A characterization of nonnegative stationary solutions for certain 1D degenerate parabolic equations and their applications
The paper examines two parabolic differential equations. The first is the spatial one-dimensional degenerate equation \N\[\NU_t=U^p(U_{xx}+\mu U)-\delta U,\;\;t>0,\;\;x\in\mathbf{R},\N\]\Nwhere \(p>0, \mu>0,\) and \(\delta=0\) or \(1.\) The second is \N\[\NV_t=(V^m)_{xx}+\mu V^m-\delta(1-p)V,\;\;t>0,\;\;x\in{\mathbf{R}}, \;\;m>1,\N\]\Nobtained from the first one by the transformation \N\[\NU(t, x)=(1-p)^{-\frac{1}{p}}(V(t,x))^{\frac{1}{1-p}},\N\]\Nwhere \(0<p<1.\)\N\NThe author point out in the abstract: ``The characterization of stationary solutions given in this paper refers to the enumeration of those that exist and the presentation of solution information such as the profile and asymptotic behavior of each of them. Due to the influence of terms derived from the degeneracy of the equations, it is not easy to classify and characterize the existence of stationary solutions of the equations considered in this paper. These results are obtained by using dynamical systems theory and geometric approaches (in particular, Poincaré compactification). In addition, an application of the results obtained in this paper is given. The results of the characterization of nonnegative weak stationary solutions of the spatial one-dimensional porous medium equation with special nonlinear terms are shown. These can be obtained by carefully using transformations known from previous studies.''
scientific article; zbMATH DE number 7896652
scientific article