Ninth MSU-UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 20 (2013), pp. 65-78.

Modified quasi-reversibility method for nonautonomous semilinear problems

Matthew A. Fury

We prove regularization for the ill-posed, semilinear evolution problem $du/dt=A(t, D)u(t)+h(t, u(t))$, $0 \leq s \leq t < T$, with initial condition $u(s)=\chi$ in a Hilbert space where D is a positive, self-adjoint operator in the space. As in recent literature focusing on linear equations, regularization is established by approximating a solution u(t) of the problem by the solution of an approximate well-posed problem. The approximate problem will be defined by one specific approximation of the operator A(t,D) which extends a recently introduced, modified quasi-reversibility method by Boussetila and Rebbani. Finally, we demonstrate our theory with applications to a wide class of nonlinear partial differential equations in $L^2$ spaces including the nonlinear backward heat equation with a time-dependent diffusion coefficient.

Published October 31, 2013.
Math Subject Classifications: 46C05, 47D06.
Key Words: Regularization for ill-posed problems; semilinear evolution equation; backward heat equation.

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Matthew Fury
Division of Science & Engineering, Penn State Abington
1600 Woodland Road
Abington, PA 19001, USA
email:, Tel: 215-881-7553, Fax: 215-881-7333

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