Existence and uniqueness for the population dynamics with nonlinear diffusion
The author treats the existence and uniqueness problem for an age structured population equation with nonlinear spatial diffusion. The equation is \[ u_ t+u_ a+\mu(a)u-\Delta_ x\beta(u)=f,\quad 0<a<A,\quad x\in\Omega,\quad 0<t<T, \] \[ \beta(u(a,x,t))=0,\quad 0<a<A,\quad x\in\partial\Omega,\quad 0<t<T, \] \[ u(0,x,t)=\int^ A_ 0 b(a)u(a,x,t)da,\quad x\in\Omega,\quad 0<t<T, \] \[ u(a,x,0)=u_ 0(a,x),\qquad 0<a<A,\qquad x\in\Omega. \] Here \(\Omega\) is a bounded open set in \(\mathbb{R}^ N\) with sufficiently smooth boundary. The method used involves nonlinear dissipative operator theory. The particular case that \(\beta(u)=| u|^{m-1} u\), \(m>(N-2)/N\) is also considered.
scientific article; zbMATH DE number 11962