Xuming XieAMAL HAMAD AHMED ALDAKHIL2022-06-012022-06-01https://drepo.sdl.edu.sa/handle/20.500.14154/55973In this thesis, we considered a free boundary problem that arise in the mathematical modeling of cancer and its treatment [1]. The free boundary problem consists a system of two semi-linear partial differential equations that represents the densities of cancer cells and T cells. We proved the existence of stationary spherical solutions if the killing rate η was in the interval determined by the other parameters in the model. We also proved that the stationary solution was unstable. Biologically, these results imply that if the initial cancer cell density is at the level of the steady state, the cancer will not grow. However, if the initial cancer density is a little different than the level of steady state, the cancer will grow indefinitely.en