Malignant Tumours Spread Model Balancing Concentrations of the Invasive Cells and Extracellular Matrix

This paper investigates a class of malignant tumor spread models that balance the concentrations of invasive cells and the extracellular matrix. Using symmetry methods, invariant solutions are derived for the reaction-diffusion system governing tumor invasion. For a particular solution, numerical analysis reveals oscillatory spatial behavior governed by two parameters, with exponential growth in invasive cell concentration and linear growth in matrix degradation, which is characteristic of malignant tumors. New global existence results for classical solutions are established using fixed point theorems for the sum of two operators, proving the existence of one or more nonnegative solutions under suitable conditions. The approach combines the Kuratowski measure of noncompactness with expansive operator theory. A concrete mathematical example illustrates that the assumptions and conditions of the results can be satisfied.