Abstract
This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem \t\t\t◊{ut=Δu−∇⋅(f(u)∇v) in Ω×(0,Tmax),vt=Δv−v+g(u) in Ω×(0,Tmax),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u_{t}= \\Delta u - \\nabla \\cdot (f(u) \\nabla v) & \\text{ in } \\Omega \\times (0,T_{max}), \\\\ v_{t}=\\Delta v-v+g(u) & \\text{ in } \\Omega \\times (0,T_{max}), \\end{cases} $$\\end{document} where Omega is a bounded and smooth domain of mathbb{R}^{n}, for ngeq 2, and f(u) and g(u) are reasonably regular functions generalizing, respectively, the prototypes f(u)=u^{alpha } and g(u)=u^{l}, with proper alpha , l>0. After having shown that any sufficiently smooth u(x,0)=u_{0}(x)geq 0 and v(x,0)=v_{0}(x)geq 0 produce a unique classical and nonnegative solution (u,v) to problem (◊), which is defined on Omega times (0,T_{max}) with T_{max} denoting the maximum time of existence, we establish that for any lin (0,frac{2}{n}) and frac{2}{n}leq alpha <1+frac{1}{n}-frac{l}{2}, T_{max}=infty and u and v are actually uniformly bounded in time.The paper is in line with the contribution by Horstmann and Winkler (J. Differ. Equ. 215(1):52–107, 2005) and, moreover, extends the result by Liu and Tao (Appl. Math. J. Chin. Univ. Ser. B 31(4):379–388, 2016). Indeed, in the first work it is proved that for g(u)=u the value alpha =frac{2}{n} represents the critical blow-up exponent to the model, whereas in the second, for f(u)=u, corresponding to alpha =1, boundedness of solutions is shown under the assumption 0< l<frac{2}{n}.
Highlights
Introduction and MotivationsMost of this article is dedicated to the following Cauchy boundary problem ⎧ ⎪⎪⎪⎨vutt = =u − ∇ · (f (u)∇v) v − v + g(u)⎪⎪⎪⎩uν = vν u(x, 0) =0 = u0 (x ), v(x, 0)v0(x) in × (0, Tmax ), in × (0, Tmax ), on ∂ × (0, Tmax ), x∈ ̄, (1)
Our result positively addresses this issue in the sense that independently of the initial data, by weakening in an inversely proportional way to the dimension the impact associated to the production rate of the chemical, the uniform-in-time boundedness of solutions to model (1) is ensured, even for superlinear thrusts from the chemoattractant
The fact that in the simplified version the values of v(·, t) only depend on the values of u(·, t) at the same time, is a strong modeling assumption. It corresponds to the situation where the signal responses to the concentration of the particles much faster than the organisms do to the signal; in particular, such difference in the relative adjustment of the bacteria and the chemoattractant makes that the last one reaches its equilibrium instantaneously
Summary
By indicating with u = u(x, t) a certain particle density at the position x and at the time t , the equations describe how the aggregation impact from the coupled cross term u∇v, related to the chemical signal v = v(x, t) (initially distributed to the law v0(x) = v(x, 0), as in (1)), may contrast the natural diffusion (associated to the Laplacian operator, u) of the cells, organized at the initial time through the configuration u0(x) = u(x, 0). For analyses concerning a chemotaxis model with signal-dependent sensitivity and sublinear production.)
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