Nonplanar traveling fronts for nonlocal dispersal equations with bistable nonlinearity. (Q2034056)

In this paper, the existence of nonplanar traveling wave solutions of the following equation is studied:\[ u_{t}=\int_{\mathbb{R}^{N}}J(x-y)u(y,t)dy-u+f(u),\, \, \, \, (x,t)\in\mathbb{R}^{N}\times(0,\infty),\] where \(J\) and \(f\) satisfy some conditions. For example, these conditions are satisfied if:\[ J(x)=\left(\frac{1}{4\pi\lambda}\right)^{\frac{N}{2}}e^{-\, \,\frac{\left|x\right|^{2}}{4\lambda}},\, \, \, f(u)=u(u-a)(1-u),\, \, \, a\in(0,1/2),\] for any given \(\lambda >0\) or if \(J(x)\) is compactly supported. The paper is organized as follows. Section 1 is an Introduction. In Section 2, the existence of traveling wave solutions with pyramidal shapes in \(\mathbb{R}^{3}\) is established. Some discussions are given in final Section 3.