Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with variable logistic source

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 73; no. 5
• Main Authors: Ayazoglu, Rabil, Akkoyunlu, Ebubekir
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2022; Springer Nature B.V
• Subjects: Boundary conditions; Engineering; Exponents; Mathematical Methods in Physics; Theoretical and Applied Mechanics; Variable logistic source; 35B35; 92C17; 35B40; Global boundedness; Chemotaxis; 35K55
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-022-01847-0

This paper deals with the higher dimension quasilinear parabolic–parabolic chemotaxis model involving a source term of logistic type u t = ∇ · ϕ ( u ) ∇ u - ∇ · ψ ( u ) ∇ υ + g ( x , u ) , τ υ t = Δ υ - υ + u , in ( x , t ) ∈ Ω × ( 0 , T ) , subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is a smooth and bounded domain in R N , N ≥ 1 and ψ , ϕ , g are smooth, positive functions satisfying ν s q ≤ ψ ≤ χ s q , ϕ ≥ σ s p , p , q ∈ R , ν , χ , σ > 0 when s ≥ s 0 > 1 , g ( x , s ) ≤ η s k ( x ) - μ s m ( x ) for s > 0 , η ≥ 0 , μ > 0 constants and g ( x , 0 ) ≥ 0 , x ∈ Ω , where k , m are measurable functions with 0 ≤ k - : = e s s inf x ∈ Ω k x ≤ k ( x ) ≤ m + : = e s s sup k ( x ) x ∈ Ω < + ∞ , 1 < m - : = e s s inf x ∈ Ω m x ≤ m ( x ) ≤ m + : = e s s sup m ( x ) x ∈ Ω < + ∞ . We extend the constant exponents k = 0 , 1 , m > 1 which in logistic source term g ( s ) ≤ η s k - μ s m for s > 0 , η ≥ 0 , μ > 0 as variable exponents k ( · ) ≥ 0 , m ( · ) > 1 with k + < m - . It is proved that if q = m - - 1 (critical case) with μ properly large that μ > μ 0 for some μ 0 > 0 , then there exists a classical solution which is global in time and bounded. Furthermore, if q < m - - 1 , we prove that the classical solutions to the above system are uniformly in-time-bounded without restriction on μ .