2007 Fiscal Year Final Research Report Summary
Studies of solvability for hyperbolic Volterra equations
| Project/Area Number |
17540143
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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 | Ibaraki University |
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Principal Investigator |
OKA Hirokazu Ibaraki University, College of Engineering, Associate Professor (90257254)
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| Co-Investigator(Kenkyū-buntansha) |
TANAKA Naoki Shizuoka University, Faculty of Science, Professor (00207119)
NAKAMOTO Ritsuo Ibaraki University, College of Engineering, Professor (80007799)
NISHIO Katsuyoshi Ibaraki University, College of Engineering, Associate Professor (40001698)
HIRASAWA Go Ibaraki University, College of Engineering, Associate Professor (10434002)
SAKAKIBARA Nobuhisa Shibaura Institute of Technology, Faculty of Engineering, Associate Professor (30235139)
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| Project Period (FY) |
2005 – 2007
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| Keywords | finite difference approximations / stability condition / semilinear evolution equations / semierouos of locally Lipschitz operators / Cauchy problems / mild solutions / dissiiativity condition / nonlinear hyperbolic systems |
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| Research Abstract |
1. When we consider the initial-boundary problem for a partial differential equation in the space of continuous functions, the operator defined naturally from the equation has possibly non dense domain because of its boundary condition. This fact motivates us to study the initial-value problem for the equation of evolution governed by a family of closed linear operators whose common domain is not necessarily dense in the underlying Banach space. It is shown that an evolution operator is generated by such a family of operators, under the stability condition introduced from the viewpoint of finite difference approximations.
2. We discuss the global solvability fir a class of semilinear evolution equations which is the abstract version of the quasilinear wave equation with strong damping. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives.
3. We introduce the notion of semigroups of locally Lipschitz operators which provide us with mild solutions to the Cauchy problem for semilinear evolution equations, and characterize such semigroups of locally Lipschitz operators. This notion of the semigroups is derived from the well-posedness concept of the initial-boundary value problem for differential equations whose solution operators are not quasi-contractive even in a local sense but locally Lipschitz continuous with their initial data.
4. A new dissipativity condition is proposed in terms of a family of metric-like functionals, and a necessary and sufficient condition is given of the existence of semigroups of locally Lipschitz operators which provide us with mild solution of Cauchy problem for nonlinear evolution equations. The advantage of using a family of metric-like functionals instead of the metric induced by the original norm lies in the fact that the obtained result may be applied to some nonlinear hyperbolic systems.
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