Cauchy problem for nonlinear parabolic equations.

*(English)*Zbl 0902.35060The authors study the Cauchy problem for a degenerate parabolic equation with bounded measurable coefficients of the form
\[
u_t = (a_{ij}| \nabla | ^{p-2}u_{x_i})_{x_j}, \quad p>2
\]
in \(\mathbb{R}^n\times (0,T]\) with initial data
\[
u(x,0) = u_0(x)\in L_{loc}^1(\mathbb{R}^n).
\]
They extend results which have been obtained by R. DiBenedetto and M. A. Herrero for the evolutionary \(p\)-Laplace equation [Trans. Am. Math. Soc. 314, 187-224 (1989; Zbl 0691.35047)] to the case of a degenerate parabolic equation as above. In particular, they estimate the interior Lipschitz norm in terms of the \(L^p\) norm of \(u\), show the growth rate of a weak solution \(u\) in terms of \(t\) and prove a Harnack type estimate. Then the existence and uniqueness of the initial trace of the nonnegative weak solution is proved by the Harnack inequality. The regularity questions of the interface are also considered, and the asymptotic behavior of solutions is investigated. They show that the interface is a Hölder continuous graph if the interface is moving.

Reviewer: V.N.Grebenev (Novosibirsk)

##### MSC:

35K65 | Degenerate parabolic equations |

35B65 | Smoothness and regularity of solutions to PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |