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  1. ; ; ; ( , Archive for Rational Mechanics and Analysis)
    Abstract

    We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then the linear diffusion overcomes its effect, either attractive or repulsive, for large times independently of the initial data, and solutions behave like the fundamental solution of the heat equation with some rate. The potential$$W(x) \sim \log |x|$$W(x)∼log|x|for$$|x| \gg 1$$|x|≫1appears as the natural limiting case when the intermediate asymptotics change. In order to obtain such a result, we produce uniform-in-time estimates in a suitable rescaled change of variables for the entropy, the second moment, Sobolev norms and the$$C^\alpha $$Cαregularity with a novel approach for this family of equations using modulus of continuity techniques.

     
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  2. ; ; ( , Journal of the European Mathematical Society)
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  3. ; ( , Communications on Pure and Applied Mathematics)
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  4. ; ; ( , SIAM Journal on Mathematical Analysis)
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  5. ; ; ( , Journal of Differential Equations)
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