Critical mass on the Keller-Segel system with signal-dependent motility
Hai‐Yang Jin, Zhi‐An Wang
Abstract
This paper is concerned with the global boundedness and blow-up of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility function decays exponentially, then a critical mass phenomenon similar to the minimal Keller-Segel model will arise. That is, there is a number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript asterisk Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m_*>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm of the initial value of cell density) is less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>m</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">m_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , while the solution may blow up if the initial cell mass is greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>m</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">m_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .