Existence of Weak Solutions for Some Singular Parabolic Equations with Gradient Terms

A class of singular parabolic equations with gradient terms is studied in this paper, with Dirichlet boun-dary condition and initial condition, of the from $\left\{\begin{array}{l}y_t-y^{\prime \prime}-\frac{\kappa}{r} y^{\prime}+\lambda \frac{\left|y^{\prime}\right|^2}{y^m}=f(r, t) \quad(y \geqslant 0, (r, t) \in(0, 1) \times(0, T]), \\y(0, t)=y(1, t)=0 \ \ \ \ \ \quad(t \in(0, T]), \\y(r, 0)=\varphi(r) \quad(r \in(0, 1)), \end{array}\right.$ where T>0, κ≥0, λ>0, m∈(0, 2). Since singular parabolic equations containing gradient terms have singular terms and nonlinear terms, the parabolic regularization method is used to regularize the equation, and then combined with the sub-super solutions method, the existence of weak solutions of the equations under different assumptions is proved. Finally, the weak comparison principle of the equations is proved.