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Analysis Of Some Deterministic and Stochastic Evolution Equations With Solutions Taking Values In An Infinite Dimensional Hilbert Manifold

Hussain, Javed (2015) Analysis Of Some Deterministic and Stochastic Evolution Equations With Solutions Taking Values In An Infinite Dimensional Hilbert Manifold. PhD thesis, University of York.

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Abstract

The objective of this thesis is threefold: Firstly, to deal with the deterministic problem consisting of non-linear heat equation of gradient type. It comes out as projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearly of degree 2n-1, onto the tangent space of a sphere M in a Hilbert space H. We are going to deal with questions of the existence and the uniqueness of a global solution, and the invariance of manifold M i.e. if the suitable initial data lives on M then all trajectories of solutions also belong to M. Secondly, to generalize the deterministic model to its stochastic version i.e. stochastic non-linear heat equation driven by the noise of Stratonovich type. We are going to show that if the suitable initial data belongs to manifold $M$ then M-valued unique global solution to the generalized stochastic model exists. Thirdly, to investigate the small noise asymptotics of the stochastic model. A Freidlin-Wentzell large deviation principle is established for the laws of solutions of stochastic heat equation on Hilbert manifold.

Item Type: Thesis (PhD)
Keywords: Evolution equations, projections, Hilbert manifold, Stochastic evolution equations, large deviation principle, weak convergence approach.
Academic Units: The University of York > Mathematics (York)
Identification Number/EthosID: uk.bl.ethos.677378
Depositing User: Mr Javed Hussain
Date Deposited: 25 Jan 2016 15:44
Last Modified: 08 Sep 2016 13:33
URI: http://etheses.whiterose.ac.uk/id/eprint/11563

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