Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion (Q2822714)

The paper is devoted to the existence of transition fronts for a class of reaction diffusion equations with inhomogeneous Kolmogorov-Petrovskii-Piskunov type non-linearities and non-local diffusion. Under consideration is the equation NEWLINE\[NEWLINE u_{t}(t,x) = \int_{\mathbb R}J(y)(u(t,x-y)-u(t,x))\,dy + f(x, u), \;\;t,x\in {\mathbb R}.\eqno{(1)} NEWLINE\]NEWLINE The kernel \(J\in C^{1}({\mathbb R})\) is an even nondecreasing function with compact support such that \(\int_{\mathbb R}J(y)\,dy=1\). The reaction function \(f\in C^{2}({\mathbb R}\times [0,1])\) is nonnegative and \(f(x,0)=f(x,1)=0\). By a transition front to the equation (1), the authors mean a solution \(u(t,x)\) to (1) such that \(0\leq u(t,x)\leq 1\) for all \((t,x)\), \(\lim_{x\to -\infty}u(t,x)=1\), \(\lim_{x\to +\infty}u(t,x)=0\), and \(\sup_{t\in {\mathbb R}}\text{diam}\{x\in {\mathbb R}:\varepsilon \leq u(t,x)\leq 1-\varepsilon\}<\infty\) for every \(\varepsilon>0\). The main results of the article are the existence theorem of transition fronts (under some additional constraints on the functions occurring in (1)) and the methods of their construction.