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The Cauchy problem of some nonlinear doubly degenerate parabolic equations. (Chinese) Zbl 0828.35071
The authors consider the problem: \[ u_t = \text{div} \bigl( |\nabla u^m |^{p - 2} \nabla u^m \bigr), \text{ in } S_T = \mathbb{R}^N \times (0,T), \quad u(x,0) = u_0 (x),\;x \in \mathbb{R}^N, \tag{1} \] where \(p > 1\), \(m > 0\), \(u_0 \geq 0\). 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 \(v_0\) \((u_0, v_0 \in L^1 (\mathbb{R}^N))\), then \[ \int_{\mathbb{R}^N} \bigl |u (x,t) - v(x,t) \bigr |dx \leq \int_{\mathbb{R}^N} \bigl |u_0(x) - v_0(x) \bigr |dx \] holds.

MSC:
35K65 Degenerate parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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