Entire solutions in nonlocal monostable equations: asymmetric case. (English) Zbl 1409.35117
Summary: This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., \(u_{t}=J*u-u+f(u)\). Here the kernel \(J\) is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds \(c^{*}\) and \(\hat{c}^{*}\), where \(c^*\) and \(\hat{c}^{*}\) are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel \(J\) is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B08 | Entire solutions to PDEs |
| 34C14 | Symmetries, invariants of ordinary differential equations |
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