| Popis: |
In this paper we are concerned with the existence of stable stationary solutions for the problem \begin{document}$ u_t = \epsilon^2(k_1^2(x) u_x)_x+k_2^2(x)g(u,x) $\end{document} , \begin{document}$ (t,x)\in\mathbb{R}^+\times (0,1) $\end{document} subject to Neumann boundary condition. We suppose that \begin{document}$ k_1,k_2\in C^1(0,1) $\end{document} are positive functions and \begin{document}$ g $\end{document} is an unbalanced bistable function. We prove the existence of a family of stable stationary solutions developing internal transition layers in a specific sub-interval of \begin{document}$ (0,1) $\end{document} . For this, we provide a general variational method inspired by the \begin{document}$ \Gamma $\end{document} -convergence theory. |
