Global solutions of a doubly tactic resource consumption model with logistic source

Abstract

We study a doubly tactic resource consumption model \bess \left\{\begin{array}{lll} u_t=\tr u-\nabla\cd(u\nabla w),\\[1mm] v_t=\tr v-\nabla\cd(v\nabla u)+v(1-v^{β-1}),\\[1mm] w_t=\tr w-(u+v)w-w+r \end{array}\right. \eess in a smooth bounded domain $\oo\in\R^2$ with homogeneous Neumann boundary conditions, where $r\in C^1(\bar{\Omega }\times [0,\infty ))\cap L^{\infty }(\Omega \times (0,\infty ))$ is a given nonnegative function fulfilling \bess \int_t^{t+1}\ii|\nn\sqrt{r}|^2<\yy\ \ \ \ \ \ for\ all\ t>0. \eess It is shown that, firstly, if $\beta >2$, then the corresponding Neumann initial-boundary problem admits a global bounded classical solution. Secondly, when $\beta =2$, the Neumann initial-boundary problem admits a global generalized solution.

21 pages

References in corpus (2)

• Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis
• Global generalized solutions to a forager-exploiter model with superlinear degradation and their eventual regularity properties