This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system \begin{equation}\label{eq:abstract} \begin{cases} u_{t}= u_{xx} + u(1-u-av), & \qquad \ x\in\mathbb{R} \cr v_{t}= d v_{xx}+ rv(1-v-bu), & \qquad \ x\in\mathbb{R} \end{cases} \quad (1) \end{equation} where $d,r,a$, and $b$ are positive constants with $a\neq 1$ and $b\neq 1$. We prove the existence of some entire solutions $(u(t,x),v(t,x))$ of $(1)$ corresponding to $(\Phi_{c}(\xi),0)$ at $t=-\infty$ (where $\xi=x-ct$ and $\Phi_c$ is a traveling wave solution of the scalar Fisher-KPP defined by the first equation of $(1)$ when $a=0$). Moreover, we also describe the asymptotic behavior of these entire solutions as $t\to+\infty$. We prove existence of new entire solutions for both the weak and strong competition case. In the weak competition case, we prove the existence of a class of entire solutions that forms a 4-dimensional manifold.

中文翻译：

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扩散 Lotka-Volterra 系统的完整解决方案

这项工作涉及扩散 Lotka-Volterra 竞赛系统的完整解的存在性 \begin{equation}\label{eq:abstract} \begin{cases} u_{t}= u_{xx} + u(1-u -av), & \qquad \ x\in\mathbb{R} \cr v_{t}= d v_{xx}+ rv(1-v-bu), & \qquad \x\in\mathbb{R} \end{cases} \quad (1) \end{equation} 其中 $d,r,a$ 和 $b$ 是正常数，$a\neq 1$ 和 $b\neq 1$。我们证明了 $(\Phi_{c}(\xi),0)$ 对应于 $(\Phi_{c}(\xi),0)$ 的 $(1)$ 的一些全解 $(u(t,x),v(t,x))$ t=-\infty$（其中$\xi=x-ct$ 和$\Phi_c$ 是$a=0$ 时$(1)$ 的第一个方程定义的标量Fisher-KPP 的行波解） . 此外，我们还将这些整个解决方案的渐近行为描述为 $t\to+\infty$。我们证明对于弱竞争和强竞争情况都存在新的完整解决方案。