Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source

This paper deals with a fully parabolic chemotaxis system with consumption of chemoattractant and logistic sourceu t = Δ u − ∇ · ( u χ ( v ) ∇ v ) + f ( u ) ,( x , t ) ∈ Ω × ( 0 , ∞ ) ,v t = Δ v − u v ,( x , t ) ∈ Ω × ( 0 , ∞ ) ,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R N( N ≥ 1 ) . The functions χ and f are assumed to generalize the chemotactic sensitivity function χ ( s ) = χ 0( 1 + α s ) 2 , s ≥ 0 ,withχ 0 > 0 , α ≥ 0 , and logistic source f ( s ) = a s − b s 2 , s ≥ 0 ,witha , b > 0 , respectively. Under some conditions, we obtain that the corresponding initial‐boundary value problem possesses a unique global classical solution that is uniformly bounded.