\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 334, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/334\hfil Decay estimates for solutions] {Decay estimates for solutions of abstract wave equations with general damping function} \author[T. B\'arta \hfil EJDE-2016/334\hfilneg] {Tom\'a\v{s} B\'arta} \address{Tom\'a\v{s} B\' arta \newline Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovsk\'a 83, 18675 Praha 8, Czech Republic} \email{Tomas.Barta@mff.cuni.cz} % barta@karlin.mff.cuni.cz \thanks{Submitted March 13, 2016. Published December 28, 2016.} \subjclass[2010]{35L90, 35L10, 37L15} \keywords{Abstract wave equation; convergence to equilibrium; \hfill\break\indent decay estimates; {\L}ojasiewicz inequality} \begin{abstract} In this article we prove convergence to equilibrium and decay estimates for a class of damped abstract wave equations. We focus on the damping term to be as general as possible, including functions that oscillate between two positive functions in a neighborhood of the origin and/or behave differently in each direction. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this article, we prove convergence to equilibrium and show decay estimates for solutions of the second-order equation \begin{equation} \label{DSOE} \ddot u + g(\dot u) + M(u) = 0 \end{equation} on a Hilbert space $H$ for a broad class of damping functions $g$ and (unbounded) nonlinear operators $M=E'$ satisfying Kurdyka-{\L}ojasiewicz-Simon estimates. There are many convergence results for second-order equations with linear damping and various operators $M$, see \cite{Har86,Jen98,HJ99} for $M$ in the form $-\Delta u + f(x,u)$ and \cite{HR92} for a more general theory. Some decay estimates were shown in \cite{HJ01} for $-\Delta u + f(x,u)$, and in \cite{CHJ09} for a general nonlinear operator $M=E'$ satisfying the {\L}ojasiewicz gradient inequality. Convergence and decay estimates for nonlinear damping and a linear operator $M = -\Delta u$ and the right-hand side $h(x,t)$ was shown in \cite{HZ88}. An example, where bounded solutions do not converge to equilibrium, can be found in \cite{JP03} (a nonlinear wave equation on a bounded domain with Dirichlet boundary conditions and linear damping). Concerning nonlinear damping and a nonlinear operator $M$, the equation \begin{equation} \label{DSOEalpha} u_{tt}+|u_t|^{\alpha}u_t - \Delta u = f(x,u) \end{equation} was studied by Chergui \cite{Che09}, where convergence to equilibrium was proved. Later, Ben Hassen and Haraux \cite{BHH11} proved convergence to equilibrium and decay estimates in the abstract setting \eqref{DSOE} with $M=E'\in C^1(V,V^*)$ where $V\hookrightarrow H\hookrightarrow V^*$ are Hilbert spaces, and for damping functions $g:V\to V^*$ satisfying $$ c_1\|v\|^{\alpha+2}\le\langle g(v),v\rangle_{V^*,V}\quad \text{and}\quad \|g(v)\|_*\le c_2\|v\|^{\alpha+1}, $$ which implies \begin{equation} \label{relest} c_1\|v\|^{\alpha+1} \frac{\|v\|}{\|v\|_V}\le \|g(v)\|_*\le c_2\|v\|^{\alpha+1}. \end{equation} In \cite{BF16}, Fa\v{s}angov\' a and the author of this paper showed that the upper and lower estimates for $g$ can be independent, they proved convergence to equilibrium (without decay estimates) for pointwise damping operators $g(v)(x)=G(v(x))$ on $V=H^1_0(\Omega)$ with $G$ estimated from below and above by two independent functions. In this article we combine ideas from \cite{BHH11} and \cite{BF16} to prove convergence and decay estimates for $g:V\to V^*$ where $V$ is an arbitrary Hilbert space, $g$ satisfying $$ h(\|v\|)\|v\|\le\langle g(v),v\rangle_{V^*,V}\quad \text{and}\quad \|g(v)\|_*\le c_2\|v\|, $$ where $h$ is a positive function (not necessarily a power $s^{\alpha+1}$). We also show that the upper estimate for $g$ can be replaced by $\gamma(\|g(v)\|_*)\le \langle g(v),v\rangle_{V^*,V}$, which is satisfied by a wide class of poinwise damping operators. Moreover, we assume that $M=E'$ satisfies Kurdyka-{\L}ojasiewicz-Simon inequality (see Kurdyka \cite{Kur98}) $$ \Theta(E(u))\le \|M(u)\|_*, $$ which is a generalization of the {\L}ojasiewicz gradient inequality (see {\L}ojasiewicz \cite{Loj62}) considered in \cite{BHH11,Che09}. This conditions on $g$ allow much more general damping functions than the previous results. In particular, if we focus on the special case $g(v)(x)=G(v(x))$, then the following cases are covered in this article and not in \cite{BHH11}: \begin{itemize} \item growth of $G$ near zero and near infinity are different, e.g. $G(s)=|s|^{a}s$ for small $s$ and $G(s)=|s|^bs$ for large $s$, \item steeper growth of $G$ in infinity than in \cite[Example 3.1]{BHH11}, e.g. $G(s)=|s|^bs$ for $b\le \frac4{N-2}$, \item $G$ with different behavior in every direction around zero, e.g. for a scalar valued $v$ one allows $G(s)=|s|^a s$ for $s>0$ and $G(s)=|s|^b s$ for $s<0$, $a\ne b$, \item $G$ with non-power-like behavior, e.g. $G(s)=|s|^a\ln^b(1/|s|)\ln^c(\ln(1/|s|))s$ for small $s$. \end{itemize} Moreover, our results \begin{itemize} \item show that the decay estimates depend on the growth of $G$ near zero only (this is not obvious since $\|v\|<\varepsilon$ does not imply that $|v(x)|$ is small for every $x\in\Omega$), \item yield more delicate decay estimates, e.g. in the logarithmic scales $\|u(t)-\varphi\|\le C |t|^a\ln^b(1/|t|)\ln^c(\ln(1/|t|))$. \end{itemize} In fact, similar decay estimates (based on Kurdyka-{\L}ojasiewicz-Simon inequality) were shown in \cite{BBJ15,BarP} for second order ordinary differential equations, and in \cite{CF06} for first-order partial differential equations. We present two kinds of results. The first kind (Theorems \ref{main} and \ref{main3}) applies if we know a-priori that the whole solution (for all $t\ge t_0$) lies in a ball where the Kurdyka-{\L}ojasiewicz-Simon estimates are satisfied. In the second kind (Theorems \ref{main2} and \ref{main3}) we have Kurdyka-{\L}ojasiewicz-Simon estimates only in a small neighborhood $U$ of an omega-limit point of the solution and we assume that the solution is relatively compact, but we do not know a-priory that it is contained in $U$ for all $t\ge t_0$. This article is organized as follows. In Section 2 we introduce our settings and assumptions and formulate the main results. Sections 3 and 4 are devoted to proofs of the two main Theorems. In Section 5, the results are applied to some semilinear wave equations. Section 6 is an appendix where we prove some technical lemmas. \section{Assumptions and statement of main results} Let $V\hookrightarrow H \hookrightarrow V^*$ be Hilbert spaces with the embedding being dense, we identify $ \langle v,u\rangle_{V^*,V}=\langle v,u \rangle_H$ for $u\in V\subset H,\ v\in H\subset V^*$. The norm and the scalar product on $V^*$ (resp. on $H$, $V$) are denoted by $\|\cdot \|_*$ and $\langle\cdot, \cdot \rangle_*$ (resp. $\|\cdot \|$ and $\langle\cdot, \cdot \rangle$, $\|\cdot \|_V$ and $\langle\cdot, \cdot \rangle_V$). By $B(0,R)$ we denote the ball in $H$ of radius $R$ centered in 0, while $B_V(0,R)$ is the corresponding ball in $V$. In the whole paper, $C$ denotes a generic constant which may change from line to line or from expression to expression. Now, we define several properties of real functions. We say that a differentiable function $f:\mathbb{R}_+\to\mathbb{R}_+$ \begin{itemize} \item is \textit{admissible} if $f$ is nondecreasing and there exists $ c_A\ge 1$ such that $f(s)>0$ and $sf'(s)\le c_A f(s)$ for all $s>0$. \item has \textit{property (K)} if for every $K>0$ there exists $C(K)>0$ such that $f(Ks)\le C(K)f(s)$ holds for all $s>0$. \item is \textit{C-sublinear} if there exists $C>0$ such that $f(t+s)\le C(f(t)+f(s))$ holds for all $t$, $s>0$. \end{itemize} It is shown in the Appendix that the first property implies the other two. It is easy to see that any nonnegative increasing concave function is admissible with $c_A=1$ provided it is everywhere differentiable (otherwise $sf'_{\pm}(s)\le f(s)$ holds, which would be also sufficient for our purpose). Let us introduce our assumptions on the operator $E$. \begin{itemize} \item[(A1)] Let $E\in C^2(V)$, $M=E'\in C^1(V,V^*)$ and let $B$ be a fixed ball in $V$. Assume that: \begin{itemize} \item[(e1)] $E$ is nonnegative on $B$ and there exists an admissible function $\Theta$ such that $\Theta(s)\le C_\Theta\sqrt s$ for all $s\ge 0$ and some $C_{\Theta}>0$, $\frac{1}{\Theta}$ is integrable in a neighbourhood of zero and \begin{equation} \label{KLS} \|M(u)\|_* \ge \Theta(E(u)),\quad \text{for all $u\in B$,} \end{equation} i.e., $E$ satisfies the Kurdyka-{\L}ojasiewicz-Simon gradient inequality with function $\Theta$ on $B$. \item[(e2)] There exists $C_M\ge 0$ such that $$ |\langle M'(u)v,v \rangle_*|\le C_M\|v\|^2 \quad\text{for all $u\in B$, $v\in V$,} $$ \item[(e3)] There exists a nondecreasing function $G:\mathbb{R}_+\to \mathbb{R}_+$ such that \begin{equation} \label{invKLS} \|M(u)\|_* \le G(E(u)),\quad \text{for all $u\in B$.} \end{equation} \end{itemize} \end{itemize} Let us comment on the above assumptions. Chergui \cite{Che09} worked with $H=L^2(\Omega)$, $V=H_0^1(\Omega)$, $E'(u)=\Delta u + f(x,u)$ which corresponds to $E(u)=\int_{\Omega}\frac12|\nabla u(x)|^2$ $+ F(x,u)\,\mathrm{d}x$, where $F(x,u):=\int_0^u f(x,s)\,\mathrm{d}s$. By \cite[Corollary 1.2]{Che09}, this function $E$ satisfies the {\L}ojasiewicz gradient inequality \begin{equation} \label{LI} \|E'(u)\|_* \ge C |E(u)-E(\varphi)|^{1-\theta} \end{equation} with some $\theta\in [0,1/2)$ in a neighbourhood of stationary points, provided $f$ satisfies certain assumptions. The {\L}ojasiewicz inequality \eqref{LI} is a special case of the Kurdyka-{\L}ojasiewicz-Simon inequality \eqref{KLS} with the function $\Theta(s)=s^{1-\theta}$, $\theta$ being the {\L}ojasiewicz exponent. It is easy to see that Chergui's operator satisfies (e2) as well. The conditions (e1) and (e2) (with \eqref{LI} instead of \eqref{KLS}) appear also in \cite{CHJ09}, where linear damping is considered. Concerning assumption (e3), there is one more condition (g4) below, which connects functions $G$ and $\Theta$ with a function $h$ defined below. Let us mention that (e3) is often satisfied with $G(s)=C\sqrt s$, in particular in all applications in \cite{BHH11} and in finite-dimensional case for any $E\in C^{1,1}_{\rm loc}(\mathbb{R}^n)$ satisfying that $E(u)=0$ for all critical points $u$ (see \cite[Lemma 2.7]{BBJ15}). We now formulate the assumptions on the damping function. \begin{itemize} \item[(A2)] The function $g:V\to V^*$ is continuous and there exists an admissible function $h$ such that \begin{itemize} \item[(g1)] there exists $C_2>0$ such that $\|g(v)\|_*\le C_2\|v\|$ on $V\cap B(0,R)$ for any $R>0$ with $C_2$ depending on $R$, \item[(g2)] $\langle g(v),v\rangle_{V^*,V}\ge h(\|v\|)\|v\|^2$ on $V$, \item[(g3)] the function $s\mapsto \frac{1}{\Theta(s) h(\Theta(s))}$ belongs to $L^1((0,1))$, \item[(g4)] there exists $C_G>0$ such that $G(s)\le C_G \frac{\sqrt s}{h(\Theta(s))}$ on $(0,K]$ for any $K>0$ with $C_G$ depending on $K$, \item[(g5)] the function $\psi: s\mapsto sh(\sqrt s)$ is convex for all $s>0$. \end{itemize} \end{itemize} Let us comment on these assumptions. If we take $(g(v))(x)=|v(x)|^{\alpha}$, we obtain equation \eqref{DSOEalpha} studied by Chergui \cite{Che09}, and (g2) holds with $h(s)=s^{\alpha}$. Chergui's condition $\alpha<\frac{4}{N-2}$ (and also condition (g3) in \cite{BF16}) implies $g(v)\in V^*$. Moreover, taking $\Theta(s)=s^{1-\theta}$ (e.g. the {\L}ojasiewicz inequality instead of \eqref{KLS}), then (g3) corresponds to condition $0<\alpha<\frac{\theta}{1-\theta}$ in \cite{Che09} and \cite{BHH11}. Condition (g3) is a condition coupling the damping function $g$ with the operator $E$. Another condition coupling $g$ and $E$ is (g4). But (as was said above) in many applications $G(s)=C\sqrt s$, and in this case (g4) holds for any $h$ and $\Theta$ since $h(\Theta(s))$ is bounded on $(0,1)$. In \cite{BHH11} the authors work with (g2) for $h(s)=s^{\alpha}$ and (g1) replaced by $\|g(v)\|_*\le C_2\|v\|^{1+\alpha}$. It is easy to modify the proof in \cite{BHH11} in such a way that the upper bound for $\|g(v)\|_*$ can be relaxed to (g1) (it is easy to show that $\|v\|\to 0$, so $\|v\|^{1+\alpha}<\|v\|$). After doing this, one can apply the result in \cite{BHH11} e.g. to $$ g(v)(x)=|v(x)|^{\alpha}\ln(1/|v(x)|)v(x) $$ with $h(s)=s^{1+\alpha}$. However, applying Theorem \ref{main} below one can take $h(s)=s^{1+\alpha}\ln(1/s)$ in (g2) and get better convergence rates. One can show (by differentiating), that functions $$ h(s)=s^{a}\ln^{r_1}(1/s)\ln^{r_2}(\ln(1/s))\cdots \ln^{r_k}(\ln\cdots \ln(1/s)) $$ are positive increasing and concave on $(0,\varepsilon)$ for $a\in(0,1)$, $r_i\in \mathbb{R}$. So, they become admissible with $c_A=1$ after redefining them appropriately on $(\varepsilon,+\infty)$. In Section 5 we give some examples of decay estimates in these scales of functions. Our main results are formulated for solutions in the following sense. We say that $u\in W^{1,1}_{\rm loc}([0,+\infty),V)\cap W^{2,1}_{\rm loc}([0,+\infty),H)$ is \textit{a strong solution} to \eqref{DSOE} if \eqref{DSOE} holds in $V^*$ for almost every $t>0$. \begin{theorem} \label{main} Let $E$ and $G$ satisfy {\rm(A1)} and {\rm (A2)}. Let $u$ be a strong solution to \eqref{DSOE} and there exists $t_1>0$ such that $u(t)\in B$ for all $t\ge t_1$. Then there exist $\varphi\in B$ and $t_0\ge 0$ such that \begin{gather} \label{Eest} E(u(t))\le 2\Psi^{-1}(t-t_0), \\ \label{uest} \|u(t)-\varphi\|\le \Phi(\Psi^{-1}(t-t_0)), \\ \label{vest} \|\dot u(t)\|\le \sqrt{\Psi^{-1}(t-t_0))} \end{gather} hold for all $t>t_0$, some $C_{\Phi}$, $C_{\Psi}>0$ and \begin{equation} \label{SO_defphipsi} \Phi(t)= C_{\Phi}\int_0^t\frac1{\Theta(s)h(\Theta(s))}ds \quad\text{and}\quad \Psi(t)= C_{\Psi}\int_t^{1/2}\frac1{\Theta^2(s)h(\Theta(s))}ds. \end{equation} \end{theorem} If we take $\Theta(s)=s^{1-\theta}$ and $h(s)=s^{\alpha}$ in Theorem \ref{main}, we obtain the same convergence rate as in \cite[Theorem 2.2]{BHH11}. The next result combines the method from \cite{Che09} (resp. \cite{BF16}) and \cite{BHH11} to obtain decay estimates for relatively compact solutions with \eqref{KLS} satisfied only on a small neighborhood of some $\varphi\in \omega_V(u)$, where $$ \omega_V(u)=\{\varphi\in V:\, \exists\, t_n\nearrow +\infty,\text{ s.t. } \|u(t_n)-\varphi\|_V\to 0 \}. $$ \begin{theorem} \label{main2} Let $u$ be a strong solution to \eqref{DSOE} with $U_T:=\{(u(t),\dot u(t)),t\ge T\}$ relatively compact in $V\times H$ and $\varphi\in \omega_V(u)$ with $E(\varphi)=0$. Let {\rm(A1)} and {\rm (A2)} hold with the following changes: \begin{itemize} \item \eqref{KLS} and \eqref{invKLS} hold with $B$ replaced by $B_V(\varphi,\delta)$ for some $\delta>0$, \item (e2) holds with $B$ replaced by `any compact subset of $V$ with $C_M$ depending on the subset', \item $h$ is admissible with $c_A=1$, \end{itemize} Then $\lim_{t\to +\infty}\|u(t)-\varphi\|_V=0$ and there exists $t_0\ge 0$ such that the decay estimates \eqref{Eest}, \eqref{uest} and \eqref{vest} hold for all $t>t_0$, some $C_{\Phi}$, $C_{\Psi}>0$ and $\Phi$, $\Psi$ defined in \eqref{SO_defphipsi}. \end{theorem} \begin{theorem} \label{main3} Theorems \ref{main} and \ref{main2} remain valid if we replace {\rm (g1)} by \begin{itemize} \item[(g1')] for every $R>0$ there exists a convex function $\gamma:\mathbb{R}_+\to \mathbb{R}_+$ with property (K) and such that $\gamma(0)=0$, $\lim_{s\to+\infty}\gamma(s)=+\infty$, $\gamma(s)\ge cs^2$ for some $c>0$ and all $s$ small enough, and $\gamma(\|g(v)\|_*) \le \langle g(v),v\rangle_{V^*,V}$ on $V\cap B(0,R)$. \end{itemize} \end{theorem} Let us mention, that condition (g1) implies boundedness of $\|g(v(t))\|_*$, while condition (g1') does not. We show in Section 5 that (g1') is useful in many examples. It was mentioned in \cite{BarP} and also in \cite{BHH11} that estimating $\|u(t)-\varphi\|$ by the lenght of the trajectory $\int_t^{+\infty}\|\dot u(s)\|ds$ often does not yield an optimal result. In fact, the trajectory can be much longer than the distance $\|u(t)-\varphi\|$ if it has a shape of a spiral (which is typically the case for second order equations with weak damping). In many applications, one can obtain a better estimate by estimating $\|u-\varphi\|$ by $E(u)$ directly. \begin{corollary} \label{main2cor} Let the assumptions of Theorems \ref{main}, \ref{main2} or \ref{main3} are satisfied and $\alpha:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing function such that $\alpha(E(u)-E(\varphi))\ge \|u-\varphi\|$ on a neighborhood of $\varphi$. Then $$ \|u(t)-\varphi\|\le \alpha(2\Psi^{-1}(t-t_0)) $$ holds for some $t_0$ and all $t> t_0$. \end{corollary} The above corollary follows from $\|u(t)-\varphi\|\le \alpha(E(u(t))-E(\varphi))\le \alpha(2\Psi^{-1}(t-t_0))$. \section{Proof of Theorem \ref{main}} For the strong solution $u$ from the Theorem let us denote $v(t):=\dot u(t)$ and $$ E_1(t):=\frac12\|v(t)\|^2 + E(u(t)). $$ Then \begin{equation} \label{toint} E_1'(t)=\langle v(t),\dot v(t) \rangle_{V,V^*} + \langle M(u(t)),\dot u(t) \rangle_{V^*,V} = - \langle v(t),g(v(t)) \rangle_{V,V^*} \end{equation} It follows from (g2) that $E_1$ is nonincreasing, so it is either positive for all $t\ge 0$ or $v(t)=0$ for all $t\ge t_0$. In the latter case, $u(t)=\varphi$ for $t\ge t_0$ and there is nothing to prove. So, we may assume that $E_1(t)>0$ for all $t\ge 0$. Moreover, it follows that $\|v(t)\|$ and $E(u(t))$ are bounded and by (e3) also $\|M(u)\|_*$ is bounded. Further, for $s$ and $t\ge 0$ we define $$ B(s):=h(\Theta(s)), \quad H(t)=E_1(t) + \varepsilon B(E_1(t)) \langle M(u(t)),v(t)\rangle_*, $$ where $\varepsilon>0$ will be specified later. We first show that for all $t\ge t_1$ the inequality \begin{equation} \label{EHineq} \frac12E_1(t)\le H(t)\le 2E_1(t) \end{equation} holds if $\varepsilon>0$ is small enough. Both inequalities follow immediately from the estimate \begin{equation} \label{BEest} \begin{aligned} |\varepsilon B(E_1(t)) \langle M(u(t)),v(t)\rangle_*| &\le \varepsilon C B(E_1(t)) G(E_1(t)) \sqrt{2E_1(t)}\le \varepsilon C E_1(t) \\ &\le \frac12 E_1(t), \end{aligned} \end{equation} where the first inequality is a consequence of definition of $E_1$ and (e3) if applied the Cauchy-Schwarz inequality and $H\hookrightarrow V^*$, the second inequality is due to (g4) and definition of $B(\cdot)$ and in the third inequality we take $\varepsilon<1/(2C)$. We now derive some estimates for $H'(t)$. Let us fix $t>t_1$ and write $(u,v)$ instead of $(u(t),v(t))$ and also $E$, $E_1$ instead of $E(t)$, $E_1(t)$. We start with \begin{equation} \label{der_H1} \begin{aligned} &H'(t) \\ &= E_1' + \varepsilon B'(E_1)E_1' \langle M(u),v\rangle_* + \varepsilon B(E_1) \langle M'(u)v,v\rangle_* + \varepsilon B(E_1) \langle M(u),\dot v\rangle_* \\ & = - \langle g(v),v \rangle_{V^*,V} - \varepsilon B'(E_1) \langle g(v),v \rangle_{V^*,V} \langle M(u),v \rangle_* + \varepsilon B(E_1) \langle M'(u) v,v \rangle_* \\ & \quad - \varepsilon B(E_1) \langle M(u), g(v)\rangle_* - \varepsilon B(E_1) \langle M(u), M(u)\rangle_* \\ & = - \langle g(v),v \rangle_{V^*,V} - \varepsilon B(E_1) \|M(u)\|_*^2 + \varepsilon B(E_1) \langle M'(u) v,v \rangle_* \\ & \quad - \varepsilon B'(E_1) \langle v, g(v)\rangle_{V,V^*} \langle M(u), v\rangle_* - \varepsilon B(E_1) \langle M(u), g(v)\rangle_* \end{aligned} \end{equation} In the above expression we keep the first two terms and estimate the other terms from above. By admissibility of $h$ and $\Theta$ we have \[ B'(s)=h'(\Theta(s))\Theta'(s)\le C\frac{h(\Theta(s))}{\Theta(s)} \cdot \frac{\Theta(s)}s = C\frac{B(s)}s. \] So, $B(\cdot)$ is admissible. Then the fourth term on the right-hand side in \eqref{der_H1} can be estimated (with help of \eqref{BEest}) by \begin{align*} |\varepsilon B'(E_1) \langle v, g(v)\rangle_{V,V^*} \langle M(u), v\rangle_*| &\le \frac1{E_1}|\varepsilon B(E_1) \langle v, g(v)\rangle_{V,V^*} \langle M(u), v\rangle_*| \\ &\le \frac12 \langle v, g(v)\rangle_{V,V^*}\,. \end{align*} The third term on the right-hand side in \eqref{der_H1} is estimated as follows ($\psi^*$ being the convex conjugate to the function $\psi$ from condition (g5)) \begin{equation} \label{thirdterm} \begin{aligned} &|\varepsilon B(E_1) \langle M'(u) v,v \rangle_* |\\ &\le \varepsilon B(E_1) C \|v\|^2 \\ & \le \varepsilon C \Big(\frac1K\psi^*(B(E_1)) + C(K)\psi(\|v\|^2)\Big) \\ &\le \varepsilon C \Big(\frac CK\psi(\Theta^2(E_1)) + C(K)\psi(\|v\|^2)\Big) \\ &\le \varepsilon C \Big(\frac CK\psi(\Theta^2(E)) + \frac CK\psi(\Theta^2(\|v\|^2)) + C(K)\psi(\|v\|^2)\Big)\\ &\le \varepsilon C \Big(\frac CK\Theta^2(E)h(\Theta(E)) + 2C(K)\|v\|^2 h(\|v\|)\Big)\\ &\le \varepsilon C \Big(\frac CK\|M(u)\|_*^2 h(\Theta(E_1)) + 2C(K)\|v\|^2 h(\|v\|)\Big)\\ &\le \frac14\varepsilon B(E_1) \|M(u)\|_*^2 + \varepsilon C \langle v, g(v)\rangle_*. \end{aligned} \end{equation} Here we used (e2) (first inequality), Young inequality (second), Lemma \ref{psistar} (third), $C$-sublinearity of $\psi(\Theta^2(\cdot))$ (fourth), definition of $\psi$ and $\Theta(s)\le\sqrt s$ (fifth), \eqref{KLS} inequality and $E\le E_1$ (sixth) and we have taken $K=\frac1{4C^2}$ and used (g2) in the last inequality. The fifth term on the right-hand side of \eqref{der_H1} is estimated by \begin{align*} \varepsilon |B(E_1) \langle M(u), g(v)\rangle_*| &\le \varepsilon B(E_1) (\frac14\|M(u)\|_*^2 + C\|g(v)\|^2_*) \\ &\le \frac14\varepsilon B(E_1)\|M(u)\|_*^2 + \varepsilon CB(E_1) \|v\|^2\\ &\le \frac24\varepsilon B(E_1)\|M(u)\|_*^2 + \varepsilon C \langle v, g(v)\rangle_*, \end{align*} where we used the Cauchy-Schwarz and Young inequalities (first step), (g1) (second step) and \eqref{thirdterm} (last step). Altogether, we have \begin{equation} \label{estHder} \begin{aligned} H'(t) & \le - (1- \frac12 - 2\varepsilon C)\langle v, g(v)\rangle_* - \frac14 \varepsilon B(E_1)\|M(u)\|_*^2 \\ & \le -c (h(\|v\|))\|v\|^2 + B(E_1)\|M(u)\|_*^2). \end{aligned} \end{equation} Denoting $\chi(s):=B(s)\Theta^2(s)$ we obtain \begin{align*} -H'(t) & \ge c B(E)\|M(u)\|_*^2 \\ & \ge c B(E)\Theta(E)^2 \\ & = c\chi(E) \\ & = c\chi(E_1 - \frac12 \|v\|^2)\\ & \ge C_1 \chi(E_1) - C\chi(1/2\|v\|^2))\\ & = C_1 \chi(E_1) - C\Theta^2(1/2\|v\|^2) h(\Theta(1/2\|v\|^2))\\ & \ge C_1 \chi(E_1) - C\|v\|^2 h(\|v\|)\\ & \ge C_1 \chi(E_1) + C H'(t). \end{align*} Here we used \eqref{estHder} (in the first step), \eqref{KLS} inequality (second step), definition of $\chi$ (third), definition of $E_1$ (fourth), $C$-sublinearity of $\chi$ (fifth), definition of $\chi$ and $B$ (sixth), $\Theta(s)\le C\sqrt s$ and property (K) for $h$ (seventh) and \eqref{estHder} (last step). It follows that \begin{align*} -(C+1)H'(t) \ge C_1\chi(E_1(t)) \ge \frac12 C_1 \chi(H(t)). \end{align*} Take $C_{\Psi}=2(C+1)/C_1$. Then $$ \frac d{dt}\Psi(H(t))=C_{\Psi}\frac{-1}{\chi(H(t))}H'(t)\ge 1 $$ and we have \begin{align*} \Psi(H(t)) - \Psi(H(t_0)))\ge t-t_0. \end{align*} It follows that $\lim_{t\to+\infty}\Psi(H(t))=+\infty$, so we can take $t_0$ such that $\Psi(H(t_0))\ge 0$ and we obtain $\Psi(H(t))\ge t-t_0$. Since $\Psi$ is decreasing (by definition) we obtain \begin{align*} H(t)\le \Psi^{-1}(t-t_0). \end{align*} Now, \eqref{Eest} and \eqref{vest} follow immediately. To show the estimate \eqref{uest}, let us compute \begin{equation} \label{Phider} \begin{aligned} -\frac1{C_{\Phi}}\frac d{dt}\Phi(H(t)) &\ge C \frac{h(\|v\|)\|v\|^2 + B(E_1)\|M(u)\|_*^2}{\Theta(H(t))B(H(t))} \\ & \ge C \frac{h(\|v\|)\|v\|^2 + B(E_1)\|M(u)\|_*^2}{(\Theta(\|v\|^2) + \|M(u)\|_*)B(E_1)}\\ & \ge C \|v\| \frac{h(\|v\|)\|v\|^2 + B(E_1)\|M(u)\|_*^2}{B(E_1) \|v\|^2 + B(E_1)\|v\| \|M(u)\|_*}. \end{aligned} \end{equation} In the first inequality we used the definition of $\Phi$ and \eqref{estHder}. In the second inequality we used $H\le 2E_1$, $C$-sublinearity of $\Theta$, \eqref{KLS} inequality and $C$-sublinearity of $B$. In the last inequality we used $\Theta(s)\le c\sqrt s$ only. We estimate the two terms in the last denominator by the nominator. Using \eqref{thirdterm} we obtain \begin{equation} \label{denom1} B(E_1)\|v\|^2 \le C (B(E_1) \|M(u)\|_*^2 + \|v\|^2 h(\|v\|)) \end{equation} and (using Young inequality and \eqref{denom1}) \begin{equation} \label{denom2} \begin{aligned} B(E_1(t)) \|v\|\|M(u)\|_* &\le B(E_1)\|M(u)\|_*^2 + B(E_1)\|v\|^2 \\ &\le (1+C)B(E_1)\|M(u)\|_*^2 + C \|v\|^2 h(\|v\|). \end{aligned} \end{equation} From \eqref{Phider}, \eqref{denom1} and \eqref{denom2} we obtain $-\frac d{dt}\Phi(H(t))\ge \frac C{C_{\Phi}}\|v\| = \|v\|$ (choosing $C_{\Phi}=C$) and integrating from $t$ to $+\infty$ we conclude that $$ \int_t^{+\infty}\|v(s)\| ds \le \Phi(H(t)) - \lim_{s\to+\infty}\Phi(H(s)) \le \Phi(\Psi^{-1}(t-t_0)). $$ Hence $\dot u\in L^1([0,+\infty))$, so $u$ has a limit $\varphi$ and \eqref{uest} holds since $\|u(t)-\varphi\|\le \int_t^{+\infty} \|v(s)\|ds$. \section{Proofs of Theorems \ref{main2} and \ref{main3}} \begin{proof}[Proof of Theorem \ref{main2}] We may assume $\varphi=0$ and denote $v(t):=\dot u(t)$. We show below that $\|u(t)\|_V\to 0$ by the same method as in \cite{BF16}. So, we know that there exists $t_1$ such that $u(t)\in B_V(\varphi,\delta)$ for all $t>t_1$ and the assumptions of Theorem \ref{main} are satisfied with $B=B_V(\varphi,\delta)$. So, we apply Theorem \ref{main} and obtain the desired decay estimates. So, it only remains to show $\|u(t)\|_V\to 0$. By \cite[Theorem 2.6]{Bar14}, it is sufficient to find a function $\mathcal{E}\in C(V\times H, \mathbb{R})$, such that $t\mapsto \mathcal{E}(u(t), v(t))$ is nondecreasing for $t\ge 0$ and satisfies \begin{equation} \label{Eder} -\frac{\,\mathrm{d}}{\,\mathrm{d} t} \mathcal{E}(u(t), v(t)) \ge c \|\dot u(t)\|_* \end{equation} whenever $u(t)\in B_V(0,\eta)$ for some fixed $\eta>0$. We show that these conditions are satisfied by the function $$ \mathcal{E}(u,v):=\Phi(H(u,v)), $$ where $$ H(u,v)=\frac12\|v\|^2 + E(u) + \varepsilon h(\|v\|_*) \langle M(u),v\rangle_*, \quad u\in V,\; v\in H $$ with $\varepsilon$ small enough. Let us write for short $\mathcal{E}(t)$ (resp. $H(t)$) for $\mathcal{E}(u(t), v(t))$ (resp. $H(u(t),v(t))$) and $u$, $v$ instead of $u(t)$, $v(t)$. By relative compactness of $U_T$, quantities $\|v\|$ and $\|M(u)\|_*$ are bounded, so we can use (g1), resp. (g1'). We have (in the following, if $v=0$ then any term containing $\frac{1}{\|v\|_*}$ has to be replaced by $0$) \begin{align*} H'(t) &= \langle v,\dot v\rangle_{V,V^*} + \langle M(u), v\rangle_{V^*,V} + \varepsilon h'(\|v\|_*) \frac{\langle v, v_t\rangle_*}{\|v\|_*} \langle M(u), v\rangle_* \\ &\quad + \varepsilon h(\|v\|_*) \langle M'(u) v,v \rangle_* + \varepsilon h(\|v\|_*) \langle M(u), \dot v\rangle_* \\ & = - \langle g(v),v \rangle_{V^*,V} - \varepsilon h'(\|v\|_*) \frac{1}{\|v\|_*} \langle M(u),v\rangle_*^2 \\ & \quad - \varepsilon h'(\|v\|_*) \frac{1}{\|v\|_*} \langle g(v), v\rangle_* \langle M(u), v\rangle_* + \varepsilon h(\|v\|_*) \langle M'(u) v,v \rangle_* \\ &\quad -\varepsilon h(\|v\|_*) \langle M(u),M(u)\rangle_* - \varepsilon h(\|v\|_*) \langle g(v),M(u)\rangle_* \end{align*} and by positivity of the second term on the right \begin{equation} \label{Hder} \begin{aligned} H'(t) & \le - \langle g(v),v \rangle_{V^*,V} - \varepsilon h(\|v\|_*) \|M(u)\|^2_* - \varepsilon h(\|v\|_*) \langle g(v),M(u)\rangle_* \\ & \quad - \varepsilon h'(\|v\|_*) \frac{1}{\|v\|_*} \langle g(v), v\rangle_* \langle M(u), v\rangle_* + \varepsilon h(\|v\|_*) \langle M'(u) v,v \rangle_*. \end{aligned} \end{equation} We show that the third, fourth and fifth terms in the last expression are dominated by the first and second terms. The last term in \eqref{Hder} is estimated (with help of (e2) and (g2)) by \begin{equation*} \begin{aligned} |\varepsilon h(\|v\|_*) \langle M'(u) v,v \rangle_* | \le \varepsilon h(\|v\|_*) C\|v\|^2\le \varepsilon C \langle g(v),v \rangle_{V^*,V} \le \frac14 \langle g(v),v \rangle_{V^*,V} \end{aligned} \end{equation*} if $\varepsilon$ is small enough. The third term on the right-hand side of \eqref{Hder} is estimated by $$ |\varepsilon h(\|v\|_*) \langle g(v),M(u)\rangle_*|\le \varepsilon h(\|v\|_*) \|M(u)\|_* \|g(v)\|_*. $$ and the fourth term (applying the Cauchy-Schwarz inequality and admissibility of~$h$) by $$ \big|\varepsilon h'(\|v\|_*) \frac{1}{\|v\|_*} \langle g(v), v\rangle_* \langle M(u), v\rangle_* \big| \le \varepsilon c_A h(\|v\|_*) \|M(u)\|_* \|g(v)\|_*. $$ By Young's inequality and (g1) we have $$ \|M(u)\|_* \|g(v)\|_* \le \frac{1}{K} \|M(u)\|^2_* + C(K)\|g(v)\|_*^2 \le \frac{1}{K} \|M(u)\|^2_* + C(K)\|v\|^2. $$ So, the third and fourth terms from \eqref{Hder} are estimated by \begin{align*} &\varepsilon (1+c_A) h(\|v\|_*) \Big(\frac{1}{K} \|M(u)\|^2_* + C(K)\|v\|^2\Big) \\ & \le \frac12 \varepsilon h(\|v\|_*) \|M(u)\|^2_* + \varepsilon C h(\|v\|_*) \|v\|^2 \\ & \le \frac12 \varepsilon h(\|v\|_*) \|M(u)\|^2_* + \frac14 \langle g(v),v\rangle_{V^*,V} \end{align*} (we first took $K$ large enough and then $\varepsilon$ small enough). Altogether, we have \begin{equation} \label{est1} \begin{aligned} - H'(t) &\ge \frac12 \langle g(v),v \rangle_{V^*,V} + \varepsilon \frac12 h(\|v\|_*) \|M(u)\|^2_* \\ &\ge c h(\|v\|_*) \left(\|v\|^2 + \|M(u)\|^2_*\right) \end{aligned} \end{equation} where we used (g2) in the second inequality. Now we compute \begin{equation} \label{der_E} \mathcal{E}'(t) = \frac{C_{\Phi} H'(t)}{\Theta(H(t)) h(\Theta(H(t)))} \le - C \frac{ h(\|v\|_*) \left(\|v\|^2 + \|M(u)\|^2_*\right)} {\Theta(H(t)) h(\Theta(H(t)))} \end{equation} and see that $\mathcal{E}$ is nonincreasing along solutions for $t>0$. Now, we assume that $\|u\|_V$ is small and apply (e1) to obtain \eqref{Eder}. We compute \begin{align*} \Theta(H(u,v)) & \le C\Big( \Theta(\frac{1}{2}\|v\|^2) + \Theta(E(u)) + \Theta(\|M(u)\|_* \|v\|_*)\Big) \\ & \le C\left(\Theta(\|v\|^2) + \|M(u)\|_* + \Theta(\|M(u)\|_*^2) + \Theta(\|v\|^2)\right) \\ & \le C(\|v\| + \|M(u)\|_*) \, , \end{align*} where we used $C$-sublinearity and monotonicity of $\Theta$, boundedness of $h$ on compact intervals and property (K) for $\Theta$ and the Cauchy--Schwarz inequality (first step), Young's inequality, \eqref{KLS}, $H\hookrightarrow V^*$ and again C-sublinearity and property (K) (second step), and $\Theta(s)\le C\sqrt s$ (third step). Since $ h$ is nondecreasing and has property (K) we have \begin{equation} \label{Theta_est} \Theta(H(u,v)) h(\Theta(H(u,v)))\le C(\|v\| + \|M(u)\|_*) h(\|v\| + \|M(u)\|_*). \end{equation} Since $ h$ is admissible with $c_A=1$ we have $$ \Big(\frac{s}{h(s)}\Big)' = \frac{h(s)-sh'(s)}{h^2(s)}\ge 0, $$ i. e., $\frac{s}{h(s)}$ is nondecreasing. From $\|v\| + \|M(u)\|_* \ge c^*\|v\|_*$ we obtain \begin{equation}\label{th_est} \frac{\|v\| + \|M(u)\|_*}{ h(\|v\| + \|M(u)\|_*)} \ge \frac{c^*\|v\|_*}{ h(c^*\|v\|_*)} \ge \frac{c^*\|v\|_*}{C(c^*) h(\|v\|_*)}. \end{equation} Altogether, inserting the estimates \eqref{Theta_est} and \eqref{th_est} into \eqref{der_E} we obtain \begin{equation*} \begin{aligned} -\mathcal{E}'(t) \ge C\cdot \frac{ h(\|v\|_*)(\|v\| + \|M(u)\|_*)^2} {(\|v\| + \|M(u)\|_*) h(\|v\| + \|M(u)\|_*)} \ge C \|v(t)\|_* \end{aligned} \end{equation*} for all $t$ where $\|u(t)\|_V<\eta$ and the proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{main3}] The proofs of Theorems \ref{main} and \ref{main2} remain valid except that we have to be more careful by estimating the term $\|M(u)\|_* \|g(v)\|_*$. Take $R>0$ such that $\|v(t)\|\le R$ for all $t\ge 0$ and $\gamma$ corresponding to this $R$. Let $\gamma^*$ be the convex conjugate to $\gamma$. By \cite[Lemma 3.2]{BF16} we have $\gamma^*(s)\le Cs^2$ for all $s$ small enough. Then using Young's inequality we obtain \begin{equation} \label{gammaEst} \|M(u)\|_* \|g(v)\|_* \le \gamma^*\Big(\frac1K\|M(u)\|_*\Big) + \gamma(K\|g(v)\|_*). \end{equation} Since we know that $\|M(u)\|_*$ is bounded, taking $K$ large enough yields $$ \|M(u)\|_* \|g(v)\|_* \le \frac C{K^2}\|M(u)\|^2_* + C(K)\langle g(v),v\rangle_{V^*,V}, $$ where we also used property (K) for function $\gamma$. The rests of the proofs remain unchanged. \end{proof} \section{Applications} In this section we show that Theorem \ref{main3} applies to the damping functions from \cite{BF16}, i.e., we consider a bounded open set $\Omega\subset \mathbb{R}^n$, $H=L^2(\Omega,\mathbb{R}^N)$, $V=H^1_0(\Omega,\mathbb{R}^N)$ (or $V=H^1(\Omega,\mathbb{R}^N)$, $\Omega$ with Lipschitz boundary) and a function $G:\mathbb{R}^n\to \mathbb{R}^n$ satisfying the following conditions \begin{itemize} \item[(A3)] There exist $\tau >0$ and an admissible function $h:\mathbb{R}_+\to \mathbb{R}_+$ satisfying (g3), (g4), (g5) such that \begin{itemize} \item[(gg1)] there exists $C_2>0$ such that $|G(z)|\le C_2|z|$ for all $z\in B(0,\tau)$, \item[(gg2)] there exists $C_3>0$ such that $C_3|z|\le |G(z)|$ for all $z\in \mathbb{R}^n\setminus B(0,\tau)$, \item[(gg3)] if $n=2$ then there exist $C_4>0$, $\alpha>0$ such that $|G(z)|\le C_4|z|^{\alpha+1}$ for all $z\in\mathbb{R}^n\setminus B(0,\tau)$; if $n>2$ then the inequality holds with $\alpha=\frac{4}{n-2}$, \item[(gg4)] there exists $C_5>0$ such that $\langle G(z),z\rangle\ge C_5|G(z)| |z|$ for all $z\in \mathbb{R}^n$. \item[(gg5)] $|G(z)|\ge h(|z|)|z|$ for all $z\in B(0,\tau)$. \end{itemize} \end{itemize} \begin{proposition} Let $G:\mathbb{R}^n\to\mathbb{R}^n$ satisfy {\rm (A3)} and define $(g(v))(x):=G(v(x))$ for $v\in V$. Then $g(V)\subset V^*$ and $g$ satisfies {\rm (A2)} with {\rm (g1)} replaced by {\rm (g1')}. \end{proposition} \begin{proof} We first show that $g(v)\in V^*$. Since $L^p(\Omega,\mathbb{R}^N)\hookrightarrow V^*$ for $p=\frac{\alpha+2}{\alpha+1}$ it is enough to show that $g(v)\in L^p(\Omega,\mathbb{R}^N)$. We have \begin{align*} \int_{\Omega}|G(v(x))|^p &= \int_{\{|v(x)|\ge\tau\}}|G(v(x))|^p + \int_{\{|v(x)|<\tau\}}|G(v(x))|^p \\ &\le \int_{\{|v(x)|\ge\tau\}}C_4^p|v(x)|^{p(\alpha+1)} + \int_{\{|v(x)|<\tau\}}C_2^p|v(x)|^p \\ &\le C_4^p \int_{\Omega}|v(x)|^{\alpha+2} + |\Omega| C_2^p \tau^p \\ &\le C \|v\|_V^{\alpha+2} + |\Omega| C_2^p \tau^p, \end{align*} where the second inequality follows from (gg3) and (gg1) and the last inequality from $V\hookrightarrow L^{\alpha+2}(\Omega)$. Now we show (g2). We define $$ \tilde h(s):=\begin{cases} \frac{h(s)}{2} & \text{for } s\in[0,\delta)\\ \frac{h(\delta)}{2} + (\frac{1}{\delta} - \frac{1}{s}) \frac{h'(\delta)\delta^2}{2} & \text{for } s\in [\delta,+\infty) \end{cases} $$ as in \cite[proof of Proposition 3.3]{BF16}. It is easy to show that $\tilde h$ is admissible and $|G(z)|\ge \tilde h(|z|)|z|$ holds for all $z\in\mathbb{R}^n$ if $\delta>0$ is small enough and such that $h'(\delta)>0$. Moreover, $\tilde h$ is bounded and $\tilde \psi$ defined by $\tilde \psi(s)=s\tilde h(\sqrt s)$ is convex on $\mathbb{R}_+$ (see \cite[proof of Proposition 3.3]{BF16}). Then we have \begin{align*} \langle g(v),v\rangle_{V^*,V} &=\int_{\Omega} \langle G(v(x)),v(x)\rangle \\ &\ge \int_{\Omega} C_5 \tilde h(|v(x)|)|v(x)|^2 \\ &= C_5|\Omega|\int_{\Omega} \tilde \psi(|v(x)|^2) \frac{dx}{|\Omega|}\\ &\ge C_5|\Omega|\tilde\psi\Big(\int_{\Omega} |v(x)|^2 \frac{dx}{|\Omega|}\Big)\\ &\ge C\tilde\psi(\|v\|^2) \\ &= C\tilde h(\|v\|)\|v\|^2 \\ &\ge C h(\|v\|)\|v\|^2, \end{align*} where we used Jensen's inequality in the fourth step, property (K) in the fifth step and inequality $h(s)\le C\tilde h(s)$ on compact intervals $[0,K]$ in the sixth step. We show (g1'). By \cite[Proposition 3.3]{BF16} there exists a function $\gamma:\mathbb{R}_+\to \mathbb{R}_+$ such that $\gamma(G(s))\le CG(s)s$ and $s\mapsto \gamma(s^{1/p})$ is convex for $s\ge0$ and $\gamma(s)\ge Cs^2$ for small $s\ge 0$. Then we have \begin{align*} \gamma(\|g(v)\|_*) &\le C \gamma\Big( \Big(\int_{\Omega} |G(v(x))|^p\Big)^{1/p} \Big) \\ &\le C \int_{\Omega} \gamma(|G(v(x))|) \\ &\le C \int_{\Omega} |G(v(x))| |v(x)| \\ &\le C \int_{\Omega} \langle G(v(x)), v(x)\rangle \\ &= C\langle g(v),v\rangle_{V^*,V}. \end{align*} The first inequality follows from $L^p\hookrightarrow V^*$, monotonicity and property (K) of $\gamma$, the second inequality is Jensen's inequality applied to $s\mapsto \gamma(s^{1/p})$ together with property (K), the third follows from $\gamma(G(s))\le CG(s)s$ and the fourth from (gg4). \end{proof} Let us consider the following examples taken from \cite{BHH11}. \smallskip \noindent\textbf{A critical semilinear wave equation.} Let $\Omega\subset\mathbb{R}^n$ be bounded open and connected. We consider the Dirichlet problem \begin{equation} \label{ex1} \begin{gathered} u_{tt}+g(u_t)-\Delta u - \lambda_1 u + |u|^{p-1}u=0 \quad \text{in $\mathbb{R}_+\times \Omega$,} \\ u(t,x) = 0 \quad \text{on $\mathbb{R}_+\times \partial\Omega$,} \end{gathered} \end{equation} where $\lambda_1$ is the first eigenvalue of $-\Delta$ and $p>1$ with $(N-2)p1$ with $(n-2)p0$, $|s|$ large, $|s|$ small, e.g. $$ G(s)=\begin{cases} |s|^{b_1}s, & s>1, \\ |s|^{a_1}s, & s\in [0,1], \\ |s|^{a_2}s, & s\in [-1,0), \\ |s|^{b_2}s, & s< -1, \end{cases} $$ with $0\le a_1 1, \end{cases} $$ with $b<\frac4{n-2}$, $0< a<\frac1p$, $r\in\mathbb{R}$ or $a=\frac1p$, $r>1$. If $a>\frac1p$ and $r\ge 0$ then one can apply Theorem \ref{main3} with $h(s)=s^a$ to obtain $$ \|u(t)-\varphi\|\le Ct^{-\frac{1-ap}{(a+1)p-1}} $$ as in the previous example. If $a<\frac1p$, $r<0$, we can apply Theorem \ref{main3} with $h(s)=s^{a+\varepsilon}$ (for $\varepsilon>0$ small enough) to obtain $$ \|u(t)-\varphi\|\le Ct^{-\frac{1-(a+\varepsilon)p}{(a+\varepsilon+1)p-1}}. $$ If $a=1/p$, we cannot estimate $G$ by any power such that (g3) holds. However, in all cases, one can take $h(s)=s^a \ln^r(1/s)$ and obtain better decay estimates if $a<\frac1p$ and obtain some decay estimates even for $a=\frac1p$. In fact, we have $\Theta^2(s)h(\Theta(s))=s^{(1-\theta)(2+a)}(1-\theta)^r\ln^{r}(1/s)$ and by Lemma \ref{IntAsymp} \begin{equation} \label{psiasymp} \Psi(t)=C\int_t^{1/2}\frac1{s^{(1-\theta)(2+a)}\ln^{r}(1/s)}ds \sim t^{1-(1-\theta)(2+a)}\ln^{-r}(1/t),\quad t\to 0+, \end{equation} where $f\sim g$ means $f=O(g)$ and $g=O(f)$. Then by Lemma \ref{InverseAsymp} \begin{equation} \label{psi1asymp} \Psi^{-1}(t) \sim t^\frac1{1-(1-\theta)(2+a)} \ln^{\frac{r}{1-(1-\theta)(2+a)}}(t),\quad t\to +\infty. \end{equation} For equation \eqref{ex1} by Corollary \ref{main2cor} we have $$ \|u(t)-\varphi\|_V\le C\left(\Psi^{-1}(t-t_0)\right)^{\frac1{p+1}} \le Ct^{-\frac{1}{(a+1)p-1}}\ln^{-\frac{r}{(a+1)p-1}}(t). $$ For equation \eqref{ex2} in the case $a<\frac1p$ by Lemma \ref{IntAsymp} we have \begin{equation} \label{phiasymp} \Phi(t)=C\int_0^{t}\frac1{s^{(1-\theta)(1+a)}\ln^{r}(1/s)}ds \sim t^{1-(1-\theta)(1+a)}\ln^{-r}(1/t),\quad t\to 0+, \end{equation} which for large $t$ yields \begin{equation} \|u(t)-\varphi\|\le \Phi(\Psi^{-1}(t-t_0)) \le C t^{-\frac{1-ap}{(a+1)p-1}}\ln^{-\frac{pr}{(a+1)p-1}}(t). \end{equation} If $a=1/p$, then we have \begin{equation} \Phi(t)=C\int_0^{t}\frac1{s^{(1-\theta)(1+a)}\ln^{r}(1/s)}ds= C\int_0^{t}\frac1{s\ln^{r}(1/s)}ds\sim \ln^{1-r}(1/t) \end{equation} for $t\to 0+$ and therefore for large $t$, \begin{equation} \|u(t)-\varphi\|\le \Phi(\Psi^{-1}(t-t_0)) \le C \ln^{1-r}(t). \end{equation} \end{example} By similar computations as above with the help of Lemmas \ref{IntAsymp}, \ref{InverseAsymp}, we have: if $$ G(s)\ge |s|^a\ln^{r_1}(1/|s|)\cdots \ln^{r_k}(\ln\cdots \ln(1/|s|)) $$ on a neighborhood of zero, then for large $t$ we obtain $$ \|u(t)-\varphi\|\le C t^{-\frac{1-ap}{(a+1)p-1}}\ln^{-\frac{pr_1}{(a+1)p-1}}(t) \ln^{-\frac{pr_2}{(a+1)p-1}}(\ln(t))\cdots \ln^{-\frac{pr_k}{(a+1)p-1}}(\ln\cdots\ln(t)) $$ provided $a>1/p$ and $$ \|u(t)-\varphi\|\le C \ln^{1-r_j}(\ln\cdots \ln(t)) \ln^{-r_{j+1}}(\ln\cdots\ln(t))\cdots \ln^{-r_k}(\ln\cdots\ln(t)) $$ provided $a=\frac1p$, $r_1=\dots=r_{j-1}=1$, $r_j>1$, $r_{j+1}$, \dots, $r_k\in\mathbb{R}$. \section{Appendix} \begin{lemma} If $f$ is admissible, then it has property {\rm (K)}. \end{lemma} \begin{proof} For $K\le 1$ it is sufficient to take $C(K)=1$ since $f$ is nondecreasing. Now, let us fix $t\ge 0$. Then for $s>t$ we have $\frac{f'(s)}{f(s)}\le \frac {c_A}{s}$ and integrating from $t$ to $T>t$ we obtain $$ \ln(f(T))-\ln(f(t)) = \ln \frac{f(T)}{f(t)}\le c_A\ln\frac Tt, $$ so $f(T)\le f(t)\left(\frac{T}{t}\right)^{c_A}$ and taking $T=Kt$ for $K>1$ we have property (K) with $C(K)=K^{c_A}$. \end{proof} \begin{lemma} Let $f$ be nonnegative, nondecreasing and $f$, $g$ have property (K). Then the composition $f(g(\cdot))$ has property (K). \end{lemma} \begin{proof} We have $f(g(Kx))\le f(C(K)g(x))\le C(C(K))f(g(x))$. \end{proof} \begin{lemma} Let $f$ be nonnegative, nondecreasing and has property {\rm (K)}. Then it is $C$-sublinear, i.e., there exists $C>0$ such that $$ f(x+y)\le C(f(x)+f(y)) \quad\text{for all $x$, $y\ge 0$.} $$ \end{lemma} \begin{proof} We have \begin{align*} f(x+y)&\le f(2\max\{x,y\})\le C(2) f(\max\{x,y\}) \\ &\le C\max\{f(x),f(y)\} \le C(f(x)+f(y)). \end{align*} \end{proof} \marginpar{Where did you define (h3)?} \begin{lemma} \label{psistar} Let $\psi^*$ be convex conjugate to the function $\psi$ from {\rm (h3)}. Then $\psi^*(h(\sqrt s))\le c\psi(s)$ for all $s\ge 0$. \end{lemma} \begin{proof} It holds that $$ \psi^*(h(\sqrt{s}))=\psi^*(\psi(s)/s)\le \psi^*(\psi'(s)) = s\psi'(s) - \psi(s). $$ Further, $$ \psi(2s)-\psi(s)=\int_s^{2s}\psi'(r) dr \ge s\cdot \psi'(s). $$ So, $$ \psi^*(h(\sqrt{s})) \le \psi(2s)-2\psi(s) \le (K-2)\psi(s) $$ since $\psi$ has property {\rm (K)}. \end{proof} \begin{lemma} \label{IntAsymp} Let $F$ be a primitive function to $$ f(t)=t^a \ln^{r_1}(1/t) \ln^{r_2}(\ln(1/t))\cdots \ln^{r_k}(\ln\cdots \ln(1/t)) $$ on $(0,\varepsilon)$, $a\ne -1$. Moreover, if $a>-1$, we assume $\lim_{t\to 0+}F(t)=0$. Then \begin{equation} \label{Fasymp} |F(t)|\sim t^{1+a} \ln^{r_1}(1/t) \ln^{r_2}(\ln(1/t))\cdots \ln^{r_k}(\ln\cdots \ln(1/t)) \quad\text{as $t\to 0+$}, \end{equation} where $F\sim g$ means $F=O(g)$ and $g=O(F)$. If $a=-1$, $r_1=\dots=r_{j-1}=-1$, $r_j<-1$, then \begin{equation} \label{Fasymp2} |F(t)|\sim \ln^{r_j+1}(\ln\cdots \ln(1/t)) \ln^{r_{j+1}}(\ln\cdots \ln(1/t)) \cdots \ln^{r_k}(\ln\cdots \ln(1/t)) \end{equation} as $t\to 0+$. \end{lemma} \begin{proof} Let us denote the right-hand side of \eqref{Fasymp} by $G(t)$ and differentiate $$ G'(t)=(a+1)f(t) + \sum_{i=1}^k tf(t) \frac{r_i}{\ln(\cdots \ln(1/t))\cdots \ln(1/t)\frac1t}\cdot\frac{-1}{t^2} = f(t)(1+a+o(1)). $$ If $a>-1$, then $\frac1CG'(s)\le f(s)\le CG'(s)$ on $(0,\varepsilon)$ for some $C>1$ and $$ F(t)=\int_0^t f(s) \le C \int_0^t G'(s)ds = CG(t) $$ and similarly $F(t)\ge \frac1C G(t)$. If $a<-1$, then $\frac1CG'(s)\le f(s)\le CG'(s)$ on $(0,\varepsilon)$ for some $C<-1$. $$ |F(t)|=\int_t^c f(s)ds + d \le C \int_t^c G'(s)ds + d = CG(c) - CG(t)+d \le \tilde C G(t), $$ where the last inequality holds since $G(t)\to +\infty$ as $t\to 0+$ and $C<0$. Analogously we can estimate $|F(t)|$ from below. So, \eqref{Fasymp} is proven and \eqref{Fasymp2} can be proven by the same method. \end{proof} \begin{lemma} \label{InverseAsymp} Let $$ f(t)=t^a \ln^{r_1}(1/t) \ln^{r_2}(\ln(1/t))\cdots \ln^{r_k}(\ln\cdots \ln(1/t)) $$ on $(0,\varepsilon)$, $a<0$. Then \begin{equation} \label{fminus1asymp} f^{-1}(t)\sim t^{1/a} \ln^{-r_1/a}(t) \ln^{-r_2/a}(\ln(t))\cdots \ln^{-\frac{r_k}a}(\ln\cdots \ln(t)) \quad\text{as $t\to +\infty$}. \end{equation} \end{lemma} \begin{proof} Let us denote by $g(t)$ the right-hand side of \eqref{fminus1asymp} and let us assume that $r_i\ge 0$ for all $i=1,2,\dots, k$. We show that $f(g(t))\le Ct$ for large $t$. Since $$ \frac1{g(t)}=t^{-1/a}o(t^{-1/a}),\quad \text{as } t\to+\infty, $$ for $t$ large enough we have $$ \ln\Big(\frac1{g(t)}\Big)\le \ln\Big(t^{-\frac2a}\Big)=-\frac2a\ln(t). $$ Further, if $h(t)\to +\infty$, then for $c>0$ and large $t$ it holds that $\ln(ch(t))= \ln c + \ln h(t)\le 2\ln h(t)$. Therefore, $$ \ln^{r_i}\Big(\ln\cdots \ln\Big(\frac1{g(t)}\Big)\Big) \le \ln^{r_i}\Big(\ln\dots \frac{-2}a\ln(t)\Big) \le 2^{r_i}\ln^{r_i}\Big(\ln\cdots \ln(t)\Big). $$ Now, we can compute \begin{align*} f(g(t)) &= g(t)^a \prod_{i=1}^k \ln^{r_i} \Big(\ln\cdots \ln\Big(\frac1{g(t)}\Big)\Big) \\ &= t \ln^{-r_1}(t)\cdots \ln^{-r_k}(\ln\cdots \ln(t)) \cdot \prod_{i=1}^k \ln^{r_i}\Big(\ln\cdots \ln\Big(\frac1{g(t)}\Big)\Big)\\ &\le t \ln^{-r_1}(t)\cdots \ln^{-r_k}(\ln\cdots \ln(t)) \cdot \big(-\frac{2}{a}\big)^{r_1}\prod_{i=2}^k 2^{r_i}\ln^{r_i} \left(\ln\cdots \ln(t)\right)\\ &\le t\cdot \big(-\frac{1}{a}\big)^{r_1}\prod_{i=1}^k 2^{r_i}. \end{align*} We can easily modify the estimates above to obtain $f(g(t))\ge t (-\frac{1}{a})^{r_1}\prod_{i=1}^k 2^{-r_i}$ and similarly if we omit the assumption that $r_i$ are positive, we obtain $$ \frac tK \le f(g(t))\le Kt \quad\text{with } K:=C^{r_1}\prod_{i=1}^k 2^{|r_i|},\quad C:=\max\{-\frac1a,-a\}. $$ Applying $f^{-1}$ (which is decreasing for large $t$) to these inequalities with $s=t/K$, we obtain $$ f^{-1}(s)\ge f^{-1}(f(g(Ks))) = g(sK) \ge \frac{K^{1/a}}C g(s), $$ resp. with $s=Kt$ $$ f^{-1}(s)\le f^{-1}(f(g(s/K))) = g(s/K) \le \frac C{K^{1/a}}g(s). $$ \end{proof} \subsection*{Acknowledgements} The author is a member of the Ne\v cas Center for Mathematical Modeling. \begin{thebibliography}{10} \bibitem{Bar14} T.~B\' arta; \emph{Convergence to equilibrium of relatively compact solutions to evolution equations}, Electron. 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