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An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems

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  • This paper mainly considers the uniform bound on solutions of non-degenerate Keller-Segel systems on the whole space. In the case that the domain is bounded, Tao-Winkler (2012) proved existence of globally bounded solutions of non-degenerate systems. More precisely, they gave the result on boundedness in quasilinear parabolic equations by using the $L^p$-bounds on the solution for some large $p>1$. In Ishida-Yokota (2012), they dealt with the same system as this paper on the whole space, however, their $L^\infty$-estimate possibly grows up even if the solutions have the uniform bounds in $L^p(\mathbb{R}^N)$ for all $p\in[1,\infty)$. The present work asserts the uniform in time $L^\infty$-bound on solutions. Moreover, this paper covers the degenerate Keller-Segel systems and constructs the uniformly bounded weak solutions.
    Mathematics Subject Classification: Primary: 35K51; Secondary: 35D30, 35Q32, 92C17.

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