For the 1D hyperbolic-parabolic chemotaxis model, smooth initial data with sufficiently steep negative slope produce finite-time shock-type blow-up: density and velocity stay bounded while their gradients diverge, forming a C^{1/3} cusp at a unique point.
Formation of point shocks for 3D compressible Euler Communi- cations on Pure and Applied Mathematics , 76(9):2073–2191, 2023
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Shock-type singularity of the hyperbolic-parabolic chemotaxis system
For the 1D hyperbolic-parabolic chemotaxis model, smooth initial data with sufficiently steep negative slope produce finite-time shock-type blow-up: density and velocity stay bounded while their gradients diverge, forming a C^{1/3} cusp at a unique point.