Analysis for a system of coupled reaction-diffusion parabolic equations arising in biology
The author investigates a model which describes the movement of the single-cell amoebae towards the region of relatively high concentration of a chemical which is produced by the amoebae themselves (the ``chemotaxis'' phenomen). It is considered the system \[ {\partial u\over\partial t}-d_1\Delta u=-\text{div}(u\nabla\psi(a)),\;{\partial a\over\partial t}-d_2\Delta a+k_2a=k_1u\quad\text{in }]0,T[\times\Omega, \] \[ {\partial u\over\partial n}-u {\partial\psi(a)\over\partial n}=0,\;{\partial a\over\partial n}=0\quad\text{on }\Gamma=\partial\Omega, \] \[ u(0,\cdot)=u_0,\;a(0,\cdot)= a_0\quad\text{in }\Omega, \] where \(\Omega\) is a smoothly bounded domain in \(\mathbb{R}^3\), \(u\) and \(a\) represent the population density and chemical concentration, respectively, \(\psi\) is a Lipschitzian function in \(\mathbb{R}\), \(d_1\), \(d_2\) are positive diffusion coefficients, \(d_1<d_2\), and \(k_1\), \(k_2\) are positive constants. In the general case the local existence and uniqueness of the solution is proved and in particular cases, for every fixed \(T>0\), global existence and uniqueness on \([0,T]\times\Omega\).
scientific article; zbMATH DE number 950646