\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 28, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/28\hfil Logarithmic regularization] {Logarithmic regularization of non-autonomous non-linear ill-posed problems in Hilbert spaces} \author[M. Fury \hfil EJDE-2018/28\hfilneg] {Matthew Fury} \address{Matthew Fury \newline Division of Science \& Engineering, Penn State Abington, 1600 Woodland Road, Abington, PA 19001, USA} \email{maf44@psu.edu, Tel 215-881-7553, Fax 215-881-7333} \dedicatory{Communicated by Jerome A Goldstein} \thanks{Submitted December 14, 2017. Published January 19, 2017.} \subjclass[2010]{47D03, 35K91} \keywords{Non-linear ill-posed problem; backward heat equation; \hfill\break\indent non-autonomous problem; semigroup of linear operators; regularization} \begin{abstract} The regularization of non-autonomous non-linear ill-posed problems is established using a logarithmic approximation originally proposed by Boussetila and Rebbani, and later modified by Tuan and Trong. We first prove continuous dependence on modeling where the solution of the original ill-posed problem is estimated by the solution of an approximate well-posed problem. Finally, we illustrate the convergence via numerical experiments in $L^2$ spaces. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} \label{intro} In this paper, we study a class of non-linear non-autonomous ill-posed problems. In recent literature, the regularization of ill-posed problems is a topic of substantial investigation with applications to various natural phenomena, especially inverse processes such as backward diffusion (cf. \cite{Skaggs}). Ill-posed problems such as the backward heat equation \begin{equation} \label{BHE} \begin{gathered} u_t = -u_{xx}, \quad x\in \mathbb{R}, \; t > 0, \\ u(x,0) = \varphi(x) \end{gathered} \end{equation} may lack existence and/or uniqueness of solutions corresponding to certain initial data, or may possess solutions that do not depend continuously on the initial data. The regularization of ill-posed problems involves defining an ``$\epsilon$-close" well-posed problem whose solutions approximate solutions of the original ill-posed problem. Let us set $A=-\Delta$ and consider functions $t \mapsto u(t)$ having range in $L^2(\mathbb{R})$. Then \eqref{BHE} becomes the abstract Cauchy problem \begin{equation}\label{ACP} \begin{gathered} \frac{du}{dt} = Au, \quad t > 0, \\ u(0)= \varphi. \end{gathered} \end{equation} Lattes and Lions \cite{LandL} define the perturbation $f_{\beta}(A)=A-\beta A^2$, $\beta >0$ yielding an approximate well-posed problem \begin{equation} \label{f(A)CP} \begin{gathered} \frac{dv}{dt} = f_{\beta}(A)v, \quad t > 0, \\ v(0)= \varphi. \end{gathered} \end{equation} Moreover, if $\varphi$ is replaced with $\varphi_{\delta}$ satisfying $\|\varphi-\varphi_{\delta}\|_2\leq \delta $, one may find $\beta = \beta(\delta)$ such that $\beta \to 0$ as $\delta \to 0$, and $\|v_{\beta}^{\delta}(t) -u(t)\|_2 \to 0$ as $\delta \to 0$ for each $t\geq 0$ (Here $v_{\beta}^{\delta}(t)$ is the solution of \eqref{f(A)CP} corresponding to initial data $\varphi_{\delta}$). Many other authors including Miller \cite{Miller1}, Showalter \cite{Showalter}, and Mel'nikova \cite{Melnikova} pioneered similar methods of regularization; for example, Showalter applies a bounded approximation $f_{\beta}(A)=A(I+\beta A)^{-1}$ in \cite{Showalter}. More recently, extensions to variations of \eqref{ACP} have been established by Ames and Hughes \cite{AmesandHughes}, Long and Dinh \cite{LongandDinh}, Trong and Tuan \cite{Trong1, Trong3}, Huang and Zheng \cite{HuangZheng2, HuangZheng}, Boussetila and Rebbani \cite{Bouss}, and Fury \cite{Fury, Furylog}. For instance, Trong and Tuan \cite{Trong3} consider the non-linear problem \begin{equation}\label{NLACP} \begin{gathered} \frac{du}{dt} = Au+h(t,u(t)), \quad 00, \; p\geq 1, \end{equation} which is of milder error order than $f_{\beta}(A)=A-\beta A^2$ or $f_{\beta}(A)=A(I+\beta A)^{-1}$, Trong and Tuan establish regularization for \eqref{NLACP} where $h$ satisfies a global Lipschitz condition. In a more recent paper \cite{Tuan3}, taking $p=T=1$, Tuan and Trong modify \eqref{BR} to \begin{equation} \label{TT} f_{\beta}(A)=-\ln (\beta A+e^{-A}), \quad 0<\beta<1 \end{equation} in order to treat the case where $h$ is locally Lipschitz. In this paper, we apply a version of \eqref{TT} to problems that are both non-linear and \emph{non-autonomous}. We consider the problem with non-constant operators, \begin{equation}\label{semilin1} \begin{gathered} \frac{du}{dt} = A(t,D)u(t)+h(t,u(t)) \;\;\;\; 0\leq s < t< T \\ u(s) = \varphi \end{gathered} \end{equation} in a Hilbert space $H$ where $D$ is a positive, self-adjoint operator in $H$, $A(t,D)=\sum_{j=1}^ka_j(t)D^j$ with $a_j \in C([0,T]: \mathbb{R}^+)\cap C^1([0,T])$ for each $1\leq j\leq k$, and $h:[s,T]\times H \to H$ satisfies (H1)--(H2) below (Section~\ref{semilinear_equations}). Problem \eqref{semilin1} is ill-posed since $\{A(t,D)\}_{t\in [0,T]}$ is not a stable family of generators; in fact since each $a_j(t)>0$, none of the operators $A(t,D)$ generates a $C_0$ semigroup on $H$ (cf. \cite[Section~5.2]{Pazy}, \cite[Theorem~2.1.2]{Goldstein}). Also, note that taking $D=-\Delta$, $k=1$ and $a_k(t)=a_1(t)\equiv 1$, i.e. $A(t,D)=-\Delta$, problem \eqref{semilin1} reduces to the non-linear backward heat equation \eqref{NLACP} which is certainly ill-posed. Based on \eqref{semilin1}, consider the approximate well-posed problem \begin{equation}\label{semilin2} \begin{gathered} \frac{dv}{dt} = f_{\beta}(t,D)v(t)+h(t,v(t)) \quad 0\leq s < t< T \\ v(s) = \varphi \end{gathered} \end{equation} where following Tuan and Trong \cite{Tuan3}, we define $f_{\beta}(t,D)$ by \eqref{f_def}--\eqref{spec_thm} below. We show that if $u(t)$ is a solution of \eqref{semilin1} adhering to certain stabilizing conditions, then \begin{equation}\label{CDM} \|u(t)-v_{\beta}(t)\|\leq C'\beta^{\frac{T-t}{T-s}} \left[1-\ln \beta \right]^{\frac{s-t}{T-s}} \quad \text{for } 0\leq s\leq t\leq T \end{equation} where $v_{\beta}(t)$ is the unique solution of \eqref{semilin2} and $C'$ is a nonnegative constant independent of both $\beta$ and $t$. Note that by letting $t=T$ in \eqref{CDM}, we have $\|u(T)-v_{\beta}(T)\| \leq C'(1-\ln \beta)^{-1} \to 0$ as $\beta \to 0$. Thus, the estimate \eqref{CDM} is a considerable improvement over other H\"{o}lder-continuous dependence results such as $\|u(\tau)-v_{\beta}(\tau)\|\leq C\beta^{1-\frac{\tau}{T}}$, $0\leq \tau 0$ independent of $t\in [s,T]$ and every $\varphi_1, \varphi_2 \in H$, \item[(H2)] for each $\varphi\in H$, $h(t,\varphi)$ is continuous from $[s,T]$ into $H$. \end{itemize} Set $\tau=T-s$. For $(t,\lambda)\in [0,T]\times[0,\infty)$, define the function \begin{equation}\label{f_def} f_{\beta}(t,\lambda)=\max\{0,-\frac{1}{\tau}\ln(\beta \tau A(t,\lambda) +e^{-\tau A(t,\lambda)})\}, \quad 0<\beta<1. \end{equation} Then for each $0\leq t\leq T$, $f_{\beta}(t,D)$ is defined by means of the functional calculus for self-adjoint operators in the Hilbert space $H$. Particularly, since $f_{\beta}(t,\lambda)$ is a Borel function defined for $\lambda \in [0,\infty)$, the operator $f_{\beta}(t,D)$ is then defined by \begin{equation} \label{spec_thm} \begin{gathered} \operatorname{Dom}(f_{\beta}(t,D)) =\{\varphi\in H : \int_{\sigma(D)}|f_{\beta}(t,\lambda)|^2 d(E(\lambda)\varphi,\varphi)<\infty\}, \\ f_{\beta}(t,D)\varphi = \int_{\sigma(D)} f_{\beta}(t,\lambda)dE(\lambda)\varphi \quad \text{for } \varphi \in \operatorname{Dom}(f_{\beta}(t,D)), \end{gathered} \end{equation} where $\{E(\cdot)\}$ denotes the resolution of the identity associated with the operator $D$ and $\sigma(D)$ is its spectrum (cf. \cite[Theorem~XII.2.3, Theorem~XII.2.6]{DandS}). Note that since $D$ is positive, self-adjoint, we have $\sigma(D) \subseteq [0,\infty)$. Let us find the maximum and minimum values of $f_{\beta}(t,\lambda)$ on $[0,T]\times [0,\infty)$. Note, the function $F(x)=-\frac{1}{\tau}\ln(\beta \tau x + e^{-\tau x})$, $x\geq 0$ has $F'(x)=\frac{e^{-\tau x}-\beta}{\beta \tau x +e^{-\tau x}}$. Hence, $F(x)$ attains an absolute maximum at $x_{M}=-\frac{1}{\tau}\ln \beta$ so that $F(x)\leq F(x_M) = -\frac{1}{\tau} \ln \left[ \beta (1-\ln \beta)\right]$ for $x\geq 0$. Furthermore, since $F(x_M)>0$ and $\lim_{x \to \infty}F(x) = -\infty$, we obtain a unique $x_{\beta} > x_M$ such that $F(x)\geq 0$ on $[0,x_{\beta}]$ and $F(x)<0$ on $(x_{\beta}, \infty)$. By \eqref{f_def}, it follows that \begin{equation} \label{f_bdd} 0\leq f_{\beta}(t,\lambda) \leq -\frac{1}{\tau} \ln \left[\beta (1-\ln \beta)\right] \quad \text{for } (t,\lambda) \in [0,T]\times [0,\infty) \end{equation} and so for each $t \in [0,T]$, $f_{\beta}(t,D)$ is a bounded operator on $H$ satisfying \begin{equation} \label{f(t,D)_bdd} \|f_{\beta}(t,D)\| \leq -\frac{1}{\tau} \ln \left[\beta (1-\ln \beta)\right] \quad \text{for all } 0\leq t\leq T. \end{equation} \begin{proposition}\label{well-posed_prop} Let $H$ be a Hilbert space and for $0<\beta<1$, let the operators $f_{\beta}(t,D), 0\leq t\leq T$ be defined by $\eqref{f_def}$--$\eqref{spec_thm}$. Assume the function $h:[s,T]\times H \to H$ satisfies conditions {\rm (H1)} and {\rm (H2)}. Then \eqref{semilin2} is well-posed, with unique classical solution $v_{\beta}(t)$ for every $\varphi \in H$ where $v_{\beta}(t)$ satisfies the integral equation \begin{equation} \label{integral_eqn} v_{\beta}(t)=e^{\int_s^tf_{\beta}(q,D)dq}\varphi +\int_s^te^{\int_r^tf_{\beta}(q,D)dq}h(r,v_{\beta}(r))dr. \end{equation} \end{proposition} \begin{proof} See \cite[Proposition~2.1]{Furylog}. In particular, $e^{\int_s^tf_{\beta}(q,D)dq}$ is an evolution system on $H$ which by \eqref{f(t,D)_bdd}, satisfies \begin{equation} \label{V_bdd} \|e^{\int_s^tf_{\beta}(q,D)dq}\|\leq \left[\beta (1-\ln \beta) \right]^{\frac{s-t}{T-s}} \quad \text{for all } 0\leq s\leq t\leq T. \end{equation} Well-posedness follows immediately from \eqref{V_bdd}. \end{proof} The following lemma will aid in establishing continuous dependence on modeling and is motivated by the approximation condition, Condition A, of Ames and Hughes (cf. \cite[Definition~1]{AmesandHughes}, and also \cite[Definition p. 4]{Trong3}). \begin{lemma}\label{CondA} Let $H$ be a Hilbert space and for $0<\beta<1$, let the operators $f_{\beta}(t,D), 0\leq t\leq T$ be defined by $\eqref{f_def}$--$\eqref{spec_thm}$. Define $B(\lambda) =\sum_{j=1}^kB_j\lambda^j$ where $B_j=\max_{t\in [0,T]}a_j(t)$ for each $1\leq j\leq k$. Then for each $t\in [0,T]$, \begin{gather*} \operatorname{Dom}(B(D)e^{\tau B(D)})\subseteq \operatorname{Dom}(A(t,D)) \cap \operatorname{Dom}(f_{\beta}(t,D)), \\ \|(-A(t,D)+f_{\beta}(t,D))\varphi\|\leq \sqrt{2}\beta \|B(D) e^{\tau B(D)}\varphi\| \end{gather*} for all $\varphi \in \operatorname{Dom}(B(D)e^{\tau B(D)})$. \end{lemma} \begin{proof} Let $t\in [0,T]$. For $\lambda\geq 0$, we have $0\leq A(t,\lambda)\leq B(\lambda) \leq B(\lambda)e^{\tau B(\lambda)}$ which by \eqref{spec_thm} shows that $\operatorname{Dom}(A(t,D))\supseteq \operatorname{Dom}(B(D)e^{\tau B(D)})$. Certainly, $\operatorname{Dom}(f_{\beta}(t,D))=H\supseteq \operatorname{Dom}(B(D)e^{\tau B(D)})$ as well since $f_{\beta}(t,D)$ is a bounded operator. Now let $\varphi \in \operatorname{Dom}(B(D)e^{\tau B(D)})$ and let $x_{\beta}$ be as in the paragraph preceding inequality \eqref{f_bdd}. Set $e_{\beta}= \{\lambda \geq 0 : B(\lambda) \leq x_{\beta}\}$ and let $e_{\beta}'$ be the complement of $e_{\beta}$ in $[0,\infty)$. We have \begin{align*} & \int_{e_{\beta}} |-A(t,\lambda)+f_{\beta}(t,\lambda)|^2 d(E(\lambda)\varphi,\varphi) \\ &= \int_{e_{\beta}} |A(t,\lambda)+\frac{1}{\tau} \ln (\beta \tau A(t,\lambda)+e^{-\tau A(t,\lambda)})|^2d(E(\lambda)\varphi,\varphi) \\ &= \int_{e_{\beta}} |\frac{1}{\tau}\ln (e^{\tau A(t,\lambda)}) +\frac{1}{\tau}\ln (\beta \tau A(t,\lambda)+e^{-\tau A(t,\lambda)})|^2 d(E(\lambda)\varphi,\varphi) \\ &= \int_{e_{\beta}} |\frac{1}{\tau}\ln (\beta \tau A(t,\lambda) e^{\tau A(t,\lambda)}+1)|^2d(E(\lambda)\varphi,\varphi). \end{align*} Applying the fact that $\ln (x+1)\leq x$ for $x\geq 0$, we get \begin{align*} \int_{e_{\beta}} |-A(t,\lambda)+f_{\beta}(t,\lambda)|^2 d(E(\lambda)\varphi,\varphi) &\leq \int_{e_{\beta}} |\beta A(t,\lambda) e^{\tau A(t,\lambda)}|^2 d(E(\lambda)\varphi,\varphi) \\ &\leq \int_0^{\infty} |\beta B(\lambda) e^{\tau B(\lambda)}|^2 d(E(\lambda)\varphi,\varphi) \\ &= \beta^2 \| B(D) e^{\tau B(D)}\varphi\|^2. \end{align*} Also, since $x_\beta>-\frac{1}{\tau}\ln \beta$, we have \begin{align*} \int_{e_{\beta}'} |-A(t,\lambda)+f_{\beta}(t,\lambda)|^2 d(E(\lambda)\varphi,\varphi) &= \int_{e_{\beta}'} |A(t,\lambda)|^2d(E(\lambda)\varphi,\varphi) \\ &\leq \int_{e_{\beta}'} |e^{-\tau B(\lambda)}e^{\tau B(\lambda)} B(\lambda)|^2d(E(\lambda)\varphi,\varphi)\\ &< \int_{e_{\beta}'} |\beta B(\lambda)e^{\tau B(\lambda)}|^2 d(E(\lambda)\varphi,\varphi) \\ &\leq \int_0^{\infty} |\beta B(\lambda) e^{\tau B(\lambda)}|^2 d(E(\lambda)\varphi,\varphi) \\ &= \beta^2 \| B(D) e^{\tau B(D)}\varphi\|^2. \end{align*} Combining yields $\|(-A(t,D)+f_{\beta}(t,D))\varphi\|^2 \leq 2\beta^2 \| B(D) e^{\tau B(D)}\varphi\|^2$, which proves the desired result. \end{proof} Following Lemma~\ref{CondA}, let us define for $(t,\lambda) \in [0,T]\times [0,\infty)$, \begin{equation}\label{g} g_{\beta}(t,\lambda)=-A(t,\lambda)+f_{\beta}(t,\lambda). \end{equation} Note, $\ln (\beta \tau A(t,\lambda)+ e^{-\tau A(t,\lambda)}) \geq \ln (e^{-\tau A(t,\lambda)})= -\tau A(t,\lambda)$ which, after dividing through by $-\tau$, yields $f_{\beta}(t,\lambda) \leq A(t,\lambda)$ and hence \begin{equation} \label{g_beta_contraction} g_{\beta}(t,\lambda) \leq 0 \quad \text{for } (t,\lambda) \in [0,T]\times [0,\infty). \end{equation} For each natural number $n$, set \begin{equation}\label{gn_bdd} e_n=\{\lambda \geq 0 : B(\lambda)\leq n\}. \end{equation} Then by \eqref{f_bdd} and \eqref{g}, we have $|g_{\beta}(t,\lambda)| \leq n-\frac{1}{\tau}\ln [\beta(1-\ln \beta)]$ for all $(t,\lambda) \in [0,T]\times e_n$. Thus, if we set $E_n=E(e_n)$, then each of $A(t,D)E_n$, $f_{\beta}(t,D)E_n$, and $g_\beta(t,D)E_n$ is a bounded operator on $H$ for all $t \in [0,T]$. Following \cite[Lemma~2.3, Corollary~2.4]{Furylog}, we obtain evolution systems $U_n(t,s)$, $V_{\beta,n}(t,s)$, and $W_{\beta,n}(t,s)$ satisfying the following for all $\varphi_n\in E_nH$ and all $0\leq s\leq t\leq T$: \begin{itemize} \item[(S1)] $U_n(t,s)\varphi_n = e^{\int_s^t A(q,D)dq}\varphi_n$, $V_{\beta,n}(t,s)\varphi_n = e^{\int_s^t f_{\beta}(q,D)dq}\varphi_n$, and \\ $W_n(t,s)\varphi_n = e^{\int_s^t g_{\beta}(q,D)dq}\varphi_n$ \item[(S2)] $U_n(t,s)W_{\beta,n}(t,s)\varphi_n=V_{\beta,n}(t,s) \varphi_n=W_{\beta,n}(t,s)U_n(t,s)\varphi_n$. \end{itemize} \section{Continuous dependence on modeling}\label{CDM_section} In this section, we use the results from Section~\ref{semilinear_equations} to prove continuous dependence on modeling for the ill-posed problem \eqref{semilin1} (Theorem~\ref{approx_thm} below). \begin{lemma} \label{E_nu=u_n} Let $u(t)$ and $v_{\beta}(t)$ be classical solutions of \eqref{semilin1} and \eqref{semilin2} respectively where the operators $f_{\beta}(t,D),0\leq t\leq T$ are defined by \eqref{f_def}--\eqref{spec_thm} and $h:[s,T]\times H\to H$ satisfies the hypotheses of Proposition~\ref{well-posed_prop}. Also, set $\varphi_n=E_n\varphi$ and $h_n(t,\varphi)=E_nh(t,\varphi)$ for all $(t,\varphi)\in [s,T]\times H$. Then \begin{gather*} E_nu(t)=U_n(t,s)\varphi_n+\int_s^tU_n(t,r)h_n(r,u(r))dr, \\ E_nv_{\beta}(t)=V_{\beta,n}(t,s)\varphi_n+\int_s^t V_{\beta,n}(t,r)h_n(r,v_{\beta}(r))dr \end{gather*} for all $t\in [s,T]$. \end{lemma} \begin{proof} The first identity follows from uniqueness of solutions since both sides of the equation are classical solutions of the linear inhomogeneous problem \begin{equation}\label{lin_hom} \begin{gathered} \frac{dw}{dt} = A(t,D)E_nw(t)+h_n(t,u(t)) \quad 0\leq s \leq t < T \\ w(s) = \varphi_n. \end{gathered} \end{equation} The second identity holds by a similar argument with $A(t,D)E_n$ replaced by $f_{\beta}(t,D)E_n$ in \eqref{lin_hom}. \end{proof} As in Lemma~\ref{CondA}, set $B(\lambda) =\sum_{j=1}^kB_j\lambda^j$ where $B_j=\max_{t\in [0,T]}a_j(t)$ for each $1\leq j\leq k$. We have \begin{theorem} \label{approx_thm} Let $u(t)$ and $v_{\beta}(t)$ be classical solutions of \eqref{semilin1} and \eqref{semilin2} respectively where the operators $f_{\beta}(t,D), 0\leq t\leq T$ are defined by \eqref{f_def}--\eqref{spec_thm} and $h:[s,T]\times H\to H$ satisfies the hypotheses of Proposition \ref{well-posed_prop}. Then if there exist constants $M' ,M''\geq 0$ such that $\|B(D)e^{(T-s)B(D)}e^{\int_s^tA(q,D)dq}\varphi\|\leq M'$ and $\|B(D)e^{(T-s)B(D)}e^{\int_s^tA(q,D)dq}h(t,u(t))\|\leq M''$ for all $t\in [s,T]$, then there exist constants $C$ and $L$ independent of $\beta$ such that \begin{equation}\label{gold} \|u(t)-v_{\beta}(t)\|\leq \beta^{\frac{T-t}{T-s}} (1-\ln \beta)^{\frac{s-t}{T-s}} C e^{L(T-s)} \quad \text{for } 0\leq s\leq t\leq T. \end{equation} \end{theorem} \begin{proof} Set $\varphi_n =E_n\varphi$ and $h_n(t,\varphi)=E_nh(t,\varphi)$ for all $(t,\varphi)\in [s,T]\times H$. From Lemma~\ref{E_nu=u_n}, for $0\leq s\leq t\leq T$, \begin{align} & \|E_nu(t)-E_nv_{\beta}(t)\| \nonumber \\ &\leq \|U_n(t,s)\varphi_n-V_{\beta,n}(t,s)\varphi_n\| \nonumber \\ &\quad +\int_s^t\|U_n(t,r)h_n(r,u(r))-V_{\beta,n}(t,r)h_n(r,v_{\beta}(r)) \|dr \nonumber \\ \label{diff_a} &\leq \|U_n(t,s)\varphi_n-V_{\beta,n}(t,s)\varphi_n\| \\ \label{diff_b} &\quad + \int_s^t\|U_n(t,r)h_n(r,u(r))-V_{\beta,n}(t,r)h_n(r,u(r)) \|dr \\ \label{diff_c} &\quad + \int_s^t\|V_{\beta,n}(t,r)h_n(r,u(r))-V_{\beta,n}(t,r)h_n(r,v_{\beta}(r)) \|dr. \end{align} For the first expression, by (S2) and \cite[Theorem~5.1.2]{Pazy}, we have \begin{align*} \eqref{diff_a} &= \|(I-W_{\beta,n}(t,s))U_n(t,s)\varphi_n\| \\ &= \|(W_{\beta,n}(t,t)-W_{\beta,n}(t,s))U_n(t,s)\varphi_n\| \\ &= \big\|\int_s^t\frac{\partial}{\partial p}W_{\beta,n}(t,p)U_n(t,s) \varphi_n dp\big\| \\ &= \big\|\int_s^t(-W_{\beta,n}(t,p)g_{\beta}(p,D)E_n)U_n(t,s)\varphi_n dp\big\| \\ &\leq \int_s^t\|W_{\beta,n}(t,p)g_{\beta}(p,D)U_n(t,s)\varphi_n\| dp. \end{align*} Next from \eqref{gn_bdd} and \eqref{spec_thm}, note that $U_n(t,s)\varphi_n \in \operatorname{Dom}(B(D)e^{(T-s)B(D)})$. Therefore, by (S1), \eqref{g_beta_contraction}, and Lemma~\ref{CondA}, we have \begin{align*} \eqref{diff_a} &\leq \int_s^t\|g_{\beta}(p,D)U_n(t,s)\varphi_n\| dp. \\ &\leq \sqrt{2} \beta (t-s)\|B(D)e^{(T-s)B(D)}U_n(t,s)\varphi_n\|. \end{align*} Similarly, for the second expression, \begin{align*} \eqref{diff_b} &= \int_s^t\|(I-W_{\beta,n}(t,r))U_n(t,r)h_n(r,u(r))\|dr \\ &\leq \int_s^t \sqrt{2}\beta (t-r)\|B(D)e^{(T-s)B(D)}U_n(t,r)h_n(r,u(r))\|dr. \end{align*} Combining the above we have \begin{equation}\label{diff_a&b} \begin{aligned} & \|U_n(t,s)\varphi_n-V_{\beta,n}(t,s)\varphi_n\| \\ & + \int_s^t\|U_n(t,r)h_n(r,u(r))-V_{\beta,n}(t,r)h_n(r,u(r)) \|dr \leq \beta C \end{aligned} \end{equation} where $C$ is a constant independent of $\beta$ and also independent of $n$ and $t$ by our stabilizing constants $M'$ and $M''$. Finally, by (S1), \eqref{V_bdd}, and (H1), the third expression satisfies \begin{equation}\label{c_better} \begin{aligned} \eqref{diff_c} &= \int_s^t\|V_{\beta,n}(t,r)(h_n(r,u(r))-h_n(r,v_{\beta}(r))) \|dr \\ &\leq \int_s^t\left[\beta (1-\ln \beta)\right]^{\frac{r-t}{T-s}}\|h_n(r,u(r)) -h_n(r,v_{\beta}(r)) \|dr \\ &\leq L \int_s^t \left[\beta (1-\ln \beta)\right]^{\frac{r-t}{T-s}} \|u(r)-v_{\beta}(r)\| dr. \end{aligned} \end{equation} Combining \eqref{diff_a&b} and \eqref{c_better}, we have shown that \[ \|E_nu(t)-E_nv_{\beta}(t)\|\leq \beta C +L \int_s^t \left[\beta (1-\ln \beta)\right]^{\frac{r-t}{T-s}}\|u(r)-v_{\beta}(r)\| dr, \] and since all constants on the right are independent of $n$, we may let $n \to \infty$ to obtain \begin{equation} \label{gron} \|u(t)-v_{\beta}(t)\|\leq \beta C +L \int_s^t \left[\beta (1-\ln \beta)\right]^{\frac{r-t}{T-s}}\|u(r)-v_{\beta}(r)\| dr. \end{equation} Note that $0<\beta<1$ implies \begin{equation} \label{betalnbeta} 0<\left[\beta (1-\ln \beta)\right]^{\frac{t-s}{T-s}}<1 \quad \text{for all } t\in [s,T]. \end{equation} Hence multiplying \eqref{gron} through by $\left[\beta (1-\ln \beta)\right]^{\frac{t-s}{T-s}}$ and applying \eqref{betalnbeta}, we obtain \[ \left[\beta (1-\ln \beta)\right]^{\frac{t-s}{T-s}}\|u(t)-v_{\beta}(t)\| \leq \beta C + L \int_s^t \left[\beta (1-\ln \beta) \right]^{\frac{r-s}{T-s}}\|u(r)-v_{\beta}(r)\| dr. \] Gronwall's inequality (cf. \cite[Theorem~6.1.2]{Pazy}) then yields the estimate \[ \left[\beta (1-\ln \beta)\right]^{\frac{t-s}{T-s}}\|u(t)-v_{\beta}(t)\| \leq \beta C e^{L(T-s)} \] which is equivalent to \eqref{gold}. \end{proof} \section{Regularization for problem \eqref{semilin1}} \label{reg_section} Below, Theorem~\ref{reg_thm} establishes the main result of the paper, that is regularization for \eqref{semilin1}. Its proof uses our estimate from Theorem~\ref{approx_thm}. \begin{theorem} \label{reg_thm} Let $u(t)$ be a classical solution of \eqref{semilin1} and assume the hypotheses of Theorem~\ref{approx_thm}. Then given $\delta>0$, there exists $\beta=\beta(\delta) >0$ such that \begin{itemize} \item[(i)] $\beta \to 0$ as $\delta \to 0$, \item[(ii)] $\|u(t)-v_{\beta}^{\delta}(t)\|\to 0$ as $\delta \to 0$ for $s\leq t\leq T$ whenever $\|\varphi - \varphi_{\delta}\|\leq \delta$ \end{itemize} where $v_{\beta}^{\delta}(t)$ is the solution of $\eqref{semilin2}$ with initial data $\varphi_{\delta}$. \end{theorem} \begin{proof} Let $\delta >0$ be given and let $\|\varphi-\varphi_{\delta}\|\leq \delta$. Also, let $v_{\beta}(t)$ be the solution of \eqref{semilin2} as in Theorem~\ref{approx_thm}. For $s\leq t\leq T$, by Theorem~\ref{approx_thm}, then \begin{equation} \label{epsilon_over_two} \begin{aligned} \|u(t)-v_{\beta}^{\delta}(t)\| &\leq \|u(t)-v_{\beta}(t)\|+\|v_{\beta}(t)-v_{\beta}^{\delta}(t)\| \\ &\leq \beta^{\frac{T-t}{T-s}} (1-\ln \beta)^{\frac{s-t}{T-s}} C e^{L(T-s)}+\|v_{\beta}(t)-v_{\beta}^{\delta}(t)\|. \end{aligned} \end{equation} Consider the second quantity in \eqref{epsilon_over_two}. By \eqref{V_bdd} and (H1), we have \begin{align*} & \|v_{\beta}(t)-v_{\beta}^{\delta}(t)\| \\ &\leq \|e^{\int_s^tf_{\beta}(q,D)dq}(\varphi-\varphi_{\delta})\| + \int_s^t\|e^{\int_r^tf_{\beta}(q,D)dq}(h(r,v_{\beta}(r)) -h(r,v_{\beta}^{\delta}(r)))\|dr \\ &\leq \delta \left[\beta (1-\ln \beta)\right]^{\frac{s-t}{T-s}} + L\int_s^t \left[\beta (1-\ln \beta)\right]^{\frac{r-t}{T-s}}\|v_{\beta}(r) -v_{\beta}^{\delta}(r)\|dr. \end{align*} Hence, \[ \left[\beta (1-\ln \beta)\right]^{\frac{t-s}{T-s}}\|v_{\beta}(t) -v_{\beta}^{\delta}(t)\| \leq \delta +L\int_s^t \left[\beta (1-\ln \beta)\right]^{\frac{r-s}{T-s}} \|v_{\beta}(r)-v_{\beta}^{\delta}(r)\|dr \] which by Gronwall's Inequality gives us \[ \left[\beta (1-\ln \beta)\right]^{\frac{t-s}{T-s}}\|v_{\beta}(t) -v_{\beta}^{\delta}(t)\| \leq \delta e^{L(T-s)}. \] Therefore, $\|v_{\beta}(t)-v_{\beta}^{\delta}(t)\| \leq \delta \left[\beta (1-\ln \beta)\right]^{\frac{s-t}{T-s}}e^{L(T-s)}$ and choosing $\beta=\delta$ yields \begin{equation}\label{v-vd} \|v_{\beta}(t)-v_{\beta}^{\delta}(t)\| \leq \beta^{\frac{T-t}{T-s}} (1-\ln \beta)^{\frac{s-t}{T-s}}e^{L(T-s)}. \end{equation} Thus $\beta \to 0$ as $\delta \to 0$, and combining \eqref{epsilon_over_two} with \eqref{v-vd}, we obtain \[ \|u(t)-v_{\beta}^{\delta}(t)\| \leq \beta^{\frac{T-t}{T-s}} (1-\ln \beta)^{\frac{s-t}{T-s}}(C+1)e^{L(T-s)} \to 0 \quad \text{as } \delta \to 0. \] \end{proof} \section{Examples} \label{ex_section} The theory of this paper may be applied to a wide class of ill-posed partial differential equations in $L^2$ spaces including the backward heat equation with a time-dependent diffusion coefficient. Let us examine a concrete example of higher order with $H=L^2(0,\pi)$ where for $\varphi \in L^2(0,\pi)$, $\|\varphi\|_2 = \left(\int_0^{\pi}|\varphi(x)|^2dx\right)^{1/2}$. Also define $D\varphi=-\varphi''$ for all twice-differentiable $\varphi \in L^2(0,\pi)$ whose first and second derivatives in the sense of distributions are also members of $L^2(0,\pi)$. Consider the fourth-order non-linear partial differential equation \begin{equation}\label{concrete1} \begin{gathered} u_t +u_{xx}-e^t u_{xxxx} = \psi(u) -e^{e^t}\sin x-e^{2e^t}\sin^2x, \\ (x,t) \in (0,\pi)\times (0,1)\\ u(0,t)=u(\pi,t)=0, \quad t \in [0,1]\\ u(x,0)=e\sin x, \quad x\in [0,\pi] \end{gathered} \end{equation} where $\psi(u)$ is a compactly supported continuous function which coincides with $u^2$ on a sufficiently large interval centered at the origin. For example, following \cite[Section~4]{Trong3}, let us fix $M$ large and positive, and define \[ \psi(u)= \begin{cases} u^2 & |u| \leq M\\ Mu+2M^2 & -2M\leq u <-M \\ -Mu+2M^2 & M 2M \end{cases} \] (see Figure~\ref{fig:psiu}). \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig1} %Mgraph(new) %\put(5,8){\makebox(0,0)[cc]{$R$} \put(-96, 38){$M$} \put(-170,38){$-M$} \put(-70, 38){$2M$} \put(-210, 38){$-2M$} \put(-3, 46){$u$} \put(-130, 144){$\psi$} \end{center} \caption{$\psi(u)$} \label{fig:psiu} \end{figure} Note, \eqref{concrete1} is an example of \eqref{semilin1} where $A(t,D)=D+e^tD^2$, $a_1(t)\equiv 1$, $a_k(t)=a_2(t)=e^t$, $h(x,t,u(x,t))=\psi(u(x,t))-e^{e^t}\sin x-e^{2e^t}\sin^2x$, and $\varphi(x)=e\sin x$. It is straight-forward to check that the function $h$ satisfies conditions (H1) and (H2), and that the exact solution of \eqref{concrete1} is $u(x,t)=e^{e^t}\sin x$. For the corresponding well-posed problem, following work in \cite{LongandDinh} and \cite{Trong3}, let us assume an approximate solution of the form $v_N(x,t)=\sum_{n=1}^N v_n(t)\sin (nx)$. Set $\varphi_{\delta}(x)=(e+\delta \sqrt{\frac{2}{\pi}})\sin x$ so that $\|\varphi-\varphi_{\delta}\|_2=\delta$. Then solving \eqref{semilin2} is equivalent to solving the system of $N$ differential equations \begin{equation} \label{concrete2} \begin{gathered} \begin{aligned} &v_m'(t) +\ln(\beta(m^2+e^t m^4)+e^{-(m^2+e^t m^4)})v_m(t)\\ &=\frac{2}{\pi} \int_0^{\pi} h(x,t,v(x,t)) \sin (mx) \,dx,\quad t\in (0,1), \; 1\leq m\leq N, \end{aligned}\\ v_1(0)=e+\delta \sqrt{\frac{2}{\pi}}, \quad v_2(0)=v_3(0)=\cdots =v_N(0)=0 \end{gathered} \end{equation} where $h(x,t,v(x,t))=\psi(v(x,t))-e^{e^t}\sin x-e^{2e^t}\sin^2x$. We apply a finite difference method in order to estimate the solution $v_N(x,t)$ of \eqref{concrete2}. Let \[ \Delta t = \frac{1}{100}, \quad t_i=i\Delta t, \quad 0\leq i\leq 100. \] For each $i=0,1,2, \dots$, we solve the $N$ difference equations \begin{align*} &\frac{v_m(t_{i+1})-v_m(t_i)}{\Delta t} +\ln\big(\beta(m^2+e^{t_i} m^4) +e^{-(m^2+e^{t_i} m^4)}\big)\big(\frac{v_m(t_{i+1})+v_{m}(t_i)}{2}\big) \\ &=\frac{2}{\pi} \int_0^{\pi} ([\sum_{n=1}^N v_n(t_i) \sin (nx)]^2 -e^{e^{t_i}}\sin x-e^{2e^{t_i}}\sin^2x) \sin (mx) \,dx, \quad 1\leq m\leq N \end{align*} for the unknown $v_m(t_{i+1})$. Tables \ref{table1} and \ref{table2} illustrate our calculations with $N=5$, $i=0,1,2,3,4$, and the indicated values for $\delta$. Note as in the proof of Theorem~\ref{reg_thm}, $\beta$ is chosen to be the same value as $\delta$ in each table. As expected, we find a smaller $L^2$-difference between $u(x,t)$ and $v_N(x,t)$ for each $t$ as $\delta$ is taken closer to zero. \begin{table}[htb] \caption{$\beta=\delta=10^{-3}$} \label{table1} \renewcommand{\arraystretch}{1.2} \scriptsize \begin{center} \begin{tabular}{| c c c c |} \hline $t$ & $u(x,t)$ & $v_N(x,t)$ & $\|u-v_N\|_2$ \\ \hline 0 & $e \sin x$ & $2.719079713 \sin x$ & 0.001 \\ & & & \\ 0.01 & $2.74574 \sin x$ & $2.74619 \sin x - 0.0000074548 \sin(3x)$ & 0.00056407 \\ & & $- 0.00000105443 \sin(5x)$ & \\ 0.02 & $2.77375 \sin x$ & $2.77382 \sin x - 0.0000121535 \sin(3x)$ & 0.0000890675 \\ & & $- 0.00000161556 \sin(5x)$ & \\ 0.03 & $2.80234 \sin x$ & $2.80199 \sin x - 0.0000135129 \sin(3x)$ & 0.000438992 \\ & & $- 0.00000163352 \sin(5x)$ & \\ 0.04 & $2.83151 \sin x$ & $2.8307 \sin x - 0.0000109582 \sin(3x)$ & 0.00101528 \\ & & $- 0.00000107001 \sin(5x)$ & \\ 0.05 & $2.86129 \sin x$ & $2.85997 \sin x - 0.00000372291 \sin(3x)$ & 0.00165438 \\ & & $+ 0.000000129345 \sin(5x)$ & \\ \hline \end{tabular} \end{center} \end{table} \begin{table}[htb] \caption{$\beta=\delta=10^{-6}$} \label{table2} \renewcommand{\arraystretch}{1.2} \scriptsize \begin{center} \begin{tabular}{| c c c c |} \hline $t$ & $u(x,t)$ & $v_N(x,t)$ & $\|u-v_N\|_2$ \\ \hline 0 & $e \sin x$ & $2.718282626344 \sin x$ & 0.000001 \\ & & & \\ 0.01 & $2.74574 \sin x$ & $2.74574 \sin x - 0.00000000772376 \sin(3x)$ & 0.00000000977658 \\ & & $- 0.00000000109207 \sin(5x)$ & \\ 0.02 & $2.77375 \sin x$ & $2.77375 \sin x - 0.0000000209072 \sin(3x)$ & 0.0000000264447 \\ & & $- 0.00000000284423 \sin(5x)$ & \\ 0.03 & $2.80234 \sin x$ & $2.80234 \sin x + 0.00000000765303 \sin(3x)$ & 0.00000000977704 \\ & & $+ 0.00000000151195 \sin(5x)$ & \\ 0.04 & $2.83151 \sin x$ & $2.83151 \sin x + 0.0000000017265 \sin(3x)$ & 0.00000000229526 \\ & & $+ 0.000000000610782 \sin(5x)$ & \\ 0.05 & $2.86129 \sin x$ & $2.86128 \sin x + 0.0000000201077 \sin(3x)$ & 0.0000125332 \\ & & $+ 0.00000000322633 \sin(5x)$ & \\ \hline \end{tabular} \end{center} \end{table} For a future research, it is worthwhile to examine similar partial differential equations of higher order where the function $h$ satisfies a local Lipschitz condition rather than global. 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