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Global existence and quenching phenomena for a parabolic equation of the mean curvature type with nonlinear convection term. (Chinese. English summary) Zbl 1066.35046

Summary: The paper studies the first initial-boundary value problem in a bounded domain in \(\mathbb{R}^n\) for the mean curvature equation with a nonlinear convection term: \(u_t-\text{div}\{\sigma (|\nabla u|^2)\nabla u\}+b(u)\cdot \nabla u=0\). The existence of the weak solution is derived and the quenching phenomena and the \(L^\infty\)-estimate to the solution are given.

MSC:

35K65 Degenerate parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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