Ninth MSU-UAB Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 20 (2013), pp. 65-78.
Title: Modified quasi-reversibility method for nonautonomous semilinear problems
Author: Matthew A. Fury (Penn State Abington, PA, USA)
Abstract:
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.