2000 Fiscal Year Final Research Report Summary
Qualitative Theory of Quasilinear Degenerate Elliptic Differential Equations
| Project/Area Number |
11640206
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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 |
Global analysis
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| Research Institution | Tokushima University |
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Principal Investigator |
FUKAGAI Nobuyoshi Tokushima University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90175563)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Masayuki Tokushima University, Department of Mathematics and Computer Sciences, Professor, 総合科学部, 教授 (70136034)
NARUKAWA Kimiaki Naruto University of Education, Department of Mathematics, Professor, 教授 (60116639)
NAITO Manabu Ehime University, Faculty of Science, Professor, 理学部, 教授 (00106791)
KOHDA Atsuhito Tokushima University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50116810)
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| Project Period (FY) |
1999 – 2000
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| Keywords | nonlinear / elliptic / differential equations / qualitative theory / quasilinear / weak solutions / bifurcation / eigenvalue problems |
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| Research Abstract |
We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) boundary value problems of quasilinear elliptic equations in a bounded domain ; (ii) blowup phenomena of seminilear parabolic equations ; (iii) solutions of ordinary differential equations derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.
1. Positive solutions of a class of nonlinear eigenvalue problems are investigated. For a quasilinear elliptic problem (^*)-div (φ(|∇u|) ∇u)=λf(x, u) in Ω, u=0 on ∂Ω, we assume asymptotic conditions on φ and f such as φ(t)〜t^<p_0-2>, f(x, t)〜t^<q_0-1> as t→0 andφ(t)〜t^<p_1-2>, f(x, t)〜t^<q_1-1> as t→∞. The combined effects of sub-nonlinearity (p_0>q_0) and super-nonlinearity (p_1<q_1) imply the existence of at least two positive solutions of (^*) for 0<λ<Λ.
2. Concerning an initial value problem of semilinear parabolic equations with convex nonlinearities, we obtain sufficient conditions which determine the solution to blow up. The criteria are formulated in terms of super-solution and sub-solution of the related stational problem.
3. A Sturm-Liouville equation on [α, ∞) is examined. (1) Higher order ordinary differential equations with general nonlinearities and determination of the Kiguraze class of positive sotuions. (2) Oscillation problem of first order 4-dimensional system of ordinary differential equations. (3) Eigenvalue problem of second order half-linear ordinary differential equations on [α, ∞).
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