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On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis

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  • Marat V. Markin

Abstract

Given the abstract evolution equation ,   with scalar type spectral operator in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on . The important case of the equation with a normal operator in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at , then all of them are strongly infinite differentiable on .

Suggested Citation

  • Marat V. Markin, 2018. "On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2018, pages 1-14, July.
    • Handle: RePEc:hin:jijmms:4168609
    DOI: 10.1155/2018/4168609
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    References listed on IDEAS

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    1. Marat V. Markin, 2011. "On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-27, May.
    2. Marat V. Markin, 2002. "On an abstract evolution equation with a spectral operator of scalar type," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 32, pages 1-9, January.
    3. Marat V. Markin, 2004. "On scalar type spectral operators, infinite differentiable and Gevrey ultradifferentiable C 0 -semigroups," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-22, January.
    4. Marat V. Markin, 2004. "On the Carleman classes of vectors of a scalar type spectral operator," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-17, January.
    5. Marat V. Markin, 2002. "A note on the spectral operators of scalar type and semigroups of bounded linear operators," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 32, pages 1-6, January.
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