Evolution inclusions governed by the difference of two subdifferentials in reflexive Banach spaces |
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Authors: | Goro Akagi,Mitsuharu Ô tani |
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Affiliation: | Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan |
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Abstract: | The existence of strong solutions of Cauchy problem for the following evolution equation du(t)/dt+∂?1(u(t))-∂?2(u(t))∋f(t) is considered in a real reflexive Banach space V, where ∂?1 and ∂?2 are subdifferential operators from V into its dual V*. The study for this type of problems has been done by several authors in the Hilbert space setting.The scope of our study is extended to the V-V* setting. The main tool employed here is a certain approximation argument in a Hilbert space and for this purpose we need to assume that there exists a Hilbert space H such that V⊂H≡H*⊂V* with densely defined continuous injections.The applicability of our abstract framework will be exemplified in discussing the existence of solutions for the nonlinear heat equation: where Ω is a bounded domain in RN. In particular, the existence of local (in time) weak solution is shown under the subcritical growth condition q<p* (Sobolev's critical exponent) for all initial data This fact has been conjectured but left as an open problem through many years. |
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Keywords: | Evolution equation Subdifferential Reflexive Banach space p-Laplacian Subcritical Local existence Degenerate parabolic equation |
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