  In this paper, we consider a Hele-Shaw model that describes tumor growth
subject to nutrient supply. This model was recently studied in
\cite{feng2022tumor} via asymptotic analysis. Our contributions are twofold:
Firstly, we provide a rigorous derivation of this Hele-Shaw model by taking the
incompressible limit of the porous medium reaction-diffusion equation, which
solidifies the mathematical foundations of the model. Secondly, from a
bifurcation theory perspective, we prove the existence of non-symmetric
traveling wave solutions to the model, which reflect the intrinsic boundary
instability in tumor growth dynamics.
