\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Ninth MSU-UAB Conference on Differential Equations and Computational Simulations. \emph{Electronic Journal of Differential Equations}, Conference 20 (2013), pp. 65--78.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{65} \title[\hfilneg EJDE-2013/Conf/20/ \hfil Modified quasi-reversibility method] {Modified quasi-reversibility method for nonautonomous semilinear problems} \author[M. A. Fury\hfil EJDE-2013/conf/20 \hfilneg] {Matthew A. 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} \thanks{Published October 31, 2013.} \subjclass[2000]{46C05, 47D06} \keywords{Regularization for ill-posed problems; semilinear evolution equation; \hfill\break\indent backward heat equation} \begin{abstract} We prove regularization for the ill-posed, semilinear evolution problem $du/dt=A(t, D)u(t)+h(t, u(t))$, $0 \leq s \leq t < T$, with initial condition $u(s)=\chi$ in a Hilbert space where $D$ is a positive, self-adjoint operator in the space. As in recent literature focusing on linear equations, regularization is established by approximating a solution $u(t)$ of the problem by the solution of an approximate well-posed problem. The approximate problem will be defined by one specific approximation of the operator $A(t, D)$ which extends a recently introduced, modified quasi-reversibility method by Boussetila and Rebbani. Finally, we demonstrate our theory with applications to a wide class of nonlinear partial differential equations in $L^2$ spaces including the nonlinear backward heat equation with a time-dependent diffusion coefficient. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction}\label{intro} During the previous several decades, much focus has been placed on approximating solutions of ill-posed problems such as the backward heat equation. In such problems, solutions do not depend continuously on initial data forcing numerical estimates of the solutions to become difficult. In general, many authors have established the regularization" of the ill-posed Cauchy problem \label{ACP} \begin{gathered} \frac{du}{dt} = Au(t) \quad 0\leq t0$,$f_{\beta}(A)$is defined by the quasi-reversibility method$f_{\beta}(A)=A-\beta A^2$. Another well-known approximation introduced by Showalter \cite{Showalter} may be used where$f_{\beta}(A)=A(I+\beta A)^{-1}$(cf. also \cite{AmesandHughes,HuangZheng}). Under these circumstances, approximation results are obtained in the following manner. \begin{definition}[{\cite[Definition 3.1]{HuangZheng}}] \label{regdefn} \rm Let$u(t)$be a solution of \eqref{ACP} with initial data$\chi \in X$and let$v_{\beta}^{\delta}(t)$be the solution of the well-posed problem \eqref{f(A)CP} with initial data$\chi_{\delta}$. Problem \eqref{ACP} is \emph{regularized} if for any$\delta>0$, there exists$\beta=\beta(\delta)>0$such that \begin{itemize} \item[(i)]$\beta \to 0$as$\delta \to 0$, \item[(i)]$\|u(t)-v_{\beta}^{\delta}(t)\|\to 0$as$\delta \to 0$for$0\leq t\leq T$whenever$\|\chi - \chi_{\delta}\|\leq \delta$. \end{itemize} \end{definition} Regularization results (cf. \cite{Melnikova,MelnikovaandFilinkov,AmesandHughes,HuangZheng2, HuangZheng}) and numerical estimates (cf. \cite{Trong1, Trong2}) for these results have been calculated in both Hilbert space and Banach space, and also for different variations of \eqref{ACP}. For instance, regularization has been applied to backward or final value problems (cf. \cite{Tuan1,Tuan2}), nonlinear problems (cf. \cite{Trong3,BethNonlin}), and also nonautonomous problems where the operator$A$in \eqref{ACP} is replaced by the time-dependent operator$A(t)$(cf. \cite{FuryandHughesSgF,Fury}). This paper extends recent regularization results for nonlinear ill-posed problems in \cite{Trong3} to regularization for the \emph{nonautonomous semilinear evolution problem} $$\label{semilin1} \begin{gathered} \frac{du}{dt} = A(t,D)u(t)+h(t,u(t)) \;\;\;\; 0\leq s \leq t< T \\ u(s) = \chi \end{gathered}$$ in a Hilbert space$H$where$D$is a positive, self-adjoint operator in$H$and$A(t,D)=\sum_{j=1}^ka_j(t)D^j$where$a_j \in C([0,T]:\mathbb{R}^+)\cap C^1([0,T])$for each$1\leq j\leq k$. Under certain conditions on the function$h:[s,T]\times H\to H$, we prove that the ill-posed problem \eqref{semilin1} may be regularized as in Definition~\ref{regdefn}, by considering the approximate well-posed problem $$\label{semilin2} \begin{gathered} \frac{dv}{dt} = f_{\beta}(t,D)v(t)+h(t,v(t)) \;\;\;\; 0\leq s \leq t< T \\ v(s) = \chi \end{gathered}$$ where $f_{\beta}(t,D)=-\frac{1}{T-s}\ln (\beta+e^{-(T-s)A(t,D)}), \quad 0\leq t\leq T.$ The approximation$f_{\beta}(t,D)$of$A(t,D)$extends the approximation $f_{\beta}(A)=-\frac{1}{pT}\ln (\beta+e^{-pTA}), \quad \beta>0,\; p\geq 1$ recently introduced by Boussetila and Rebanni \cite{Bouss} as a modified quasi-reversibility method and employed by Huang \cite{Huang} and Trong and Tuan \cite{Trong3} in the case of the autonomous problem \eqref{ACP}. As is discussed in \cite{Bouss}, \cite{Huang}, and \cite{Trong3}, one advantage of this more recent approximation is that the amplification factor of the error between the operators$A$and$f_{\beta}(A)$is milder than if the approximations$A-\beta A^2$or$A(I+\beta A)^{-1}$were used, both of which induce an error of order$e^{C/\beta}$. Results in the current paper may analogously be compared to regularization estimates recently established for nonautonomous problems in which the approximation$f_{\beta}(t,D)=A(t,D)-\beta D^{k+1}$of$A(t,D)$is used (cf. \cite{FuryandHughes} and also \cite{FuryandHughesSgF}). The paper is organized as follows. In Section~\ref{semilinear_equations}, we prove that the approximate problem \eqref{semilin2} is well-posed with unique classical solution for every$\chi \in H$, that is a function$v_{\beta}:[s,T]\to H$such that$v_{\beta}(t)\in \operatorname{Dom}(f_{\beta}(t,D))$for$s< t< T$,$v_{\beta} \in C([s,T]:H)\cap C^1((s,T):H)$, and$v_{\beta}$satisfies$\eqref{semilin2}$in$H$(cf. \cite[Chapter 5.1 p. 126]{Pazy}). We also in Section~\ref{semilinear_equations} discuss the nature in which the operators$f_{\beta}(t,D)$approximate the operators$A(t,D)$and in Section~\ref{reg_section}, we use these results to show that the solution of \eqref{semilin2} may be used to regularize the ill-posed problem \eqref{semilin1}. Finally, in Section~\ref{ex_section}, we apply our theory to a wide class of nonlinear partial differential equations in$L^2(\mathbb{R}^n)$with a simple application to the nonlinear backward heat equation with a time-dependent diffusion coefficient. Below,$\rho(D)$will denote the resolvent set of the operator$D$which consists of all complex numbers$\lambda$such that$(\lambda I-D)^{-1}$exists as an everywhere-defined bounded operator. The set$\sigma(D)$will denote the spectrum of$D$which is defined as the complement of$\rho(D)$. \section{Semilinear Evolution Equations} \label{semilinear_equations} Consider the generally ill-posed, semilinear evolution equation \eqref{semilin1} where$D$is a positive, self-adjoint operator in a Hilbert space$H$and$A(t,D)=\sum_{j=1}^k a_j(t)D^j$with$a_j \in C([0,T]:\mathbb{R}^+)\cap C^1([0,T])$for each$1\leq j\leq k$. Also, let$0<\beta<1$and consider the approximate problem \eqref{semilin2} where $$\label{f_beta} f_{\beta}(t,D)=-\frac{1}{T-s}\ln (\beta+e^{-(T-s)A(t,D)}), \quad 0\leq t\leq T.$$ We note that for$t\in [0,T]$,$f_{\beta}(t,D)$is defined by means of the functional calculus for self-adjoint operators in the Hilbert space$H$. Specifically, since$D$is positive, self-adjoint, the spectrum$\sigma(D)$of$D$is contained in$[0,\infty)$. Furthermore, for$t\in [0,T]$, since the function$f_{\beta}(t,\lambda) =-\frac{1}{T-s}\ln (\beta+e^{-(T-s)A(t,\lambda)})$is a Borel function defined for$\lambda \in [0,\infty)$, the operator$f_{\beta}(t,D)$is then defined by$f_{\beta}(t,D)x=\int_0^{\infty}f_{\beta}(t,\lambda)dE(\lambda)x$for $$\label{spec_thm} x\in \operatorname{Dom}(f_{\beta}(t,D))=\{x\in H : \int_0^{\infty}|f_{\beta}(t,\lambda)|^2d(E(\lambda)x,x)<\infty\}$$ where$\{E(\cdot)\}$denotes the resolution of the identity, that is the unique spectral measure associated with the operator$D$satisfying the equations$\operatorname{Dom}(D)=\{x\in H : \int_0^{\infty}|\lambda|^2d(E(\lambda)x,x) <\infty\}$and$Dx=\int_0^{\infty}\lambda dE(\lambda)x$for$x \in \operatorname{Dom}(D)$(cf. \cite[Theorem~XII.2.3, Theorem~XII.2.6]{DandS}). Note that for$(t,\lambda) \in [0,T]\times [0,\infty)$, since$A(t,\lambda)\geq A(t,0)=0, \begin{align*} \ln \beta \leq \ln (\beta+e^{-(T-s)A(t,\lambda)}) \leq \ln (\beta+1). \end{align*} Multiplying through by-(T-s)^{-1}$yields $$\label{f_beta_bdd} -\frac{1}{T-s}\ln (\beta+1) \leq f_{\beta}(t,\lambda) \leq -\frac{1}{T-s}\ln \;\beta.$$ Hence by \eqref{spec_thm} and \eqref{f_beta_bdd}, we see that$\operatorname{Dom}(f_{\beta}(t,D))=H$and$f_{\beta}(t,D)$is a bounded, everywhere-defined operator on$H$for each$t \in [0,T]$. Since$f_{\beta}(t,D)$is a bounded operator on$H$for each$t\in [0,T]$, the linear version of \eqref{semilin2} is easily well-posed meaning that a unique classical solution exists for each$\chi$in a dense subset of$X$and solutions depend continuously on the initial data (cf. \cite[Chapter~2.13, p. 140]{Goldstein}). In order to show that the \emph{nonlinear} problem \eqref{semilin2} is well-posed and to ultimately prove regularization for \eqref{semilin1}, we will also require special conditions on the function$h:[s,T] \times H \to H$. We have \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$\eqref{f_beta}$. Assume the function$h:[s,T]\times H \to H$satisfies the following conditions. \begin{itemize} \item[(i)] h is uniformly Lipschitz in$H$, i.e.$\|h(t,x_1)-h(t,x_2)\|\leq L\|x_1-x_2\|$for some constant$L>0$independent of$t\in [s,T]$and every$x_1, x_2 \in H$, \item[(ii)] for each$x\in H$,$h(t,x)$is continuous from$[s,T]$into$H$. \end{itemize} Then \eqref{semilin2} is well-posed, with unique classical solution$v_{\beta}(t)$for every$\chi \in H$satisfying the integral equation $$\label{integral_eqn} v_{\beta}(t)=e^{\int_s^tf_{\beta}(\tau,D)d\tau}\chi+\int_s^t e^{\int_r^tf_{\beta}(\tau,D)d\tau}h(r,v_{\beta}(r))dr.$$ \end{proposition} \begin{proof} We first note that, as previously discussed, the inequality \eqref{f_beta_bdd} implies that$f_{\beta}(t,D)$is a bounded operator on$H$for each$t\in [0,T]$. Also, by the assumption on the functions$a_j$, it may be shown that for each$x\in H$, the function$t \mapsto f_{\beta}(t,D)x$is a continuously differentiable function. These properties together imply that the linear, homogeneous version of \eqref{semilin2} is well-posed, and also provides the existence of an evolution system$V_{\beta}(t,s)=e^{\int_s^tf_{\beta}(\tau,D)d\tau}, 0\leq s\leq t\leq T$associated with the operators$f_{\beta}(t,D), 0\leq t\leq T$(cf. \cite[Theorem~5.4.8, Theorem~5.4.3]{Pazy}, \cite[Proposition~2.10]{FuryandHughes}). As$(t,s)\mapsto V_{\beta}(t,s)$is continuous in the strong operator topology (cf. \cite[Theorem~5.4.8]{Pazy}), and by the assumptions on the function$h:[s,T]\times H \to H$, following \cite[Theorem~6.1.7]{Pazy}, we define the mapping$F: C([s,T]:H) \to C([s,T]:H)$by $(Fv)(t)=V_{\beta}(t,s)\chi + \int_s^tV_{\beta}(t,r)h(r,v(r))dr.$ Using properties of the bounded operators$V_{\beta}(t,s)$and the Lipschitz condition on$h$, it follows from an application of the Contraction Mapping Theorem that$F$has a unique fixed point$v_{\beta} \in C([s,T]:H)$(cf. \cite[Proof of Theorem~6.1.2]{Pazy}). Next, define$G(t)=h(t,v_{\beta}(t))$and consider the linear evolution problem $$\label{inhom2} \begin{gathered} \frac{dw}{dt} = f_{\beta}(t,D)w(t)+G(t) \quad 0\leq s \leq t< T \\ w(s) = \chi. \end{gathered}$$ Note$G(t)$is continuous from$[s,T]$into$H$by the following calculation which follows from our assumptions on$h$and continuity of$v_{\beta}(t): \begin{align*} \|G(t)-G(t_0)\| &= \|h(t,v_{\beta}(t))-h(t_0,v_{\beta}(t_0))\| \\ &\leq \|h(t,v_{\beta}(t))-h(t,v_{\beta}(t_0))\| + \|h(t,v_{\beta}(t_0))-h(t_0,v_{\beta}(t_0))\| \\ &\leq L\|v_{\beta}(t)-v_{\beta}(t_0)\| + \|h(t,v_{\beta}(t_0))-h(t_0,v_{\beta}(t_0))\| \\ &\to 0 \quad \text{as} \quad t \to t_0. \end{align*} Hence, by \cite[Theorem~5.5.2]{Pazy}, \eqref{inhom2} is well-posed with unique classical solution $w_{\beta}(t)=V_{\beta}(t,s)\chi+\int_s^tV_{\beta}(t,r)G(r)dr,$ implying that $w_{\beta}(t) = V_{\beta}(t,s)\chi+\int_s^tV_{\beta}(t,r)h(r,v_{\beta}(r))dr = v_{\beta}(t)$ sincev_{\beta}$is a fixed point of$F$. Because$w_{\beta}(t)$is a classical solution of \eqref{inhom2},$v_{\beta}(t)$must then be a classical solution of \eqref{semilin2}. Uniqueness follows from uniqueness of the fixed point since any classical solution of \eqref{semilin2} satisfies the integral equation \eqref{integral_eqn}. Finally, continuous dependence on initial data holds by the following calculuation. By \eqref{f_beta_bdd}, consider for$0\leq s\leq t\leq T$and$x\in H, \label{V_bdd} \begin{aligned} \|V_{\beta}(t,s)x\|^2 &= \|e^{\int_s^tf_{\beta}(\tau,D)d\tau}x\|^2 \\ &= \int_0^{\infty}|e^{\int_s^tf_{\beta}(\tau,\lambda)d\tau}|^2 d(E(\lambda)x,x) \\ &\leq \int_0^{\infty}|e^{-\frac{t-s}{T-s}ln\beta}|^2d(E(\lambda)x,x) \\ &= \int_0^{\infty}|\beta^{\frac{s-t}{T-s}}|^2d(E(\lambda)x,x) \\ &= (\beta^{\frac{s-t}{T-s}}\|x\|)^2 \end{aligned} which implies\|V_{\beta}(t,s)\|\leq \beta^{\frac{s-t}{T-s}}$. Now, let$v_1$and$v_2$be classical solutions of \eqref{semilin2} corresponding to initial data$\chi_1$and$\chi_2$respectively. Then, as$v_1$and$v_2$each satisfy \eqref{integral_eqn}, and since$0<\beta<1, we have \begin{align*} & \|v_1(t)-v_2(t)\| \\ &\leq \|V_{\beta}(t,s)\chi_1-V_{\beta}(t,s)\chi_2\| +\int_s^t\|V_{\beta}(t,r)h(r,v_1(r))-V_{\beta}(t,r)h(r,v_2(r))\|dr \\ &\leq \beta^{\frac{s-t}{T-s}} \|\chi_1-\chi_2\| +\int_s^t\beta^{\frac{r-t}{T-s}}\|h(r,v_1(r))-h(r,v_2(r))\|dr \\ &\leq \beta^{-1} \|\chi_1-\chi_2\|+L\beta^{-1}\int_s^t\|v_1(r)-v_2(r)\|dr. \end{align*} Gronwall's Inequality (cf. \cite[Proof of Theorem~6.1.2]{Pazy}) then implies \begin{align*} \|v_1(t)-v_2(t)\| &\leq \beta^{-1}\|\chi_1-\chi_2\| e^{L\beta^{-1}(T-s)} \\ &\to 0 \quad \text{as\|\chi_1-\chi_2\| \to 0$for each$t\in [s,T].} \end{align*} \end{proof} We have shown that under the assumptions of Proposition~\ref{well-posed_prop}, the approximate problem \eqref{semilin2} is well-posed with unique classical solutionv_{\beta}(t)$. In order that the solution$v_{\beta}(t)$of \eqref{semilin2} is used to regularize problem \eqref{semilin1}, we will examine the difference between the operators$A(t,D)$and the approximate operators$f_{\beta}(t,D)$. The following lemma demonstrates this 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$\eqref{f_beta}$. 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$t\in [0,T]$, $\operatorname{Dom}(e^{(T-s)B(D)})\subseteq \operatorname{Dom}(A(t,D)) \cap \operatorname{Dom}(f_{\beta}(t,D))$ and $\|(-A(t,D)+f_{\beta}(t,D))\psi\|\leq \frac{\beta}{T-s}\|e^{(T-s)B(D)}\psi\|$ for all$\psi \in \operatorname{Dom}(e^{(T-s)B(D)})$. \end{lemma} \begin{proof} Let$t \in [0,T]$and$\psi \in \operatorname{Dom}(e^{(T-s)B(D)})$. Then since$(T-s)A(t,\lambda) \leq (T-s)B(\lambda) \leq e^{(T-s)B(\lambda)}$for$\lambda \geq 0$, we have$\psi \in \operatorname{Dom}(A(t,D))$by \eqref{spec_thm}, and so$\psi \in \operatorname{Dom}(A(t,D))\cap \operatorname{Dom}(f_{\beta}(t,D))$since$\operatorname{Dom}(f_{\beta}(t,D))=H. Next, \begin{align*} & \|(-A(t,D)+f_{\beta}(t,D))\psi\|^2 \\ &= \int_0^{\infty} |-A(t,\lambda)+f_{\beta}(t,\lambda)|^2d(E(\lambda)\psi, \psi) \\ &= \int_0^{\infty} |A(t,\lambda)+\frac{1}{T-s} \ln (\beta+e^{-(T-s)A(t,\lambda)})|^2d(E(\lambda)\psi,\psi) \\ &= \int_0^{\infty} |\frac{1}{T-s}\ln (e^{(T-s)A(t,\lambda)}) +\frac{1}{T-s}\ln (\beta+e^{-(T-s)A(t,\lambda)})|^2d(E(\lambda)\psi,\psi) \\ &= \int_0^{\infty} |\frac{1}{T-s}\ln (\beta e^{(T-s)A(t,\lambda)}+1)|^2 d(E(\lambda)\psi,\psi) \end{align*} and using the fact that\ln (x+1)\leq x$for$x\geq 0, then \begin{align*} \|(-A(t,D)+f_{\beta}(t,D))\psi\|^2 &\leq \int_0^{\infty} |\frac{\beta}{T-s}e^{(T-s)A(t,\lambda)}|^2 d(E(\lambda)\psi,\psi) \\ &\leq \int_0^{\infty} |\frac{\beta}{T-s}e^{(T-s)B(\lambda)}|^2 d(E(\lambda)\psi,\psi) \\ &= \|\frac{\beta}{T-s}e^{(T-s)B(D)}\psi\|^2 \end{align*} proving the desired result. \end{proof} In light of the inequality in Lemma~\ref{CondA}, define for(t,\lambda) \in [0,T]\times [0,\infty)$, $g_{\beta}(t,\lambda)=-A(t,\lambda)+f_{\beta}(t,\lambda).$ Note, $\ln (\beta + e^{-(T-s)A(t,\lambda)}) \geq \ln (e^{-(T-s)A(t,\lambda)}) = -(T-s)A(t,\lambda)$ which, after multiplying through by$-(T-s)^{-1}$, yields$f_{\beta}(t,\lambda) \leq A(t,\lambda)$and hence $$\label{g_beta_contraction} g_{\beta}(t,\lambda) \leq 0 \quad \text{for } (t,\lambda) \in [0,T]\times [0,\infty).$$ For each natural number$n$, set$e_n=\{ \lambda \in [0,\infty) : \max_{t\in [0,T]}|A(t,\lambda)|\leq n\}$. Note by inequality \eqref{f_beta_bdd}, $$\label{fn_bdd} \lambda \in e_n \Rightarrow \max_{t\in [0,T]}|f_{\beta}(t,\lambda)| \leq M_{\beta}$$ for some constant$M_{\beta}$, and by the definition of$g_{\beta}(t,\lambda)$, then $$\label{gn_bdd} \lambda \in e_n \Rightarrow \max_{t\in [0,T]}|g_{\beta}(t,\lambda)| \leq n+M_{\beta}.$$ Set$E_n=E(e_n)$and let$\psi\in H$be arbitrary. Consider the homogeneous evolution problem $$\label{u_n} \begin{gathered} \frac{du}{dt} = A(t,D)E_nu(t) \quad 0\leq s \leq t < T \\ u(s) = \psi. \end{gathered}$$ \begin{lemma} \label{u_n_well-posed} The evolution problem \eqref{u_n} has a unique classical solution$t \mapsto U_n(t,s)\psi$for every$\psi \in H$, where$U_n(t,s)$,$0\leq s\leq t\leq T$is an evolution system on$H$such that$U_n(t,s)=e^{\int_s^tA(\tau,D)d\tau}$when acting on$E_nH$. \end{lemma} \begin{proof} For each$t\in [0,T]$,$A(t,D)E_n$is a bounded operator on$H$since$|A(t,\lambda)| \leq n$for$(t,\lambda) \in [0,T]\times e_n$. Also, the function$t\mapsto A(t,D)E_n$is continuous in the uniform operator topology since each$a_j$is a continuous function. This implies the existence of a solution operator$U_n(t,s), 0\leq s\leq t\leq T$such that$t\mapsto U_n(t,s)\psi$is a unique classical solution of the homogeneous problem \eqref{u_n} for every$\psi \in H$(cf. \cite[Theorem~5.1.1]{Pazy}). It may also be shown that$U_n(t,s)$is an evolution system with the property that$U_n(t,s)=e^{\int_s^tA(\tau,D)d\tau}$when acting on$E_nH$(cf. \cite[Theorem~5.1.2]{Pazy} and \cite[Lemma~3.2]{FuryandHughes}). \end{proof} Note by replacing$A(t,D)E_n$with either$f_{\beta}(t,D)E_n$or$g_{\beta}(t,D)E_n$in \eqref{u_n}, we obtain by \eqref{fn_bdd} and \eqref{gn_bdd}, evolution systems$V_{\beta,n}(t,s)$or$W_{\beta,n}(t,s)$, respectively, similarly as in Lemma~\ref{u_n_well-posed}. We have the following corollary. \begin{corollary}\label{UW=V=WU} Let$\psi_n\in E_nH$. Then $U_n(t,s)W_{\beta,n}(t,s)\psi_n = V_{\beta,n}(t,s)\psi_n = W_{\beta,n}(t,s)U_n(t,s)\psi_n$ for all$0\leq s\leq t\leq T$. \end{corollary} \begin{proof} Note that just as$U_n(t,s)=e^{\int_s^tA(\tau,D)d\tau}$when acting on$E_nH$, we have$V_{\beta,n}(t,s)=e^{\int_s^tf_{\beta}(\tau,D)d\tau}$and$W_{\beta,n}(t,s)=e^{\int_s^tg_{\beta}(\tau,D)d\tau}$when acting on$E_nH$as well. The identity then follows from the relation$g_{\beta}(t,\lambda)=-A(t,\lambda)+f_{\beta}(t,\lambda)$and from properties of the functional calculus for self-adjoint operators (cf. \cite[Corollary~XII.2.7]{DandS}). \end{proof} \section{Regularization for problem \eqref{semilin1}} \label{reg_section} In this section, we use the results from Section~\ref{semilinear_equations} to prove regularization for the ill-posed problem \eqref{semilin1} (Theorem~\ref{reg_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$\eqref{f_beta}$and$h:[s,T]\times H\to H$satisfies the hypotheses of Proposition~\ref{well-posed_prop}. Also, set$\chi_n=E_n\chi$and$h_n(t,x)=E_nh(t,x)$for all$(t,x)\in [s,T]\times H$. Then \begin{gather*} E_nu(t)=U_n(t,s)\chi_n+\int_s^tU_n(t,r)h_n(r,u(r))dr,\\ E_nv_{\beta}(t)=V_{\beta,n}(t,s)\chi_n+\int_s^tV_{\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 $$\label{lin_hom} \begin{gathered} \frac{dw}{dt} = A(t,D)E_nw(t)+h_n(t,u(t)) \quad 0\leq s \leq t < T \\ w(s) = \chi_n. \end{gathered}$$ The second identity holds by a similar argument with$A(t,D)E_n$replaced by the function$f_{\beta}(t,D)E_n$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 the following result. \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_beta} 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 \begin{gather*} \|e^{(T-s)B(D)}e^{\int_s^tA(\tau,D)d\tau}\chi\|\leq M',\\ \|e^{(T-s)B(D)}e^{\int_s^tA(\tau,D)d\tau}h(t,u(t))\|\leq M'' \end{gather*} for all$t\in [s,T]$, then there exist constants$C$and$L$independent of$\beta$such that \[ \|u(t)-v_{\beta}(t)\|\leq \beta^{\frac{T-t}{T-s}} C e^{L(T-s)} \quad \text{for } 0\leq s\leq t0$. Since $\lim_{t \to T^-}q(t)=-\infty$, there must then exist $t_{\beta}\in (t_0, T)$ such that $q(t_{\beta})=0$, that is $-\frac{1}{(T-t_{\beta})^2}=\ln \beta$. Hence, we have $$\label{T-t_beta} T-t_{\beta}=\sqrt{-\frac{1}{\ln \beta}}.$$ Now, consider by Lemma~\ref{E_nu=u_n}, \begin{align*} \|E_nu(T)-E_nu(t_{\beta})\| &= \| \int_{t_{\beta}}^T\frac{d}{dt}E_nu(t)\;dt\| \\ &= \| \int_{t_{\beta}}^T (A(t,D)E_{n}u(t)+h_n(t,u(t)))\;dt\| \\ &\leq \int_{t_{\beta}}^T(\|A(t,D)U_n(t,s)\chi_n\|\\ &\quad +\int_s^t\|A(t,D)U_n(t,r)h_n(r,u(r))\|dr+\|h_n(t,u(t))\|)\;dt \\ &\leq M(T-t_{\beta}) \end{align*} for some constant $M$ independent of $\beta$ and $n$ where the last inequality follows from the stabilizing constants $M'$ and $M''$ of Theorem \ref{approx_thm}. Letting $n\to \infty$, we obtain $\|u(T)-u(t_{\beta})\| \leq M(T-t_{\beta})$. Finally, since $t_{\beta} \in (s,T)$, we have by Theorem~\ref{approx_thm} and \eqref{T-t_beta}, \begin{align*} \|u(T)-v_{\beta}(t_{\beta})\| &\leq \|u(T)-u(t_{\beta})\|+\|u(t_{\beta})-v_{\beta}(t_{\beta})\| \\ &\leq M(T-t_{\beta})+\beta^{\frac{T-t_{\beta}}{T-s}} C e^{L(T-s)} \\ &= M\sqrt{-\frac{1}{\ln \; \beta}}+\beta^{\frac{1}{T-s} \sqrt{-\frac{1}{ln \beta}}} C e^{L(T-s)}, \end{align*} as desired. \end{proof} We utilize the estimates in Theorem~\ref{approx_thm} and Lemma~\ref{t=T} to prove regularization for \eqref{semilin1} as follows. \begin{theorem} \label{reg_thm} Let $u(t)$ be a classical solution of \eqref{semilin1} as in Theorem~\ref{approx_thm} and let the hypotheses of Theorem~\ref{approx_thm} be satisfied. Then for any $\delta>0$, 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 $\|\chi - \chi_{\delta}\|\leq \delta$ \end{itemize} where $v_{\beta}^{\delta}(t)$ is the solution of \eqref{semilin2} with initial data $\chi_{\delta}$. \end{theorem} \begin{proof} Let $\delta >0$ be given and let $\|\chi-\chi_{\delta}\|\leq \delta$. Also, let $v_{\beta}(t)$ be the solution of \eqref{semilin2} as in Theorem~\ref{approx_thm}. For \$s\leq t