Dead cores and effectiveness of semilinear reaction-diffusion systems (Q1206983)

The authors study the following coupled semilinear elliptic problem; \(\Delta u=\lambda g(x)h(u,v)\) in \(\Omega\), \(u=a(x)\) on \(\partial\Omega\), \(\Delta v=\lambda g(x)h(u,v)\) in \(\Omega\), \(v=b(x)\) on \(\partial\Omega\), where \(g\in C^ \gamma(\overline{\Omega})\) and \(h\in C^ \gamma((- \infty,\infty) \times (-\infty,\infty))\) such that \(h(u,v)\) is 0 if \(u\) or \(v\) is negative and nonincreasing in each component. This equation arises as a model in a chemical situation and \(u\) and \(v\) correspond to the densities of chemical materials in steady state. They prove the unique existence of a positive solution and that it decreases as \(\lambda\) increases. The main interest in this paper is the limit of the solution when \(\lambda\to\infty\). It is called a dead core problem because the limit function may vanish in a subregion of \(\Omega\). Such study has been done in the case of a single equation and several results are known concerning the dead core and effectiveness. The authors investigate the limit of \(u\) and \(v\) and properties of the dead cores in relation with the coefficients \(g\), \(k\) and the boundary condition \(a\) and \(b\).