Self-adjoint Laplace operator with translation invariance on infinite-dimensional space R∞\mathbb R^\infty

We define the translation-invariant Laplacian $-\triangle_{\mathbb R^\infty}$ on the product measurable space $\mathbb R^\infty$ as a non-negative self-adjoint operator in some Hilbert space $L^2(\mathbb R^\infty)$, which is a subset of the set $CM(\mathbb R^\infty)$ of all complex measures on $\mathbb R^\infty$. Furthermore, we show that for any $f\in L^2(\mathbb R^n)$ and any $u\in L^2(\mathbb R^\infty)$, $e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}(f\otimes u)=(e^{\sqrt{-1}\triangle_{\mathbb R^n}t}f)\otimes(e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}u) \ (t\in (-\infty,+\infty))$ and $e^{\triangle_{\mathbb R^\infty}t}(f\otimes u)=(e^{\triangle_{\mathbb R^n}t}f)\otimes(e^{\triangle_{\mathbb R^\infty}t}u) \ (t\in [0,+\infty))$ hold

Front-like entire solutions for monostable reaction-diffusion systems

This paper is concerned with front-like entire solutions for monostable reactiondiffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Combining a SIS and traveling fronts with different wave speeds and directions, the existence and various qualitative properties of entire solutions are then established using comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish some comparison arguments for the three systems. The existence of entire solutions is then proved via the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems