| Abstract: |
| In recent years, the mathematical models with nonlocal interactions have been proposed in various fields: material science, neural fields, pattern formation problem and so on.
A nonlocal interaction is given by the convolution with the suitable integral kernel, so its mathematical model is described by the integro-differential equations.
A typical example of a kernel in the neural fields is the Mexican hat type function. In the case that the kernel is non-negative, several studies have reported the existence and stability of traveling wave solutions.
However, few are known about the existence of traveling wave solutions in the case of sign-changing kernels.
Our study revealed the existence of traveling wave solutions connecting the unstable state and the stable state to a nonlocal scalar equation with sign-changing kernel.
In this talk, we first introduce a new notion of upper-lower-solution for the equation of wave profile for a given wave speed.
We construct two different pairs of upper-lower-solutions and use Schauder`s fixed point theorem to obtain traveling waves for a continuum of wave speeds under some assumptions.
Finally, we analyze wave profiles by showing the limit of the both-hand tails. |
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