Long-time behavior for a nonlocal model from directed polymers

We consider the long time behavior of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers. The model is characterized by convolution with a kernel $R$ and an $L^2$ inner product. In one spatial dimension, we extend a previous result of the authors [arXiv:2002.02799], where only the case $R =\delta$ was considered; in particular, we show that solutions spread according to a $2/3$ power law consistent with the KPZ scaling conjectured for direct polymers. In the special case when $R = \delta$, we find the exact profile of the solution in the rescaled coordinates. We also consider the behavior in higher dimensions. When the dimension is three or larger, we show that the long-time behavior is the same as the heat equation in the sense that the solution converges to a standard Gaussian. In contrast, when the dimension is two, we construct a non-Gaussian self-similar solution.