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Da Prato-Zabczyk's maximal inequality revisited. I.

Jan Seidler (1993)

Mathematica Bohemica

Existence, uniqueness and regularity of mild solutions to semilinear nonautonomous stochastic parabolic equations with locally lipschitzian nonlinear terms is investigated. The adopted approach is based on the factorization method due to Da Prato, Kwapień and Zabczyk.

Determination of a diffusion coefficient in a quasilinear parabolic equation

Fatma Kanca (2017)

Open Mathematics

This paper investigates the inverse problem of finding the time-dependent diffusion coefficient in a quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown. Finally, some numerical experiments are presented.

Determination of the unknown source term in a linear parabolic problem from the measured data at the final time

Müjdat Kaya (2014)

Applications of Mathematics

The problem of determining the source term F ( x , t ) in the linear parabolic equation u t = ( k ( x ) u x ( x , t ) ) x + F ( x , t ) from the measured data at the final time u ( x , T ) = μ ( x ) is formulated. It is proved that the Fréchet derivative of the cost functional J ( F ) = μ T ( x ) - u ( x , T ) 0 2 can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved....

Difference methods for parabolic functional differential problems of the Neumann type

K. Kropielnicka (2007)

Annales Polonici Mathematici

Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions...

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