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The Cauchy problem of some nonlinear doubly degenerate parabolic equations. (Chinese) Zbl 0828.35071
The authors consider the problem: $$u_t = \text{div} \bigl( |\nabla u^m |^{p - 2} \nabla u^m \bigr), \text{ in } S_T = \bbfR^N \times (0,T), \quad u(x,0) = u_0 (x),\ x \in \bbfR^N, \tag 1$$ where $p > 1$, $m > 0$, $u_0 \ge 0$. They study the existence and uniqueness and the long-time behavior of the solution of (1). The main result is as follows: Let $u$ and $v$ be two solutions with initial conditions $u_0$ and $v_0$ $(u_0, v_0 \in L^1 (\bbfR^N))$, then $$\int_{\bbfR^N} \bigl |u (x,t) - v(x,t) \bigr |dx \le \int_{\bbfR^N} \bigl |u_0(x) - v_0(x) \bigr |dx$$ holds.

35K65Parabolic equations of degenerate type
35K15Second order parabolic equations, initial value problems
35B40Asymptotic behavior of solutions of PDE