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Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity, and logistic source

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In this paper we study the zero‐flux chemotaxis‐system

Ω being a convex smooth and bounded domain of R n , n 1 , and where m , k R , μ > 0 and α < m + 1 2 . For any v 0 the chemotactic sensitivity function is assumed to behave as the prototype χ ( v ) = χ 0 ( 1 + a v ) 2 , with a 0 and χ 0 > 0 . We prove that for nonnegative and sufficiently regular initial data u ( x , 0 ) and v ( x , 0 ) , the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.

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