The following degenerate parabolic system modelling chemotaxis is considered. $$ {\mbox{(KS)}} \qquad\qquad \left\{ \begin{array}{llll} & u_t = \nabla \cdot \Big( \nabla u^m - u \nabla v \Big), & x \in \mathbb R^N, \ 0 <t <T, \nonumber \\ & \tau v_t = \Delta v - v + u, & x \in \mathbb R^N, \ 0 <t <T, \nonumber \\ & u(x,0) = u_0(x), \quad \tau v(x,0) = \tau v_0(x), & x \in \mathbb R^N, \end{array} \right. $$ where $m>1, \tau=0$ or 1, and $N \ge 1$. Our aim in this paper is to prove the existence of a global weak solution of (KS) under some appropriate conditions on $m$ without any restriction on the size of the initial data. Specifically, we show that a solution ($u,v$) of (KS) exists globally in time if either (i) $m \ge 2 $ for large initial data or (ii) $1 < m \le 2-\frac{2}{N}$ for small initial data. In the case of (ii), the decay properties with the optimal rate of the solution ($u,v$) are also discussed.

We consider a class of multi-dimensional models and prove the existence of non-trivial steady periodic waves. One of the main results is concerned with the spectral instability of a stationary periodic wave, proved by means of Floquet's theory and the introduction of the Evans function for a suitable related eigenvalue problem. A description of the zero set of the Evans function around the origin allows us to establish a link between the spectral stability analysis and a first-order system of conservation laws derived from the original model through homogenisation.

In this paper, we first develop some techniques and results for integrated semigroups when the generator is not a Hille-Yosida operator and is non-densely defined. Then we establish a theorem of Da Prato and Sinestrari's type for the nonhomogeneous Cauchy problem and prove a perturbation theorem. In particular, we obtain necessary and sufficient conditions for the existence of mild solutions for non-densely defined non-homogeneous Cauchy problems. Next we extend the results to $L^{p}$ spaces and consider the semilinear and non-autonomous problems. Finally we apply the results to studying age-structured models with dynamic boundary conditions in $L^{p}$ spaces. We also demonstrate that neutral delay differential equations in $L^{p}$ can be treated as special cases of the age-structured models in an $L^{p}$ space.