In this talk, I will discuss the intriguing behavior of nonlocal diffusion equations with free boundaries in high dimensions. These equations exhibit a "spreading-vanishing" dichotomy in their long-term solutions, reminiscent of local problems. However, a remarkable distinction arises in the determination of the spreading front's speed, which can reach infinity, unlike in local problems. My collaborator, Professor Du, and I have obtained improved estimates for the spreading speed, both finite and infinite, particularly focusing on kernel functions behaving like |x|^r near infinity. Our findings represent the first results of their kind for this class of free boundary problems, and we have also investigated the spreading speed in high-dimensional settings.

 

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