*(English)*Zbl 0702.35140

Consider the nonlinear degenerate parabolic equation

where subscripts denote partial differentiation. The functions a and b are hypothesized to belong to $C\left([0,\infty )\right)\cap {C}^{2}(0,\infty )$ and be such that ${a}^{\text{'}}\left(s\right)>0$ for $s>0$, and ${a}^{\text{'}\text{'}}$ and ${b}^{\text{'}\text{'}}$ are locally HĂ¶lder continuous on (0,$\infty )$ and $a\left(0\right)=0$ and $b\left(0\right)=0$. In the present paper the author has established improved existence and uniqueness theorems for the Cauchy problem, the Cauchy-Dirichlet problem, and the first boundary value problem for equation (1). The comparison principles for generalized solutions of equation (1) and the relationship between the results given in this paper and those in earlier publications is also given.

##### MSC:

35K65 | Parabolic equations of degenerate type |

35K55 | Nonlinear parabolic equations |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. (PDE) |