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PROC

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.

PROC

We prove regularization for ill-posed evolution problems that are both inhomogeneous and nonautonomous in a Hilbert Space $H$. We consider the ill-posed problem $du/dt = A(t,D)u(t)+h(t)$, $u(s)=\chi$, $0\leq s \leq t< T$ where $A(t,D)=\sum_{j=1}^ka_j(t)D^j$ with $a_j\in C([0,T]:\mathbb{R}^+)$ for each $1\leq j\leq k$ and $D$ a positive, self-adjoint operator in $H$. Assuming there exists a solution $u$ of the problem with certain stabilizing conditions, we approximate $u$ by the solution $v_{\beta}$ of the approximate well-posed problem $dv/dt = f_{\beta}(t,D)v(t)+h(t)$, $v(s)=\chi$, $0\leq s \leq t< T$ where $0<\beta <1$. Our method implies the existence of a family of regularizing operators for the given ill-posed problem with applications to a wide class of ill-posed partial differential equations including the inhomogeneous backward heat equation in $L^2(\mathbb{R}^n)$ with a time-dependent diffusion coefficient.

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