The application of mathematics to fields such as biology, medicine, and biotechnology has resulted in quantitative relations, and mathematical biology, an interdisciplinary branch, has emerged. The considered models of cell/tissue growth are determined by cross‐reaction‐diffusion equations with possible transitions between malignant cells with different growth rates or healthy cells. For the considered models, the exact solutions in the explicit form were not obtained in the existing literature to our knowledge. Our main aim is to fill this gap by revealing their exact solutions. The obtained results have a major role in the literature so that the considered models are seen in a large scale of applications.