Global existence and boundedness in a fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities and logistic source

Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system

“…In the case f (u) = λu − µu κ (λ ∈ R, µ > 0, κ > 1), finite-time blow-up was recently proved in [5] via the method in [44] when κ is sufficiently closed to 1 and χα − ξγ > 0 holds. Moreover, some related works deriving boundedness can be found in [15,28,29,30,31]; showing finite-time blow-up can be cited in [21]; dealing with nonlinear diffusion and sensitivities can be referred in [6,23,25]. Particularly, in the two-dimensional setting, Fujie-Suzuki [15] established boundedness in the fully parabolic version of (1.11) under the condition that β = δ, χα − ξγ > 0 and u 0 L 1 (Ω) < 4π χα−ξγ ; note that the authors relaxed the condition for u 0 in the radially symmetric setting and removed the condition β = δ.…”

Finite-time blow-up in the three-dimensional fully parabolic attraction-dominated attraction-repulsion chemotaxis system

“…(For the fully parabolic or the simplified systems there are several further works showing boundedness owing to additional system components which have been identified as beneficial for boundedness in the chemotaxis literature, like decaying sensitivity functions (e.g. [3,40]), nonlinear diffusion of porous-medium (e.g. [42,20]) or p-Laplacian type [21], or the addition of logistic decay terms [37,23,3].)…”

“…[3,40]), nonlinear diffusion of porous-medium (e.g. [42,20]) or p-Laplacian type [21], or the addition of logistic decay terms [37,23,3].) Concerning blow-up, however, the only results for the fully parabolic system (2) seems to be the recent note [4], where finite-time blow-up was shown for radial solutions in n ≥ 3 if β = δ.…”

Boundedness in a fully parabolic attraction-repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

“…In these literatures, the results were successfully obtained by using estimates for v, w from below and by reducing the case that χ, ξ are constants. On the other hand, in the case of linear diffusion and normal sensitivity that D(s) ≡ 1, G(s) = s, H(s) = s, global existence and boundedness in the system with τ 1 = τ 2 = 1 were proved in [4] by the method using a test function defined as a combination of an exponential function and integrals of χ, ξ. However, since the proof in [4] strongly depends on |(u + 1) m−1 ∇u| 2 = (u + 1) m−1 |∇u| 2 (this holds true only in the case m = 1!…”

“…The strategy for the proof of Theorem 1.1 is to show L p -boundedness of u. In the case that D(s) ≡ 1, G(s) = s, H(s) = s, L p -estimate for u was established in [4] by deriving the differential inequality…”