Global solutions and uniform boundedness of attractive/repulsive LV competition systems

Abstract

Abstract In this paper, we study global solutions to the following strongly coupled systems: {ut=∇⋅(D1∇u−χu∇v)+(a1−b1u−c1v)u,x∈Ω,t>0,0=D2Δv+(a2−b2u−c2v)v,x∈Ω,t>0, $$\textstyle\begin{cases} u_{t}=\nabla\cdot(D_{1} \nabla u -\chi u \nabla v)+(a_{1}-b_{1}u-c_{1}v)u,\quad x \in\Omega,t>0, \\ 0=D_{2}\Delta v+(a_{2}-b_{2}u-c_{2}v)v,\quad x \in\Omega,t>0, \end{cases} $$ over Ω⊂RN $\Omega\subset\mathbb{R}^{N}$, N≥2 $N\geq2$, subject to homogeneous Neumann boundary conditions and nonnegative initial data. Here Di $D_{i}$, ai $a_{i}$, bi $b_{i}$ and ci $c_{i}$, i=1,2 $i=1,2$, are positive constant. It is proved that this system admits global and bounded classical solutions for all χ>0 $\chi>0$. We also prove the global well-posedness for its repulsive counterpart {ut=∇⋅(D1∇u+χu∇v)+(a1−b1u−c1v)u,x∈Ω,t>0,0=D2Δv+(a2−b2u−c2v)v,x∈Ω,t>0, $$\textstyle\begin{cases} u_{t}=\nabla\cdot(D_{1} \nabla u +\chi u \nabla v)+(a_{1}-b_{1}u-c_{1}v)u,\quad x \in\Omega,t>0, \\ 0=D_{2}\Delta v+(a_{2}-b_{2}u-c_{2}v)v,\quad x \in\Omega,t>0, \end{cases} $$ provided that b1>a2b2χ(N−2)c2D2N $b_{1}>\frac{a_{2}b_{2} \chi(N-2)}{c_{2}D_{2} N}$. Our results extend (Discrete Contin. Dyn. Syst. 35:1239–1284, 2015) to higher dimensions and to its repulsive case.

Keywords

• Lotka–Volterra competition system
• Global solution
• Uniform boundedness