边流形上的双调和热算子与非线性四阶方程
The biharmonic heat operator on edge manifolds and non-linear fourth order equations
摘要：
我们构造了双调和热核的一个适当的自伴扩张的双拉普拉斯流形上的不完全边奇点。我们采用微局部描述的双调和热核建立相应的双调和热核算子在某些巴拿赫空间上的映射性质。这就得到了一类四阶半线性方程的短时存在性，例如Cahn-Hilliard方程。我们还得到了解在边缘奇异点附近的渐近性。
英文摘要：
We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. This yields short time existence for a class of semi-linear equations of fourth order, including for example the Cahn-Hilliard equation. We also obtain asymptotics of the solutions near the edge singularity.
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