Upper and lower solutions for a homogeneous Dirichlet problem with nonlinear diffusion and the principle of linearized stability (Q5940631)

This paper is devoted to a homogeneous Dirichlet problem with nonlinear diffusion of the form NEWLINE\[NEWLINE\begin{cases} \nabla\cdot \bigl(d(x,u) \nabla u \bigr)+ f(x,u) =0 \quad&\text{in }\Omega\\ u =0 \quad &\text{on }\partial\Omega \end{cases} \tag{1}NEWLINE\]NEWLINE where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^n\). The authors present linear stability (instability) for the corresponding quasilinear parabolic problems. Note that the authors study the questions mentioned above under minimal assumptions on the coefficients \(d\) and \(f\), namely, that both are sufficiently smoth and that \(d\) is positive.