Black-box Domain Adaptation (BDA) utilizes a black-box predictor of the source domain to label target domain data, addressing privacy concerns in Unsupervised Domain Adaptation (UDA). However, BDA assumes identical label sets across domains, which is unrealistic. To overcome this limitation, we propose a study on BDA with unknown classes in the target domain. It uses a black-box predictor to label target data and identify "unknown" categories, without requiring access to source domain data or predictor parameters, thus addressing both data privacy and category shift issues in traditional UDA. Existing methods face two main challenges: (i) Noisy pseudo-labels in knowledge distillation (KD) accumulate prediction errors, and (ii) relying on a preset threshold fails to adapt to varying category shifts. To address these, we propose ADU, a framework that allows the target domain to autonomously learn pseudo-labels guided by quality and use an adaptive threshold to identify "unknown" categories. Specifically, ADU consists of Selective Amplification Knowledge Distillation (SAKD) and Entopy-Driven Label Differentiation (EDLD). SAKD improves KD by focusing on high-quality pseudo-labels, mitigating the impact of noisy labels. EDLD categorizes pseudo-labels by quality and applies tailored training strategies to distinguish "unknown" categories, improving detection accuracy and adaptability. Extensive experiments show that ADU achieves state-of-the-art results, outperforming the best existing method by 3.1\% on VisDA in the OPBDA scenario.