Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion

Figure 1.  Two different asymptotic behaviors of the solution depending on the initial datum

Figure 4.  Growing of the second moment of the solution to 3 in time in the regime $1/3<\varepsilon<\|G\|_{L^1}$. $\varepsilon = 0.5, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}$, and initial datum 40 with $\delta = 0.05.$

Figure 2.  Convergence of the solution to the steady state in the aggregation-dominated regime $0<\varepsilon<\|G\|_{L^1}$. $\varepsilon = 0.002, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}.$ (a) $\rho_{0}(x) = \frac{93}{8}(1-\frac{961}{4}x^2), $ (b) $\rho_{0}(x) = \frac{21}{8}(1-\frac{49}{4}x^2).$

Figure 3.  Decay of the solution to zero in the diffusion-dominated regime $\varepsilon\geq\|G\|_{L^{1}}$. $\varepsilon = 2, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}, $ $\rho_{0}(x) = \frac{21}{8}(1-\frac{49}{4}x^2)$.

Figure 5.  Convergence of the solution to the steady state in the aggregation-dominated regime $0<\varepsilon<\|G\|_{L^{1}}$. $\varepsilon = 0.5, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}.$ (a) $\rho_{0}(x) = \frac{9}{8}(1-\frac{9}{4}x^2), $ (b) $\rho_{0}(x) = \frac{105}{400}(1-\frac{1225}{10000}x^2).$