The chapter introduces a system of quasilinear elliptic-parabolic differential equations and their initial boundary value problem. The equations are described as: \[ \begin{aligned} & \partial_t b^j(u) - V \cdot a^j(b(u), V u) = f^j(b(u)) \quad \text{in } ]0, T[ \times \Omega, \, j = 1, \ldots, m, \\ & b(u) = b^0 \quad \text{on } \{0\} \times \Omega, \\ & u = u^D \quad \text{on } ]0, T[ \times \Gamma, \\ & a^j(b(u), V u) \cdot v = 0 \quad \text{on } ]0, T[ \times (\partial \Omega \setminus \Gamma), \, j = 1, \ldots, m. \end{aligned} \] The authors discuss the conditions for the ellipticity of \(a\) and the (weak) monotonicity of \(b\), with \(b\) being a subgradient when \(m > 1\). They first treat the case where \(b\) is continuous and later extend to include Stefan problems, allowing \(b\) to have jumps. Special cases such as elliptic equations with time as a parameter and standard parabolic equations are also covered. The chapter provides examples of applications, including gas flow through a porous medium described by the equation \(\partial_t u = \Delta u^m\) with \(m > 1\). This equation can be transformed into a form that fits the class of equations studied. Solutions to this equation have been extensively studied, and it is noted that these equations differ from parabolic ones due to the non-Lipschitz continuity of \(b\). Another example is the equation \(\partial_t \max(u, 0) = \Delta u\) with constant negative Dirichlet data and positive initial data. The solution becomes stationary after a finite time, and the parabolic region collapses. The authors show that \(\partial_t b(u)\) is at least in \(L^2([0, T] \times \Omega)\). The chapter also discusses nonsteady filtration, where one incompressible fluid in a porous medium is described by the equation \(\partial_t \theta(p) = \nabla \cdot (k(\theta(p)) (\nabla p + e))\). The Kirchhoff transformation is used to simplify this to a form \(\partial_t b(u) = \nabla \cdot (\nabla u + k(b(u)) e)\).The chapter introduces a system of quasilinear elliptic-parabolic differential equations and their initial boundary value problem. The equations are described as: \[ \begin{aligned} & \partial_t b^j(u) - V \cdot a^j(b(u), V u) = f^j(b(u)) \quad \text{in } ]0, T[ \times \Omega, \, j = 1, \ldots, m, \\ & b(u) = b^0 \quad \text{on } \{0\} \times \Omega, \\ & u = u^D \quad \text{on } ]0, T[ \times \Gamma, \\ & a^j(b(u), V u) \cdot v = 0 \quad \text{on } ]0, T[ \times (\partial \Omega \setminus \Gamma), \, j = 1, \ldots, m. \end{aligned} \] The authors discuss the conditions for the ellipticity of \(a\) and the (weak) monotonicity of \(b\), with \(b\) being a subgradient when \(m > 1\). They first treat the case where \(b\) is continuous and later extend to include Stefan problems, allowing \(b\) to have jumps. Special cases such as elliptic equations with time as a parameter and standard parabolic equations are also covered. The chapter provides examples of applications, including gas flow through a porous medium described by the equation \(\partial_t u = \Delta u^m\) with \(m > 1\). This equation can be transformed into a form that fits the class of equations studied. Solutions to this equation have been extensively studied, and it is noted that these equations differ from parabolic ones due to the non-Lipschitz continuity of \(b\). Another example is the equation \(\partial_t \max(u, 0) = \Delta u\) with constant negative Dirichlet data and positive initial data. The solution becomes stationary after a finite time, and the parabolic region collapses. The authors show that \(\partial_t b(u)\) is at least in \(L^2([0, T] \times \Omega)\). The chapter also discusses nonsteady filtration, where one incompressible fluid in a porous medium is described by the equation \(\partial_t \theta(p) = \nabla \cdot (k(\theta(p)) (\nabla p + e))\). The Kirchhoff transformation is used to simplify this to a form \(\partial_t b(u) = \nabla \cdot (\nabla u + k(b(u)) e)\).