Cauchy’s Problem for Degenerate Quasilinear Parabolic Equations

Abstract

In this paper we consider the differential properties of the solution to the Cauchy problem for the quasilinear parabolic equation \[ (1)\qquad \frac{{\partial v}}{{\partial t}} = \frac{1}{2}\sum\limits_{i,j = 1}^n {c_{ij} } (t,x,v)\frac{{\partial ^2 v}}{{\partial x_i \partial x_i }} + \sum\limits_{i = 1}^n {a_i } (t,x,v)\frac{{\partial v}}{{\partial x_i }}, \] where $c_{ij} = \sum\nolimits_{k = 1}^n {b_{ik} } (t,x,v)b_{jk} (t,x,v)$.Let class $C_T^{m,\gamma } $ be a class of continuous functions, which has bounded space derivatives up to the m-th order, and its m-th derivative is Holder continuous with Holder constant $\gamma $It is proved in this paper that if \[ \{ {b(t,x,v),a(t,x,v),\psi (x)} \} \in C_T^{m,\gamma } ,\qquad m \geqq 2, \] then $v \in C_{t_0 }^{m,\gamma } $, where $t_0 > 0$ depends only on the constants of class $C_T^{m,\gamma } $. If $n = 1$, $a(t,x,v) \equiv 0$, then the above assertion will follow for all $t \in [0,T]$ and $x \in ( - \infty ,\infty )$. It is noted that $b(t,x,v)$ may b...