This paper studies initial boundary value problems for systems of quasilinear elliptic-parabolic differential equations. The equations are of the form $\partial_{t}b^{j}(u)-\nabla\cdot a^{j}(b(u),\nabla u)=f^{j}(b(u))$ in $]0,T[\times\Omega$, with boundary conditions on $\partial\Omega$. The structure conditions include ellipticity of $a$ and (weak) monotonicity of $b$, with $b$ being a subgradient when $m>1$. The paper first considers the case where $b$ is continuous and later includes Stefan problems where $b$ can have jumps. Special cases include elliptic equations with time as parameter and standard parabolic equations. The gas flow through a porous medium is described by $\partial_{t}u=\varDelta u^{m}$ with $m>1$, which can be transformed into $\partial_{t}b(u)=\varDelta u$ with $b(z):=\max{(z,0)^{1/m}}$ or $(\mathrm{sign}z)|z|^{1/m}$. Solutions to this equation have been studied by Oleinik, Kalashnikov, Yui-Lin, and Aronson. These equations differ from parabolic ones because $b$ is not Lipschitz continuous, allowing solutions with compact support. Another example is $\partial_{t}\max{(u,0)}=\varDelta u$ with constant negative Dirichlet data and positive initial data. The solution becomes stationary after a finite time, indicating that the parabolic region collapses. However, $\partial_{t}b(u)$ is shown to be in $L^{2}(]0,T[\times\Omega)$. An explicit solution is given for this case. The paper also discusses applications to nonsteady filtration, where the equation $\partial_{t}\theta(p)=\nabla\cdot(k(\theta(p))(\nabla p+e))$ is solved for incompressible fluid flow in a porous medium. The Kirchhoff transformation leads to a differential equation of the form $\partial_{t}b(u)=\nabla\cdot(\nabla u+k(b(u))e)$. The paper provides a detailed analysis of these equations and their solutions.This paper studies initial boundary value problems for systems of quasilinear elliptic-parabolic differential equations. The equations are of the form $\partial_{t}b^{j}(u)-\nabla\cdot a^{j}(b(u),\nabla u)=f^{j}(b(u))$ in $]0,T[\times\Omega$, with boundary conditions on $\partial\Omega$. The structure conditions include ellipticity of $a$ and (weak) monotonicity of $b$, with $b$ being a subgradient when $m>1$. The paper first considers the case where $b$ is continuous and later includes Stefan problems where $b$ can have jumps. Special cases include elliptic equations with time as parameter and standard parabolic equations. The gas flow through a porous medium is described by $\partial_{t}u=\varDelta u^{m}$ with $m>1$, which can be transformed into $\partial_{t}b(u)=\varDelta u$ with $b(z):=\max{(z,0)^{1/m}}$ or $(\mathrm{sign}z)|z|^{1/m}$. Solutions to this equation have been studied by Oleinik, Kalashnikov, Yui-Lin, and Aronson. These equations differ from parabolic ones because $b$ is not Lipschitz continuous, allowing solutions with compact support. Another example is $\partial_{t}\max{(u,0)}=\varDelta u$ with constant negative Dirichlet data and positive initial data. The solution becomes stationary after a finite time, indicating that the parabolic region collapses. However, $\partial_{t}b(u)$ is shown to be in $L^{2}(]0,T[\times\Omega)$. An explicit solution is given for this case. The paper also discusses applications to nonsteady filtration, where the equation $\partial_{t}\theta(p)=\nabla\cdot(k(\theta(p))(\nabla p+e))$ is solved for incompressible fluid flow in a porous medium. The Kirchhoff transformation leads to a differential equation of the form $\partial_{t}b(u)=\nabla\cdot(\nabla u+k(b(u))e)$. The paper provides a detailed analysis of these equations and their solutions.