Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two
Xiangdong Zhao
Abstract
Abstract This paper deals with the parabolic-elliptic Keller–Segel system with weakly singular sensitivity and logistic source under the homogeneous Neumann boundary: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mi>χ</mml:mi> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mfrac> <mml:mi>u</mml:mi> <mml:msup> <mml:mi>v</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mfrac> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>r</mml:mi> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mi>μ</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>v</mml:mi> <mml:mo>−</mml:mo> <mml:mi>v</mml:mi> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> </mml:math> in a smooth bounded domain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>χ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>r</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> . It is proved that the system possesses a globally bounded classical solution for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> with µ > 0 suitably large, without establishing the uniformly positive bound for v from below. In addition, we give the explicit expression of the upper bound for solution u with respect to the parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>χ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> </mml:math> via a recursive argument on α . This concludes that weakly singular sensitivity benefits to obtain the global boundedness of classical solution in dimension two.