &\quad v_t &= \Delta e^{nu} - a_2e^{\beta_1v},&&\quad (x,t)\in\Omega\times (0,T),\\ \frac{\partial u}{\partial \eta} &= e^{pv + \alpha_2u}, &\quad \frac{\partial v}{\partial\eta} &= e^{qu + \beta_2v}, &&\quad (x,t)\in\partial \Omega\times (0,T),\\ u(x,0) &= u_0(x),&\quad v(x,0) &= v_0(x), &&\quad x\in\overline\Omega, \end{alignedat} \] where \(\Omega\subset\mathbb R^N\) is a bounded domain with smooth boundary \(\partial\Omega\); constants \(m\), \(n\), \(p\), \(q\), \(a_i > 0\), \(\alpha_i\), \(\beta_i\geq 0\), \((i = 1,2)\); \(u_0\) and \(v_0\) are nonnegative functions satisfying the compatibility conditions on \(\overline\Omega\). There are three kinds of nonlinear mechanisms which influence on the dynamical processes in the problem, i.e., nonlinear diffusion, nonlinear absorption, and nonlinear boundary flux. The aim of the article is to study the interactions among them. Two parameters are used to get a simple description for the interactions among the multi-nonlinearities in the system under consideration. The existence of blow up phenomenon is determined by the signs of certain parameters and the conditions for global boundedness of solutions to the above-mentioned problem are given in the terms of signs of parameters. The case of critical signs of parameters which are used to describe the behavior of the solutions is also discussed.
scientific article; zbMATH DE number 2102520