Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary
Johannes Lankeit, Michael Winkler
Abstract
Abstract The chemotaxis system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mo>∇</mml:mo> <mml:mo>⋅</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∇</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>v</mml:mi> <mml:mo>−</mml:mo> <mml:mi>u</mml:mi> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> </mml:math> is considered under the boundary conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mi>u</mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> and v = v ⋆ on ∂Ω, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" 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:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> is a ball and v ⋆ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case n = 2, while global weak solutions are constructed when n ∈ {3, 4, 5}. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.