Facets of low regularity in cross-diffusive systems / Mario Fuest. Paderborn, 2021
Inhalt
- Introduction
- Solutions blowing up in finite-time and their blow-up profiles
- Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening
- Introduction
- Preliminaries
- The mass accumulation function w
- A supersolution to a superlinear ODE: finite-time blow-up
- Blow-up profiles in quasilinear fully parabolic Keller–Segel systems
- Introduction
- Pointwise estimates for subsolutions to equations in divergence form
- Pointwise estimates in quasilinear Keller–Segel systems
- Existence of blow-up profiles
- Uniqueness in nondegenerate quasilinear Keller–Segel systems
- On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems
- Global existence in fully cross-diffusive systems
- Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis
- Introduction
- Preliminaries
- Estimates within [0, T eta)
- Deriving W22 bounds for u and v
- Proof of Theorem 5.1.1
- Possible generalizations of Theorem 5.1.1
- Gagliardo–Nirenberg inequalities
- Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity
- Introduction
- Global weak W12-solutions to approximative systems
- The limit process k to infty: existence of weak solutions to P_(eps delta) by a Galerkin method
- The limit process delta to 0: guaranteeing nonnegativity
- The limit process epsilon to 0: proofs of Theorem 6.2.1 and Theorem 6.2.2
- Approximative solutions to (6.P)
- The limit process alpha to 0: obtaining solution candidates
- Preliminary observations
- Controlling the right-hand side of (6.4.7)
- Space-time bounds and the limit process
- Existence of global weak solutions to (6.P): proof of Theorem 6.1.1