Abstract
In this paper, we consider the following indirect signal generation and singular sensitivity n t = Δ n + χ ∇ ⋅ n / φ c ∇ c , x ∈ Ω , t > 0 , c t = Δ c − c + w , x ∈ Ω , t > 0 , w t = Δ w − w + n , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R N N = 2 , 3 with smooth boundary ∂ Ω . Under the nonflux boundary conditions for n , c , and w , we first eliminate the singularity of φ c by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.
Highlights
One of the first mathematical models of chemotaxis was introduced by Keller and Segel [1] to describe the aggregation of certain types of bacteria
The physical domain Ω ⊂ RNðN = 2, 3Þ is a bounded domain with smooth boundary
0 = Δc − c + n, x ∈ Ω, t > 0: Another important chemotaxis model is formed with singular sensitivity function, such as χðn, cÞ = χ/c
Summary
One of the first mathematical models of chemotaxis was introduced by Keller and Segel [1] to describe the aggregation of certain types of bacteria. The physical domain Ω ⊂ RNðN = 2, 3Þ is a bounded domain with smooth boundary This model describes a biological process in which cells move towards their preferred environment and a signal being produced by the cells themselves. 0 = Δc − c + n, x ∈ Ω, t > 0: Another important chemotaxis model is formed with singular sensitivity function, such as χðn, cÞ = χ/c. Tao and Wang [22] considered the global solvability, boundedness, blow-up, existence of nontrivial stationary solutions, and asymptotic behavior. Considering the singular sensitivity function, we study the following singular chemotaxis model of indirect signal generation. ΦðxÞ = arctan τÞdτ, and so x, φðxÞ = on are xα all log ð1 + xÞ, φð satisfied with conditions of (6) Under these assumptions, we give the well-posedness and asymptotic behavior results as follows. There exists ε0 > 0 such that if m satisfies for some 0 < ε < ε0, the solution of (3) has the following decay estimates: nð·, tÞ − jmΩjL∞ðΩÞ ⟶ 0, cð·, tÞ − jmΩjL∞ðΩÞ ⟶ 0, ð10Þ wð·, tÞ − jmΩjL∞ðΩÞ ⟶ 0, m
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