Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening

Abstract

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system in smooth bounded domains $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n , $$n \ge 1$$ n ≥ 1 , are known to be global in time if $$\lambda \ge 0$$ λ ≥ 0 , $$\mu > 0$$ μ > 0 and $$\kappa > 2$$ κ > 2 . In the present work, we show that the exponent $$\kappa = 2$$ κ = 2 is actually critical in the four- and higher dimensional setting. More precisely, if $$\begin{aligned} \qquad n&\ge 4,&\quad \kappa \in (1, 2) \quad&\text {and} \quad \mu > 0 \\ \text {or}\qquad n&\ge 5,&\quad \kappa = 2 \quad&\text {and} \quad \mu \in \left( 0, \frac{n-4}{n}\right) , \end{aligned}$$ n ≥ 4 , κ ∈ ( 1 , 2 ) and μ > 0 or n ≥ 5 , κ = 2 and μ ∈ 0 , n - 4 n , for balls $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n and parameters $$\lambda \ge 0$$ λ ≥ 0 , $$m_0 > 0$$ m 0 > 0 , we construct a nonnegative initial datum $$u_0 \in C^0({{\overline{\Omega }}})$$ u 0 ∈ C 0 ( Ω ¯ ) with $$\int \nolimits _\Omega u_0 = m_0$$ ∫ Ω u 0 = m 0 for which the corresponding solution (u, v) of ($$\star $$ ⋆ ) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for $$\kappa \in (1, \frac{3}{2})$$ κ ∈ ( 1 , 3 2 ) (and $$\lambda \ge 0$$ λ ≥ 0 , $$\mu > 0$$ μ > 0 ). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function $$w(s, t) = \int \nolimits _0^{\root n \of {s}} \rho ^{n-1} u(\rho , t) \,\mathrm {d}\rho $$ w ( s , t ) = ∫ 0 s n ρ n - 1 u ( ρ , t ) d ρ fulfills the estimate $$w_s \le \frac{w}{s}$$ w s ≤ w s . Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, $$s_0$$ s 0 and $$\gamma $$ γ , the function $$\phi (t) = \int \nolimits _0^{s_0} s^{-\gamma } (s_0 - s) w(s, t)$$ ϕ ( t ) = ∫ 0 s 0 s - γ ( s 0 - s ) w ( s , t ) cannot exist globally.

Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production

“…• In the system (1.1), when D(u) = 1, S(u) = u and f (u) = u, Winkler [22] showed that if 1 < κ < 3 2 + 1 2n−2 (n ≥ 5), then there exists a solution blowing up in finite time; Moreover, a similar blow-up result was obtained in the case that D(u) = (u + 1) m−1 with m ≥ 1 in [2]; Furthermore, Fuest [5] showed that solutions blow up in finite time under the conditions that 1 < κ < min 2, n 2 and µ > 0 (n ≥ 3) and that κ = 2 and µ ∈ 0, n−4 n (n ≥ 5); In the two dimensional setting and κ = 2, global existence and boundedness were established when Ω u 0 < 8π, whereas finite-time blow-up occurs when Ω u 0 < m 0 with m 0 > 8π in [4].…”

“…In [5,19,23] the critical values such that solutions remain bounded or blow up in finite time were derived. With regard to the conditions (1.9), (1.13) and (1.…”

“…In summary, in [2,4,5,22], blow-up results were derived in the chemotaxis system with logistic source and linear production. However, boundedness and blow-up results were not obtained in the quasilinear chemotaxis system with logistic source and nonlinear production (when D(u) = 1 and S(u) = u, recently, Yi et al [26] derived the blow-up result under the condition that ℓ + 1 > κ 1 + 2 n ).…”

“…In the case m = α = ℓ = 1, Fuest (NoDEA Nonlinear Differential Equations Appl. ;2021; 28; 16) obtained conditions for κ such that solutions blow up in finite time. However, in the above system boundedness and finite-time blow-up of solutions have been not yet established.…”

Reaction-Driven Relaxation in Three-Dimensional Keller–Segel–Navier–Stokes Interaction

“…Indeed, in the resulting version of (1.1) with u ≡ 0 any choice of μ > 0 is sufficient to suppress any blow-up phenomenon in two-dimensional initial value problems in the sense that for widely arbitrary initial data, global bounded solutions always exist ( [41,45]); in associated three-and higher-dimensional counterparts, however, similar findings on exclusion of explosions to date seem to rely on the stronger hypothesis that μ > μ 0 with some μ 0 = μ 0 ( ) > 0 ( [53,61]), while for small values of μ > 0 only some weak solutions are known to exist globally ( [32]). Although some studies concerned with simplified model variants have revealed some considerably strong singularity-counteracting effects of logistic damping in the sense of immediate regularization of strongly singular distributions ( [33,60]), not only results on possibly transient emergence of high population densities ( [24,56]), but moreover especially some detections of genuine blow-up both in high-dimensional systems with quadratic zero-order dissipation ( [12]), and in three-dimensional models involving some subquadratic but yet superlinear absorption ( [12]), indicate some persistence of taxis-driven destabilization also in the presence of such degradation mechanisms.…”