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

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  • In this paper we consider a one-dimensional nonlocal interaction equation with quadratic porous-medium type diffusion in which the interaction kernels are attractive, nonnegative, and integrable on the real line. Earlier results in the literature have shown existence of nontrivial steady states if the $L^1$ norm of the kernel $G$ is larger than the diffusion constant $\varepsilon$. In this paper we aim at showing that this equation exhibits a 'multiple' behavior, in that solutions can either converge to the nontrivial steady states or decay to zero for large times. We prove the former situation holds in case the initial conditions are concentrated enough and 'close' to the steady state in the $∞$-Wasserstein distance. Moreover, we prove that solutions decay to zero for large times in the diffusion-dominated regime $\varepsilon≥ \|G\|_{L^1}$. Finally, we show two partial results suggesting that the large-time decay also holds in the complementary regime $\varepsilon < \|G\|_{L^1}$ for initial data with large enough second moment. We use numerical simulations both to validate our local asymptotic stability result and to support our conjecture on the large time decay.

    Mathematics Subject Classification: Primary: 35B30, 35B35, 35B40; Secondary: 35B36, 92D25.


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  • 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) $\rho_{0}(x) = \frac{105}{400}(1-\frac{1225}{10000}x^2).$

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