Global boundedness of Lotka-Volterra competition system with cross-diffusion

Abstract

In this paper, we consider the following Lotka-Volterra competition system with cross-diffusion$\{\begin{array}{cc}& u_t=\Delta \left(\varphi \right(w\left)u\right)+{\alpha }_1\Delta \left(uv\right)-{\alpha }_{2}uv+{\alpha }_{3}uw-{\alpha }_{4}u,\\ & v_t=\Delta \left(\varphi \right(w\left)v\right)+{\beta }_1\Delta \left(uv\right)-{\beta }_{2}uv+{\beta }_{3}vw-{\beta }_{4}v,\\ & w_t=\Delta w-uw-vw+r_{3}w(1-w),\end{array}$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset R^n(n=1,2)$ with ${\alpha }_4>{\alpha }_0$ and ${\beta }_4>{\beta }_0$ (${\alpha }_0$ and ${\beta }_0$ depend on the initial data w). It is shown that the problem possesses a global classical solution for $n=1$. On the other hand, in the case $n=2$, it is proved the global existence of classical solutions for ${\alpha }_i={\beta }_i(i=1,2,3,4)$.