The numerical approximation to positive solution for some reaction-diffusion problems
The authors consider the positive solutions of the problem NEWLINE\[NEWLINE \begin{aligned} u_t & = u_{xx}+u^3, \quad 0<x<L, \;t>0 \\ \vspace{3pt} u(0,t) &= u(L,t)=0, \quad t>0\\ \vspace{3pt} u(x,0)& =u_0(x), \quad 0<x<L \end{aligned} NEWLINE\]NEWLINE and the stationary positive solutions. It is shown that in the numerical approximation of these solutions, it is important and favourable to use the condition NEWLINE\[NEWLINE \int^L_0 \overline u(x)dx=\frac \pi{\sqrt 2}, NEWLINE\]NEWLINE satisfied by the exact positive stationary solution \(\overline u\).
scientific article; zbMATH DE number 1668898