| Abstract: |
| In recent years, mathematical models with nonlocal effects described by convolution with suitable integral kernels have been proposed in various fields: material science, neural fields, and so on. Recent studies have shown that those nonlocal scalar equations can be derived as certain reduced systems of reaction-diffusion systems. One of the studies has pointed out that the properties of reaction-diffusion networks can be described by the shape of the kernel, and especially that structures such as Turing instability can be obtained using a kernel with local activation and long-range inhibition. Such sign-changing kernels have been pointed out for their importance from the viewpoint of pattern formation problems, but there have been few studies on traveling waves.
This talk will consider the existence of traveling waves connecting the unstable state and the stable state of a nonlocal scalar equation with a sign-changing kernel. We first describe the problem setup and then present numerical results on traveling waves and other spatio-temporal patterns. Next, we introduce new upper-lower solutions to prove the existence of traveling waves. Finally, we construct them and analyze asymptotic profiles of traveling waves. |
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