Global boundedness of solutions resulting from both the self-diffusion and the logistic-type source

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This paper is concerned with the zero-flux logistic chemotaxis system with nonlinear signal production: \(u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u)\), \(v_t=\Delta v-v+g(u)\) in a bounded and smooth domain \(\Omega \subset {\mathbb {R}}^n\) (\(n\ge 1\)), where \(\chi >0\) is a constant, and \(f,g\in C^1({\mathbb {R}})\) generalize the prototypes \(f(s)=s-\mu s^{\alpha }\) and \(g(s)=s(s+1)^{\beta -1}\) with \(\alpha >1\) and \(\beta ,\mu >0\). The existing studies, for the case that the self-diffusion or the logistic-type source prevails over the cross-diffusion singly, have asserted the global boundedness of solutions under the assumption that \(\beta 0\) suitably large. In the present paper, we prove that if \(\alpha -1=\beta =\frac{2}{n}\), then due to the self-diffusion and the logistic kinetics working together, the solutions are globally bounded regardless of the size of \(\mu >0\). Note that \(\alpha -1=\beta =\frac{2}{n}\) reduces to \(\alpha =2\) and \(\beta =1\) if \(n=2\). Hence, our result covers that of the two-dimensional classical chemotaxis system with logistic source: \(u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+u-\mu u^2\), \(v_t=\Delta v-v+u\) (Osaki et al. in Nonlinear Anal 51:119–144, 2002).