Finite-Time Blowup in a Supercritical Quasilinear Parabolic-Parabolic Keller-Segel System in Dimension 2
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Finite-Time Blowup in a Supercritical Quasilinear Parabolic-Parabolic Keller-Segel System in Dimension 2 Tomasz Cie´slak · Christian Stinner
Received: 20 January 2012 / Accepted: 17 May 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract In this note we show finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. We prove this result by slightly adapting M. Winkler’s method, which he introduced in (Winkler in J. Math. Pures Appl., 10.1016/j.matpur.2013.01.020, 2013) for the semilinear Keller-Segel system in dimensions at least three, to the two-dimensional setting. This is done in the case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned non-decay assumption is essential with respect to keeping the finitetime blowup result. Namely, we prove that without the non-decay assumption solutions exist globally in time, however infinite-time blowup may occur. Keywords Chemotaxis · Finite-time blowup · Infinite-time blowup Mathematics Subject Classification (2010) 35B44 · 35K20 · 35K55 · 92C17 1 Introduction In the present note we deal with solutions (u, v) of the parabolic-parabolic Keller-Segel system ⎧ ut = ∇ · (φ(u)∇u) − ∇ · (ψ(u)∇v), x ∈ Ω, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x ∈ Ω, t > 0, vt = v − v + u, (1.1) ∂u ∂v ⎪ = ∂ν = 0, x ∈ ∂Ω, t > 0, ⎪ ⎪ ∂ν ⎪ ⎪ ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ Ω, T. Cie´slak () ´ Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland e-mail: [email protected] C. Stinner Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland e-mail: [email protected]
T. Cie´slak, C. Stinner
¯ in a ball Ω = BR ⊂ R2 , R > 0, where the initial data are supposed to satisfy u0 ∈ C 0 (Ω) ¯ and v0 ∈ W 1,∞ (Ω) such that u0 ≥ 0 and v0 ≥ 0 in Ω. Moreover, let φ, ψ ∈ C 2 ([0, ∞)) such that φ(s) > 0,
ψ(s) = sβ(s),
and
β(s) > 0
for s ∈ [0, ∞)
(1.2)
are fulfilled with some β ∈ C 2 ([0, ∞)). Let us introduce the following notation. Suppose that there exist s0 > 1 and positive constants a and b such that the functions s s σ φ(τ ) σ φ(σ ) dτ dσ, s > 0, and H (s) := dσ, s ≥ 0, (1.3) G(s) := s0 s0 ψ(τ ) 0 ψ(σ ) satisfy G(s) ≤ as(ln s)μ ,
s ≥ s0 ,
(1.4)
with some μ ∈ (0, 1) as well as H (s) ≤ b
s , ln s
s ≥ s0 .
(1.5)
We remark that H in (1.3) is well-defined due to the positivity of β in [0, ∞). Moreover, assume that ψ(s) ≥ c0 s,
s ≥ 0.
(1.6)
Next we introduce the well-known Lyapunov functional for the Keller-Segel system. 1 1 F (u, v) := |∇v|2 + v2 − uv + G(u) (1.7) 2 Ω 2 Ω Ω Ω is a Lyapunov functional for (1.1) with dissipation rate
D(u, v) :=
vt2 + Ω
2 φ(u) ∇u − ∇v . ψ(u) · ψ(u) Ω
(1.8)
More precisely, any classical solution to (1.1) satisfies d F u(·, t), v(·, t) = −D u(·, t), v(·, t) dt
for all t ∈ 0, Tma
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