Asymptotic speeds of spread for a nonlocal diffusion equation
The author studies the spatial spreading dynamics of a class of nonlocal diffusion equation \[ {u_t}(t,x) = (J * u - u)(t,x) + f(u,G * * S(u))(t,x), \] where \(t \geq 0\), \(x \in \mathbb R\) and \(J \in {L^1}(\mathbb R)\) is a nonnegative symmetric kernel function. In a former work for the same equation, the author proved that there exists a critical speed \({c^ * } > 0\) such that this nonlocal diffusion equation has a unique regular traveling wave solution if and only if \(c > {c^ * }\) in the case when the investigated equation does not necessarily admit the comparison principle. The present paper establishes that this \({c^ * }\) is the asymptotic speed of propagation. In additional as an application of the obtained theoretical results, are considered some models which arising from population dynamics and neural network and the corresponding spreading speed are established for these models.
scientific article; zbMATH DE number 6921336
scientific article