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

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  • 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.
    Mathematics Subject Classification: Primary: 47D06, 46C99; Secondary: 35K05.

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