| Abstract: |
| We discuss qualitative properties as blow-up phenomena to solutions of some classes of parabolic problems. In particular, we consider a problem of Keller-Segel type under Neumann boundary conditions in a smooth and bounded domain $\Omega \subset \mathbb{R}^n, \ n \geq 3$, and we show a criterion which ensures that, under suitable conditions on the data, the solution, which blows up in finite time in the $L^{\infty}(\Omega)$-norm, it also blows up in the $L^p(\Omega)$-norm with suitable exponents $p$.
Moreover, we consider unbounded solutions of a chemotaxis system with nonlinear diffusion. Under appropriate assumptions on data, a safe interval of existence of the solution is derived with a lower bound of the lifespan. |
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