# American Institute of Mathematical Sciences

2015, 2015(special): 479-488. doi: 10.3934/proc.2015.0479

## Estimates for solutions of nonautonomous semilinear ill-posed problems

 1 Division of Science and Engineering, Penn State Abington, 1600 Woodland Road, Abington, PA 19001

Received  September 2014 Revised  January 2015 Published  November 2015

The nonautonomous, semilinear problem $\frac{du}{dt}=A(t)u(t)+h(t,u(t))$, $0 \leq s \leq t < T$, $u(s)=\chi$ in Hilbert space with a Lipschitz condition on $h$, is generally ill-posed under prescribed conditions on the operators $A(t)$. Hence, regularization techniques are sought out in order to estimate known solutions of the problem. We study two quasi-reversibility methods of approximation which have successfully established regularization in the linear case, and provide an estimate on a solution $u(t)$ of the problem under these approximations in the nonlinear case. The results apply to partial differential equations of arbitrary even order including the nonlinear backward heat equation with a time-dependent diffusion coefficient.
Citation: Matthew A. Fury. Estimates for solutions of nonautonomous semilinear ill-posed problems. Conference Publications, 2015, 2015 (special) : 479-488. doi: 10.3934/proc.2015.0479
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