2007 Fiscal Year Final Research Report Summary
Integrated Study for Nonlinear Evolution Equations and Nonlinear Elliptic Equations
| Project/Area Number |
16340043
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Waseda University |
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Principal Investigator |
OTANI Mitsuharu Waseda University, Faculty of Science and Engineering, Professor (30119656)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Yoshio Waseda University, Faculty of Science and Engineering, Professor (20111825)
TANAKA Kazunaga Waseda University, Faculty of Science and Engineering, Professor (20188288)
ISHII Hitoshi Waseda University, Faculty of Education and Integrated Arts and Sciences, Professor (70102887)
KENMOCHI Nobuyuki Chiba University, Faculty of Education, Professor (00033887)
OZAWA Tohru Hokkaido University, Graduate School of Science, Professor (70204196)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Nonlinear Evolution Equation / Nonlinear Elliptic Equation / Nonlinear PDE / Method of Variation / Subdifferential Operator |
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| Research Abstract |
(i) L^∞-energy Method, developed in this research, is applied to the nonlinear parabolic equations with nonlinear terms involving the time derivative to show the existence of the unique local solution. The verification for the uniqueness was difficult for the existing methods because of the lack of regularity. However this method makes it possible by assuring the high regularity of solutions. Furthermore this method turns out to be very effective also for nonlinear parabolic systems for chemotaxis and systems with the hysteresis effect by the fact that it can assure the existence an uniqueness of solution under much weaker conditions than ever
(ii) The infinite dimensional global attractor is constructed in L^2, which attracts all orbits for the initial boundary value problem for the quasi-linear parabolic equation governed by the p-Laplacian. The infinite dimensional global attractor is never observed for the semilinear parabolic equations, so this very new observation seems to be very
… More
important. On the other hand, the existence of the exponential attractor with finite fractal dimension , which attracts all orbits starting from some special class of initial data exponentially, is shown for some special quasilinear parabolic equations involving Laplacian and p-Laplacian., whence follows the finite dimensionality of the global attractor. These observations suggest that in contrast with semilinear equations, there should exist some structure in quasilinear parabolic equations which controls the finite-dimensionality and infinite-dimensionality of global attractors, which gives a very interesting future object ton study.
iii) It is shown that for Cauchy problem and periodic problem for the abstract evolution equation governed by time-dependent subdifferential operators, if the sequence of approximating subdifferential operators converges to the original one, then the corresponding approximating solutions converge to the solution of the original equation. As for the periodic problem, it is very meaningful to give an affirmative answer to the open problem left long. Less
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