Analysis of doubly degenerate chemotaxis-consumption systems / Duan Wu ; Researched and written at the Institute of Mathematics of Paderborn University under the supervision of Herr Univ.-Prof. Dr. Michael Winkler. Paderborn, 2026
Inhalt
- Introduction
- Global solvability and nontrivial large time behavior in a one-dimensional problem
- Introduction
- Preliminaries
- Regularized problems, local existence and basic properties
- Testing procedures and a crucial functional inequality
- Uniform Lp-boundedness of u
- Uniform L-boundedness of u
- Proofs of Theorem 2.1.1 and Corollary 2.1.2
- Enhanced migration mechanisms in two-dimensional models
- Introduction
- Preliminaries
- Uniform Lp-boundedness of u
- Uniform L-boundedness of u
- Global solvability. Proof of Theorem 3.1.1
- Harnack-type inequality for v
- Large time behavior. Proof of Theorem 3.1.4
- Effects of sublinear signal consumption rates
- Introduction
- Preliminaries
- Uniform Lp-boundedness of u
- Uniform L-boundedness of u
- Global weak solutions. Proof of Theorem 4.1.1
- Large time behavior. Proof of Theorem 4.1.2
- Nonnegative solutions in the critical case on planar domains
- Introduction
- Preliminaries
- Regularized problems and basic properties
- Two functional inequalities resulting from a planar Sobolev embedding
- Basic testing procedures
- An asymptotic energy structure
- Short time analysis
- Space-time bounds for |v|4v3, |v|2v and uv |v|2 as well as u2 v
- Bounds for u in LlogL and Llog2 L
- A pointwise lower bound for v. Estimates for u in Lp
- Large time regularity implied by eventual smallness of v
- Proofs of Theorem 5.1.1 and Corollary 5.1.2
- Global boundedness and large time behavior in systems involving growth saturation
- Introduction
- Preliminaries
- Regularized problems and basic properties
- Basic testing procedures, a functional inequality and an ODI result
- Uniform Lp-boundedness of u
- Uniform L-boundedness of u
- Global weak solution. Proof of Theorem 6.1.1
- Large time behavior. Proof of Theorem 6.1.4
- Bibliography