Stationary patterns to diffusion problems
J. Garc�a-Meli�n, José C. Sabina de Lis · Mathematical Methods in the Applied Sciences · 2000
In this paper we are giving a complete account, as $\lambda\rightarrow\infty$ opagenumbers\end, of all possible solutions u to the problem $u_{xx}+\lambda f(u)=0,\;u(0)=u(1)=0$ opagenumbers\end which satisfy $0\leq u(x)\leq M\;\hbox{in}\; 0 0$ opagenumbers\end. A complete classification of them together with a detailed analysis of their limit profile, including exact location of ‘inner layers’, is given, their stability character is also determined. The analysis is completed by studying the cases where solutions u degenerate at zeros of f and considering also the p-Laplacian version of the problem where the diffusion term is replaced by the operator $(|u_{x}|^{p-2}u_x)_x,\;p>1$ opagenumbers\end. Copyright © 2000 John Wiley & Sons, Ltd.