2001 Fiscal Year Final Research Report Summary
Study on zero sets of solutions of Heat Equations
| Project/Area Number |
10640170
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Hyogo University of Teacher Education |
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Principal Investigator |
WATANABE Kinji Hyogo Univ. of Teacher Education, Professor, 学校教育学部, 教授 (20004468)
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| Project Period (FY) |
1998 – 2001
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| Keywords | partial differential equations / Heat equations / zero sets |
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| Research Abstract |
The following results are obtained and partially published in Hyogo University of Teacher Education Journal, vol. 21, No 3, (2001), pp. 1-9.
Let u(t, x, y) be an analytic solution of second order parabolic equations defined in neigbourhood of the origin (0, 0, 0) with two spatial dimension and let Z(t) be its zero sets, dependent on time varialble t, in some neighborhood of the origin (0, 0). Then one of the main results is the following. (1). When Z(t) is topologically homeomorphic to Z(0) for sufficiently small -t > 0 , then Z(t) is topologically homeomorphic to Z(Q) for sufficiently small t > 0. (2). Converse statement is holds under some assumptions.
To prove these, it is useful to study on zero sets of Hermite polynomials with two independent variables and their conjugate Hermite polynomials, which determine the initial form of u(t, x, y) in some sense.
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