Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space

Pages: 259 - 272, Issue special, November 2013

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Matthew A. Fury - Division of Science and Engineering, Penn State Abington, 1600 Woodland Road, Abington, PA 19001, United States (email)

Abstract: 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.

Keywords:  Regularizing families of operators, ill-posed problems, evolution equations, backward heat equation.
Mathematics Subject Classification:  Primary: 47D06, 46C99; Secondary: 35K05.

Received: July 2012; Published: November 2013.