Global Boundedness of the Fully Parabolic Keller-Segel System with Signal-Dependent Motilities

• Published in: Acta applicandae mathematicae Vol. 171; no. 1
• Main Authors: Wang, Zhi-An, Zheng, Jiashan
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.02.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Diffusion; Mathematics; Mathematics and Statistics; Motility; Partial Differential Equations; Probability Theory and Stochastic Processes; Smooth boundaries; 35K20; Keller-Segel system; Density-dependent motilities; 92C17; Global boundedness; Weighted functional; 35K55
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-021-00392-8

This paper establishes the global uniform-in-time boundedness of solutions to the following Keller-Segel system with signal-dependent diffusion and chemotaxis { u t = ∇ ⋅ ( γ ( v ) ∇ u − u ϕ ( v ) ∇ v ) , x ∈ Ω , t > 0 , v t = d Δ v − v + u , x ∈ Ω , t > 0 in a bounded domain Ω ⊂ R N ( N ≤ 4 ) with smooth boundary, where the density-dependent motility functions γ ( v ) and ϕ ( v ) denote the diffusive and chemotactic coefficients, respectively. The model was originally proposed by Keller and Segel in (J. Theor. Biol. 30:225–234, 1970 ) to describe the aggregation phase of Dictyostelium discoideum cells, where the two motility functions satisfy a proportional relation ϕ ( v ) = ( α − 1 ) γ ′ ( v ) with α > 0 denoting the ratio of effective body length (i.e. distance between receptors) to the step size. The major technical difficulty in the analysis is the possible degeneracy of diffusion. In this work, we show that if γ ( v ) > 0 and ϕ ( v ) > 0 are smooth on [ 0 , ∞ ) and satisfy inf v ≥ 0 d γ ( v ) v ϕ ( v ) ( v ϕ ( v ) + d − γ ( v ) ) + > N 2 , then the above Keller-Segel system subject to Neumann boundary conditions admits classical solutions uniformly bounded in time. The main idea of proving our results is the estimates of a weighted functional ∫ Ω u p v − q d x for p > N 2 by choosing a suitable exponent p depending on the unknown v , by which we are able to derive a uniform L ∞ -norm of v and hence rule out the diffusion degeneracy.