Perturbation analysis of a semilinear parabolic problem with nonlinear boundary conditions
The objective of the paper is the study of the nonlinear, parabolic, initial-boundary value problem: \[ u_t=u_{xx}-\lambda u^p, \qquad x\in(0,1), \quad t>0, \] \[ u_x(0,t)=0, \;\;u_x(1,t)=u^q(1,t), \quad t>0, \qquad u(x,0)=u_0(x), \] where \(\lambda\), \(p\), \(q\) are constants, \(\lambda>0\), \(p>1\), \(q>1\) and \(u_0>0\). A formal asymptotic description of the steady states is presented and the stability of these steady states is studied in the case when \(p\) and \(q\) exceed unity. In particular, a perturbation analysis is made taking \(p=1+\varepsilon \alpha\) and \(q=1+\varepsilon\beta\) where \(\varepsilon\) is a small positive parameter. The analysis emphasizes the role of \(\alpha/\beta\) especially when this parameter is near one or a half. The authors expect that their response diagrams will give a reliable guide to the size of stationary solutions for larger values of \(\varepsilon\). They indicate also how the steady states can be used to give information on the solution of the unsteady problem.
scientific article; zbMATH DE number 894679