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On the multiplicity of solutions of a simplified tumor growth model with a free boundary

We consider a mathematical model of tumor growth taking into account the spatial structure and include a different phases because of the lack of the nutriments. This model is expressed as a PDE on a spherical domain describing the tumor region. The nonlinearity of this PDE is discontinuous, and our problem can be regarded as a free boundary problem. In fact, the boundary of the tumor and another part in the interior are unknown. We obtain a multiplicity result of solutions together with some properties of the associated free boundaries.