Global boundedness of a chemotaxis-consumption model with signal-dependent sensitivity

This paper deals with the global boundedness of a classical solution to a chemotaxis-consumption model with signal-dependent sensitivity. Specifically, this paper focuses on studying the existence and boundedness of a classical solution in the absence of a logistic source term when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the chemotaxis sensitivity function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi left-parenthesis v right-parenthesis equals chi v Superscript k Baseline left-parenthesis 1 less-than-or-equal-to k less-than-or-equal-to 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> χ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> χ </mml:mi> <mml:msup> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>k</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\chi (v)=\chi v^{k} (1\leq k\leq 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Due to the absence of the logistic source term, obtaining the necessary gradient estimates for global existence seems difficult because of the strongly coupled structure. To this end, we propose new energy functionals to address this difficulty.