Occurrence of blow-up under subquadratic or weak quadratic damping

Determine whether blow-up can occur in the fully parabolic chemotaxis–May–Nowak system with logistic damping when the damping exponent satisfies α<1, or when α=1 and the damping coefficient μ is sufficiently small.

Background

The paper proves global existence and uniform-in-time boundedness for the fully parabolic chemotaxis–May–Nowak system in several parameter regimes, including quadratic damping in dimensions 3 through 5 when μ is sufficiently large and certain subquadratic logarithmically modified damping in two dimensions. The authors note that these results do not determine whether blow-up is possible under genuinely subquadratic damping or under quadratic damping with a sufficiently small coefficient.

Resolving this question would determine whether the obtained boundedness results are optimal in the dimensions addressed by the fully parabolic theory, particularly dimensions 2, 3, 4, and 5.