TY - JOUR
T1 - Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension
AU - Fujie, Kentarou
AU - Senba, Takasi
N1 - Funding Information:
The first author is supported by Grant-in-Aid for Research Activity start-up (No. 17H07131 ), Japan Society for the Promotion of Science . The second author is supported by Grant-in-Aid for Scientific Research (C) (No. 26400172 ), Japan Society for the Promotion of Science .
Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2019/1/15
Y1 - 2019/1/15
N2 - This paper deals with positive solutions of the fully parabolic system {ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω⊂R4 with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0). Global existence and boundedness of solutions were shown if ‖u0‖L1(Ω)<(8π)2/χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖u0‖L1(Ω)>(8π)2/χ. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8π/χ-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R4.
AB - This paper deals with positive solutions of the fully parabolic system {ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω⊂R4 with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0). Global existence and boundedness of solutions were shown if ‖u0‖L1(Ω)<(8π)2/χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖u0‖L1(Ω)>(8π)2/χ. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8π/χ-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R4.
KW - Blow-up
KW - Chemotaxis
KW - Indirect process
KW - Lyapunov functional
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U2 - 10.1016/j.jde.2018.07.068
DO - 10.1016/j.jde.2018.07.068
M3 - Article
AN - SCOPUS:85051067784
SN - 0022-0396
VL - 266
SP - 942
EP - 976
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 2-3
ER -