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Evolution of nonparametric surfaces with speed depending on curvature. II: The mean curvature case. (English) Zbl 0808.53004
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ n\), \(n \geq 2\) with \(C^ \infty\) boundary \(\partial \Omega\) and let \(Q\) be the cylinder \(\Omega \times (0, \infty)\). The authors investigate the solvability and asymptotic behavior of solutions of the following initial-boundary value problem \(u_ t = (1 + | Du|^ 2)^{1/2} H(u)\) in \(Q\), \(u(x,t) = 0\) in \(\partial \Omega \times [0,\infty)\), \(u(x,0) = u_ 0(x)\) in \(\Omega\), \((u_ 0 \in C^ \infty _ 0(\Omega))\), where \(H(u)=\text{div} (D(u) / \sqrt{1 + | Du|^ 2})\), \(Du = \text{grad }u\). This problem describes an evolution of the surface \((x, u(x,t))\), \(x \in \overline{\Omega}\), propagating with normal speed equal to the mean curvature of the surface at the moment \(t\). The case of the Gauss curvature is studied in the first author’s paper [Part I, Indiana Univ. Math. J. 40, 237-258 (1991; Zbl 0737.53002)].

MSC:
53A05 Surfaces in Euclidean and related spaces
35G10 Initial value problems for linear higher-order PDEs
37C10 Dynamics induced by flows and semiflows
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