Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model
Dealing with reaction-diffusion systems, the starting point is in general to prove that all solutions are confined to some regions during the evolution when they start inside. The most simple example in this respect is that solutions stay nonnegative if this is true for the initial values. The term ``invariant regions'' explains itself. The author proves a general theorem applicable to strongly coupled quasilinear parabolic systems of the type \[ {\partial\overline u\over\partial t}- \sum^u_{i=1}{\partial\over\partial x_i} \Biggl(D(x,\overline u){\partial\overline u\over\partial x_i}\Biggr)=\overline F(x,t,\overline u). \] From this he derives several results for two-component cross-diffusion problems, where \[ D(u)=\begin{pmatrix} c_1+d_1 u_2+ e d_2u_1 & \alpha d_1u_1\\ \beta d_2 u_2 & c_2+ d_2u_1+f d_1u_2\end{pmatrix} \] and \(c_i\), \(d_i>0\). According to different conditions connecting \(\alpha\), \(\beta\), \(e\) and \(f\), he gets bounded or unbounded invariant regions of different shape. The importance of these results is that for certain cases one can even go further and prove a-priori estimates in Hölder-spaces (see, e.g. [\textit{M. Wiegner}, Math. Ann. 292, No. 4, 711-727 (1992; Zbl 0801.35064)] and \textit{K. H. W. Küfner}, Analysis 15, No. 4, 343-357 (1995; Zbl 0840.35015)] from which the global existence of solutions for these reaction-diffusion systems follows by results of [\textit{H. Amann}, Math. Z. 202, No. 2, 219-250 (1989; Zbl 0702.35125)]).
strongly coupled quasilinear parabolic systems
scientific article; zbMATH DE number 952685
scientific article