| Co-Investigator(Kenkyū-buntansha) |
NAKAHARA Toru Saga University, Faculty of Science and Engineering; Professor, 理工学部, 教授 (50039278)
森 通 佐賀大学, 理工学部, 教授 (60039352)
TOMISAKI Matsuyo Saga University, Faculty of Science and Engineering; Associate Professor, 理工学部, 助教授 (50093977)
KUBO Masahiro Saga University, Faculty of Science and Engineering; Assistant, 理工学部, 助手 (80205129)
OGURA Yukio Saga University, Faculty of Science and Engineering; Professor, 理工学部, 教授 (00037847)
MORI Toru Saga University, Faculty of Science and Engineering; Professor (50039278)
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| Research Abstract |
1. We studied the existence of positive solutions of the quasilinear elliptic equation (*) Lu = f(x,u,Du) in R^N, where n 2 and is a real constant. In particular, we gave sufficient conditions for (*) of for the exterior boundary value problem associated with (*) to have a positive solution converging to a nonnegative constant. In the special case f = f(x,u), we proved the existence of a maximal decaying positive solution of (*) under some additional conditions for f, and we showed the uniqueness of decaying positive solution of (*). As a preliminary lemma, we improved the theorem of A. Friedman (1973) and of E. S. Noussaior and C. A. Swanson (1988) on the existence of the solutions for linear elliptic equations.
2. For the equation (*) with f satisfying f(x,u,0) = 0 and sublinearity with respect to Du at Du = 0, we established a existing theory of nonconstant positive solutions which converge to nonnegative constants or grow to infinity as |x| .
3. We proved the existence of positive solutions of the exterior Dirichlet problem for some elliptic equations with singular coefficients on the boundary. In a special case, we found three positive solutions such that one of them blew up at some finite place and the others grew with different growth order. We also studied theexistence of solutions of the initial-boundary value problem for parabolic equations with unbounded coefficients.
4. Besides the topics mentioned above, the research of this project covered various topics in the mathematical field as follows: bi-generalized diffusion processes; Kimura's diffusion models; elliptic-parabolic variational inequatities; asymptotic behavior of elementary solutions of generalized diffusion equations; periodic solutions of congruences; Jacobi sums and cyclotomic units; Gauss map on Riemannian manifolds; geometric and stochastic mean value and first exit time from geodesic balls in Riemannian manifolds; and Umbral calculus; convex filtration on Riemannian manifolds.
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