A characterization of nonnegative stationary solutions for certain 1D degenerate parabolic equations and their applications

Figure 1.  Schematic picture of the non-trivial weak stationary solutions obtained in Theorem 2.4 (ⅰ) and (ⅱ). Here, it should be noted that the position of the singularity points $ x_{-} $ and $ x_{+} $ are not determined in our studies, however, they are shown in the figure for convenience

Figure 2.  Schematic pictures of the non-trivial weak stationary solutions obtained in Theorem 2.8. Here, it should be noted that the position of the singularity points $ x_{-} $ and $ x_{+} $ are not determined in our studies, however, they are shown in the figures for convenience. [Top left: The weak stationary solution in Theorem 2.8 (ⅲ).] [Top right: The weak stationary solution in Theorem 2.8 (ⅳ).] [Lower center: The stationary solution with periodicity in Theorem 2.8 (ⅴ).]

Figure 3.  Schematic pictures of the dynamics of the blow-up vector fields and $ E_{0} $ in the case that $ \delta = 0 $ and $ 0<p<1 $

Figure 4.  Schematic pictures of the dynamics of the blow-up vector fields and $ E_{0} $ in the case that $ \delta = 0 $ and $ p>1 $

Figure 5.  Schematic pictures of the dynamics of the blow-up vector fields and $ E_{0} $ in the case that $ \delta = 1 $ and $ 1<p<2 $

Figure 6.  Schematic pictures of the dynamics of the blow-up vector fields and $ E_{0} $ in the case that $ \delta = 1 $ and $ p>2 $

Figure A.1.  Locations of the Poincaré sphere and chart $ \overline{U}_{2} $

Figure 7.  Schematic pictures of the dynamics on the Poincaré disk in the case that $ \delta = 0 $. [Left: Case $ 0<p<1 $.] [Right: Case $ p>1 $.]

Figure 8.  Schematic pictures of the dynamics on the Poincaré disk in the case that $ \delta = 1 $. [Left: Case $ 1<p<2 $.] [Right: Case $ p>2 $.]