Asymptotic and numerical analysis of nonlocal reaction-diffusion equations - Lee, Jia Yin (2024)
This thesis consists of two parts. In the first part of the thesis, we study a local-nonlocal Fisher-Komogorov-Petrovskii-Piskunov equation, which contains a linear combination of local and nonlocal interaction terms in one spatial dimension. Similar to the local case, this equation admits two uniform steady states, one stable and the other unstable. In our study, we restrict our attention to kernels that are symmetric, strictly positive, and decreasing with distance to describe the nonlocal interaction over space. Using asymptotic and numerical analysis, we show that travelling wave solutions of this model have interesting properties when the diffusion length scale is much smaller than the interaction length scale, and the model is highly nonlocal. For example, we observe non-monotonic travelling waves behind the wavefront, connecting the two steady states. In particular, the solutions behave like Dirac delta functions where the height tends to infinity and the width tends to zero in the same limit before converging to the stable steady state.