Graph Neural Networks (GNNs) with heat kernel effectively capture the smoothness of labels and features across nodes, preventing oscillations during propagation, and denoising the graph. However, existing models typically employ a global heat kernel, where the diffusion process depends on a single, uniform diffusion time, inevitably resulting in over-smoothing. Additionally, the global heat kernel struggles to handle heterophilic graphs, where nodes exhibit varying neighbor label distributions. To address the above issues, we extend the global heat kernel by a localized scale (i.e., node-level) and integrate it with graph convolution, yielding the Localized Heat Kernel for GNN (LHK-GNN). By adaptively adjusting the diffusion time for each node, our approach enables heat diffusion to accommodate local complexity on graph. Experiments demonstrate the effectiveness of LHK-GNN in mitigating over-smoothing and handling heterophilic graphs.