Abstract
We derive uniform in time \(L^{\infty }\)-bound for solutions to an aggregation-diffusion model with attractive-repulsive potentials or fully attractive potentials. We analyze two cases: either the repulsive nonlocal term dominates over the attractive part, or the diffusion term dominates over the fully attractive nonlocal part. When the repulsive part of the potential has a weaker singularity (\(2-n\leq B< A\leq 2\)), we use the classical approach by the Sobolev and Young inequalities together with differential iterative inequalities to prove that solutions have the uniform in time \(L^{\infty }\)-bound. When the repulsive part of the potential has a stronger singularity (\(-n< B<2-n \leq A\leq 2\)), we show the uniform bounds by utilizing properties of fractional operators. We also show uniform bounds in the purely attractive case \(2-n\leq A\leq 2\) within the diffusion dominated regime.
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Acknowledgements
JAC was partially supported by the EPSRC grant number EP/P031587/1. JW is partially supported by Program for Liaoning Excellent Talents in University (Grant No. LJQ2015041) and Key Project of Education Department of Liaoning Province (Grant No. LZD201701). The authors warmly thank the referees for the very valuable comments and suggestions that allowed us to improve this paper. We also thank Edoardo Mainini for pointing us out an important reference.
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Carrillo, J.A., Wang, J. Uniform in Time \(L^{\infty }\)-Estimates for Nonlinear Aggregation-Diffusion Equations. Acta Appl Math 164, 1–19 (2019). https://doi.org/10.1007/s10440-018-0221-y
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DOI: https://doi.org/10.1007/s10440-018-0221-y