Study on zero sets of solutions of Heat Equations

Research Project

Project/Area Number	10640170
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Basic analysis
Research Institution	Hyogo University of Teacher Education
Principal Investigator	WATANABE Kinji Hyogo Univ. of Teacher Education, Professor, 学校教育学部, 教授 (20004468)
Project Period (FY)	1998 – 2001
Project Status	Completed (Fiscal Year 2001)
Budget Amount *help	¥1,800,000 (Direct Cost: ¥1,800,000) Fiscal Year 2001: ¥500,000 (Direct Cost: ¥500,000) Fiscal Year 2000: ¥400,000 (Direct Cost: ¥400,000) Fiscal Year 1999: ¥500,000 (Direct Cost: ¥500,000) Fiscal Year 1998: ¥400,000 (Direct Cost: ¥400,000)
Keywords	partial differential equations / Heat equations / zero sets / Heat equation / partial differential aquation / Partial differential equations / strong uniqueness
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.