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Reduction of critical mass in a chemotaxis system by external application of chemoattractant. (English) Zbl 1295.35136

The authors study the solutions to the parabolic-elliptic system \( u_{t}=\Delta u-\nabla \cdot (u\nabla v)\), \(0=\Delta v+u+f_{0}\delta (x)\), \(u(x,0)=u_{0}(x)\neq 0,\text{ }x\in \mathbb{R}^{2}\), where \(f_{0}>0,\) \( \delta \) is the Dirac distribution and \(u_{0}\) is a radially symmetric bounded non-negative function. In the absence of the external source, it is known that the number \(8\pi \) corresponds to a critical mass, namely when the total mass \(\mu =\int_{\mathbb{R}^{2}}u_{0}dx\) is greater than \(8\pi \), the solution blows up in finite time and collapses into a Dirac-type singularity. This behavior is not encountered if \(\mu <8\pi .\) The authors prove that a Dirac source term \(f_{0}\delta (x)\) reduces the critical mass threshold to \(\mu _{c}=8\pi -2f_{0}.\) They prove the existence of a measured-valued global-in-time weak solution, whenever \(f_{0}>0\) and \( u_{0}(x)\neq 0\), but such a solution blows up at \(x=0\) immediately. Particular properties of the solutions in the subcritical and supercritical cases are also provided.

MSC:

35B44 Blow-up in context of PDEs
92C17 Cell movement (chemotaxis, etc.)
35K45 Initial value problems for second-order parabolic systems
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