\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 77, pp. 1--21.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/77\hfil Nonlinear perturbations] {Nonlinear perturbations of the Kirchhoff equation} \author[M. Milla Miranda, A. T. Louredo, L. A. Medeiros \hfil EJDE-2017/77\hfilneg] {Manuel Milla Miranda, Aldo T. Louredo, Luiz A. Medeiros } \address{Manuel Milla Miranda \newline Universidade Estadual da Para\'iba, DM, PB, Brazil} \email{milla@im.ufrj.br} \address{Aldo T. Louredo \newline Universidade Estadual da Para\'iba, DM, PB, Brazil} \email{aldolouredo@gmail.com, Phone +55 (83) 3315-3340} \address{Luiz A. Medeiros \newline Universidade Federal do Rio de Janeiro, IM, RJ, Brazil} \email{luizadauto@gmail.com} \dedicatory{Communicated by Jerome A. Goldstein} \thanks{Submitted January 24, 2017. Published March 21, 2017.} \subjclass[2010]{35L15, 35L20, 35K55, 35L60, 35L70} \keywords{Kirchhoff equation; nonlinear boundary condition; \hfill\break\indent existence of solutions} \begin{abstract} In this article we study the existence and uniqueness of local solutions for the initial-boundary value problem for the Kirchhoff equation \begin{gather*} u'' - M(t,\|u(t)\|^{2})\Delta u + |u|^{\rho} =f \quad\text{in } \Omega \times (0, T_0), \\ u=0\quad\text{on }\Gamma_0 \times ]0, T_0[, \\ \frac{\partial u}{\partial \nu} + \delta h(u')=0 \quad\text{on } \Gamma_1 \times ]0, T_0[, \end{gather*} where $\Omega$ is a bounded domain of $\mathbb{R}^n$ with its boundary constiting of two disjoint parts $\Gamma_0$ and $\Gamma_1$; $\rho >1$ is a real number; $\nu(x)$ is the exterior unit normal vector at $x \in \Gamma_1$ and $\delta(x), h(s)$ are real functions defined in $\Gamma_1$ and $\mathbb{R}$, respectively. Our result is obtained using the Galerkin method with a special basis, the Tartar argument, the compactness approach, and a Fixed-Point method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Frist we do some preliminary considerations to justify the mixed problem we want to study. Milla Miranda and Medeiros \cite{mm-lam} analyzed the existence of solutions for problem \begin{equation}\label{I1.1} \begin{gathered} u'' - \mu(t) \Delta u=0 \quad\text{in } \Omega \times (0, \infty), \\ u=0 \quad\text{on } \Gamma_0 \times (0, \infty), \\ \mu(t) \frac{\partial u}{\partial \nu} + \delta(x) u'=0\quad\text{on }\Gamma_1 \times (0, \infty), \\ u(x,0)=u_0(x),\quad u'(x,0)=u_1(x),\quad x \in\Omega. \end{gathered} \end{equation} When $\mu$ is a positive constant, existence and uniqueness of global solutions for \eqref{I1.1} has been proved by Komornik and Zuazua \cite{Komornik}, Lasiecka and Triggiane \cite{triggiane} and Quinn and Russell \cite{Quinn e Russell33}, Goldstein \cite{Goldstein} applying semigroup theory. This method does not work for \eqref{I1.1} because the boundary condition \eqref{I1.1}$_3$ brings serious difficulties. For this reason, the authors of \cite{mm-lam} defined a special basis of the space where lie the approximations of the initial data and apply the Galerkin method. This approach works well for problem \eqref{I1.1}. Motivated by \eqref{I1.1}, Milla Miranda and Jutuca \cite{Jutuca} analized the initial-boundary value problem for the Kirchhoff equation \begin{equation}\label{I1.2} \begin{gathered} u'' - M\Big(t,\int_{\Omega}|\nabla u|^2dx \Big) \Delta u=f \quad\text{in } \Omega \times (0, \infty), \\ u=0 \quad\text{on } \Gamma_0 \times (0, \infty), \\ \mu(t) \frac{\partial u}{\partial \nu} + \delta(x) u'=0\quad\text{on } \Gamma_1 \times (0, \infty), \\ u(x,0)=u_0(x),\quad u'(x,0)=u_1(x),\quad x \in\Omega. \end{gathered} \end{equation} Following the ideas in \cite{mm-lam} but having much more difficulty, the authors of \cite{Jutuca}, succeeded in the construction of a special basis and the Galerkin method works well for \eqref{I1.2}. They proved existence and uniqueness of solutions for \eqref{I1.2}. See also \cite{Cavalcanti,Ong}. An extensive list of references about the Kirchhoff equation can be found in Medeiros, Limaco and Menezes \cite{silvano}. In Medeiros et al.\ \cite{adautociceroejuan} was investigated the existence and uniqueness of global solutions for the problem \begin{equation}\label{I1.3} \begin{gathered} u'' - \Delta u + |u|^\rho =f\quad\text{in }\Omega \times (0,\infty) \\ u=0\quad\text{on }\Gamma \times (0, \infty) \\ u(x,0)=u^0(x),\quad u'(x,0)=u^1(x),\quad x \in \Omega \end{gathered} \end{equation} There, Galerkin method and Tartar argument \cite{Tartar} were applied. Motivated by the studies of \eqref{I1.1}-\eqref{I1.3}, we investigate the existence and uniqueness of local solutions of the initial value problem for the nonlinear mixed problem of Kirchhoff type: \begin{equation}\label{I1.4} \begin{gathered} u'' - M\Big(t, \int_{\Omega}|\nabla u|^2dx \Big) \Delta u +|u|^\rho =f \quad\text{in } \Omega \times (0, T_0), \\ u=0 \quad\text{on } \Gamma_0 \times (0, T_0), \\ \frac{\partial u}{\partial \nu} + \delta (x) h(u')=0\quad\text{on }\Gamma_1 \times (0, T_0), \\ u(x,0)=u^0(x),\quad u'(x,0)=u^1(x),\quad x \in\Omega. \end{gathered} \end{equation} By applying the Galerkin method with a special basis, a modification of the Tartar approach, compactness method and fixed-point theorem, we obtain our result. Note that the existence of global solutions for \eqref{I1.4} without the term $|u|^\rho=0$, null Dirichlet boundary condition on $\Gamma$ and $u^0 \in H_0^1(\Omega) \cap H^2(\Omega)$, $u^1\in H_0^1(\Omega)$ is a open question. \section{Notation and statement of main results} Let $\Omega$ be bounded open set of $\mathbb{R}^n$ with boundary $\Gamma$ of class $C^2$. It is assumed that $\Gamma$ is constituted by two disjoint parts $\Gamma_0$ and $\Gamma_1$, $\Gamma_0$ and $\Gamma_1$ with positive measures, such that $\overline{\Gamma_0} \cap\overline{ \Gamma_1}= \emptyset$. By $\nu(x)$ represents the unit normal vector at $x \in \Gamma_1$. We denote by $H^m(\Omega)$ the Sobolev space of order $m$ and by $(u,v)$ and $|u|$, the scalar product and norm, respectively, in $L^2(\Omega)$. We define the Hilbert space $$ V=\{v\in H^1(\Omega): v=0\text{ on }\Gamma_0\}, $$ equipped with the scalar product $$ ((u,v))=\sum_{i=1}^n \int_{\Omega}\frac{\partial u}{\partial x_i}(x) \frac{\partial v}{\partial x_i}(x)\,dx $$ and norm $\|u\|^2=((u,u))$. All scalar functions considered in this article will be real-valued. To state our main result, we introduce the following hypotheses: \begin{itemize} \item[(H1)] The function $M(t, \lambda)$ satisfies $M \in W^{1,\infty }_{\rm loc}([0, \infty[^2 )$, $M(t, \lambda)\geq m_0>0$ for all $\{t, \lambda\} \in ([0, \infty[)^2$ with $m_0$ constant. \item[(H2)] The function $h$ is a Lipschitz continuous, $h(0)=0$, and $h$ is strongly monotonous, that is, for a positive constant $d_0$, \[ (h(r)-h(s))(r-s)\geq d_0(r-s)^2,\quad \forall r,s \in \mathbb{R}. \] \item[(H3)] $\delta \in W^{1, \infty}(\Gamma_1)$ and $\delta(x)\geq \delta_0$ for all $x \in \Gamma_1$ and a positive constant $\delta_0$. \item[(H4)] The real number $\rho$ satisfies the following restrictions \begin{equation}\label{q6} \rho >1 \text{ if }n=1,2; \quad \frac{n+1}{n} \leq \rho \leq \frac{n}{n-2} \text{ if } n\geq 3. \end{equation} \end{itemize} Let $h: \mathbb{R}\to \mathbb{R}$ be a Lipschitz continuous function with $h(0)=0$. In Marcus and Mizel \cite{marcus} (see also \cite{cazenave}) it is shown that $h(v) \in H^{1/2}(\Gamma_1)$ for $v \in H^{1/2}(\Gamma_1)$ and $h: H^{1/2}(\Gamma_1) \to H^{1/2}(\Gamma_1)$, $v\mapsto h(v)$, is continuous. \begin{remark}\label{obs2.1}\rm Consider the trace of order zero $\gamma_0: V \to H^{1/2}(\Gamma_1)$. Then the map \[ \widetilde{h}=h\circ \gamma_0, \quad \widetilde{h}: V \to H^{1/2}(\Gamma_1) \] is continuous. \end{remark} %\label{obs2.2} Throughout the article, to facilitate the notation, the mapping $\widetilde{h}(v)$, $v \in V$, will be denoted by $h(v)$. \begin{remark}\label{obs2.3} \rm Let $\delta: \Gamma_1 \to \mathbb{R}$ be a function such that $\delta \in W^{1, \infty}(\Gamma_1)$. Then $\delta v \in H^{1/2}(\Gamma_1)$ for $v \in H^{1/2}(\Gamma_1)$, and the linear operator \[ \delta: H^{1/2}(\Gamma_1)\to H^{1/2}(\Gamma_1), \quad v \mapsto \delta v \] is continuous. Also, the linear operators \begin{gather*} \delta: H^{1}(\Gamma_1)\to H^{1}(\Gamma_1),\quad v \mapsto \delta v, \\ \delta: L^2(\Gamma_1) \to L^2(\Gamma_1),~v \mapsto \delta v \end{gather*} are continuous. The statements in this remark follow from the theory of interpolation of Hilbert spaces, see Lions-Magenes \cite{L-M}. \end{remark} Next, we state our main result. \begin{theorem}\label{teo2.1} Assume that hypotheses {\rm (H1)--(H4)} are satisfied. Consider $\{u^0,u^1\}$ in $V \cap H^2(\Omega) \times V$ satisfying the compatibility condition \begin{equation}\label{2.1} \frac{\partial u^0}{\partial \nu} + \delta h(u^1)=0, \end{equation} and the norm condition \begin{equation}\label{q16} \|u^0\| < \lambda^{*}:=\Big(\frac{m_0}{3k_0^{\rho+1}} \Big)^{\frac{1}{\rho-1}}, \end{equation} where $k_0$ is the immersion constant of $V$ in $L^{\rho +1}(\Omega)$, and \begin{equation}\label{q15} f\in L^1(0, T; L^2(\Omega)), \quad f'\in L^1(0, T; L^2(\Omega))\,. \end{equation} Then there exist a real number $0< T_0 \leq T$, and a unique function $u$ with \begin{equation}\label{2.2} \begin{gathered} {u} \in L^{\infty}(0,T_0; V \cap H^2(\Omega) ), \\ {u'} \in L^{\infty}(0,T_0; V ), \\ u'' \in L^{\infty}(0,T_0; L^2(\Omega)) \cap L^2(0, T_0; L^2(\Gamma_1)), \end{gathered} \end{equation} such that $u$ satisfies \begin{equation}\label{2.3} u''-M(\cdot,\|u\|^{2})\Delta u + |u|^\rho=f \quad\text{in } L^{\infty}(0,T_0; L^2(\Omega)), \end{equation} \begin{equation} \label{2.4} \begin{gathered} \frac{\partial u}{\partial\nu} + \delta h(u')=0 \quad\text{in } L^{2}(0,T_0; H^{1/2}(\Gamma_1)), \\ \frac{\partial u'}{\partial\nu} + \delta h'(u')u''=0 \quad\text{in } L^{2}(0,T_0; L^{2}(\Gamma_1)), \end{gathered} \end{equation} and \begin{equation}\label{2.5} u(0)=u^0,\quad u'(0)=u^1, \end{equation} \end{theorem} \begin{remark}\label{obs2.4} \rm By Remarks \ref{obs2.1} and \ref{obs2.3}, the function $\delta h(u^1)$ belongs to $H^{1/2}(\Gamma_1)$. Then condition \eqref{2.1} makes sense. \end{remark} \section{Existence of Solutions} To apply Banach Fixed-Point Theorem in the proof of our result, we introduce an auxiliary problem related to \eqref{I1.4}. \subsection{Auxiliary Problem} Consider the problem \begin{equation}\label{3.1} \begin{gathered} u'' - \mu \Delta u + |u|^\rho=f \quad\text{in }\Omega \times (0, \infty ), \\ u=0 \quad\text{on }\Gamma_0 \times (0, \infty), \\ \frac{\partial u}{\partial \nu} + \delta h(u')=0 \quad\text{on } \Gamma_1 \times (0, \infty), \\ u(0)=u^0,\quad u'(0)=u^1\quad\text{in }\Omega. \end{gathered} \end{equation} Where $\mu(t), h(s)$ and $\delta $ are real functions defined in $[0, \infty )$, $\mathbb{R}$ and $\Gamma_1$, respectively. The existence of solutions of \eqref{3.1} is derived by applying the Galerkin method with a special basis of $V \cap H^2(\Omega)$ and a modification of the Tartar method. To obtain this basis we introduce some results. \begin{lemma}\label{lema1} Let $m$ and $n$ be functions in $L^1(0,T)$ with $m(t)\geq 0$ and $n(t)\geq 0$ a.e. $t $ in $(0,T)$ and let $a\geq 0$ be a constant. Consider $\varphi: [0,T] \to \mathbb{R}$ continuous, $\varphi(t)\geq0$, for all $t \in [0,T]$, and satisfying $$ \frac{1}{2}\varphi^2(t) \leq \frac{1}{2}a^2 + \int_0^t m(\tau)\varphi(\tau) d\tau + \int_0^t n(\tau) \varphi^2(\tau) d\tau,\quad \forall t \in [0,T]. $$ Then $$ \varphi(t) \leq \Big( a + \int_0^T m(\tau)d \tau \Big) \exp\Big( \int_0^t n(\tau)d \tau \Big), \quad \forall t \in [0,T]. $$ \end{lemma} The above result is a consequence of a lemma provided in Brezis \cite[p. 157]{Brezis}. Milla Miranda and Medeiros \cite{mm-lam} showed the following three results: \begin{proposition}\label{pr3.1} Let us consider $f \in L^2(\Omega)$ and $g \in H^{1/2}(\Gamma_1)$. Then, the solution $u$ of the problem \begin{equation}\label{3.2} \begin{gathered} -\Delta u=f \quad \text{in }\Omega, \\ u=0 \quad \text{on } \Gamma_0, \\ \frac{\partial u}{\partial \nu}=g \quad \text{on } \Gamma_1, \end{gathered} \end{equation} belongs to $V \cap H^2(\Omega)$ and satisfies $$ \|u\|^2_{H^2(\Omega)}\leq c\big[|f|^2 + \|g\|^2_{H^{1/2}(\Gamma_1)}\big], $$ where the constant $c>0$ is independent of $u, f$ and $g$. \end{proposition} \begin{proposition}\label{pr3.2} In $V \cap H^2(\Omega)$ the norms $H^2(\Omega)$ and $$ \Big[ |\Delta u|^2 + \|\frac{\partial u}{\partial \nu}\|_{H^{1/2}(\Gamma_1)}^2 \Big]^{1/2}, $$ are equivalent. \end{proposition} We equipp $V \cap H^2(\Omega)$ with the preceding norm. \begin{remark}\label{obs3.1} \rm The space $V \cap H^2(\Omega)$ is dense in $V$. In fact, we consider the operator $A=-\Delta$ defined by the triplet $\{V, L^2(\Omega), ((u,v))\}$. Then its domain $D(-\Delta)$ is $$ D(-\Delta)=\big\{v \in V \cap H^2(\Omega); \frac{\partial v}{\partial \nu}=0 \text{ on } \Gamma_1 \big\}, $$ is dense in $V$ (see \cite{Lions1}). As $D(-\Delta)$ is contained in $V \cap H^2(\Omega)$, the conclusion follows. \end{remark} \begin{lemma}\label{pr3.3} \rm Consider a function $\delta$ satisfying hypothesis (H3), and a Lipschitz continuous function $h(s)$, $s \in \mathbb{R}$, with $h(0)=0$. Take $u^0 \in V \cap H^2(\Omega)$ and $u^1 \in V$ satisfying the condition \begin{equation}\label{3.3} \frac{\partial u^0}{\partial \nu} + \delta h(u^1)=0 \quad \text{on }\Gamma_1. \end{equation} Then, for each $\varepsilon >0$, there exist $w$ and $z$ in $V \cap H^2(\Omega)$ such that \begin{gather*} \|w-u^0\|_{V \cap H^2(\Omega)}< \varepsilon,\quad \|z-u^1\|<\varepsilon, \\ \frac{\partial w}{\partial \nu} + \delta h(z)=0 \quad \text{on }\Gamma_1. \end{gather*} \end{lemma} With respect to the function $\mu$ we make the following assumptions: \begin{equation}\label{q4} \mu \in W^{1,1}_{\rm loc}(0, \infty), \quad 0< \mu_0 \leq \mu(t) \leq \mu_1,\quad \forall t \geq 0,\quad \mu'\in L^1(0,\infty) \end{equation} for some constants $\mu_0$, $\mu_1$. Consider the real number $\rho$ satisfying the restrictions (H4). Then \begin{equation}\label{q8} V \hookrightarrow L^{p^{*}}(\Omega) \hookrightarrow L^{2\rho}(\Omega) \hookrightarrow L^{\rho+1}(\Omega)\hookrightarrow L^{\rho}(\Omega) \end{equation} where $p^{*}=\frac{2n}{n-2}$, $n \geq 3$. In what follows $X \hookrightarrow Y$ denotes that injection of the space $X$ into the space $Y$ is continuous. Note that when $p>1$ and $n=1$ or $n=2$, the continuous injections \eqref{q8} without $L^{{p^{*}}}(\Omega)$ is true. With respect to the above injections, we introduce the following notation: \begin{equation}\label{q12} \begin{gathered} \|v\|_{L^{\rho+1}(\Omega)} \leq k_0 \|v\|,\quad \|v\|_{L^{\rho}(\Omega)} \leq k_1 \|v\|, \\ \|v\|_{L^{2\rho}(\Omega)} \leq k_2 \|v\|,\quad \|v\|_{L^{(\rho-1)n}(\Omega)} \leq k_3 \|v\|, \\ \|v\|_{L^{p^{*}}(\Omega)} \leq k_4 \|v\| \end{gathered} \end{equation} for all $v \in V$. Consider \begin{gather}\label{00q16} \|u^0\| < \lambda^{*}_1:=\Big(\frac{\mu_0}{3k_0^{\rho+1}} \Big)^{\frac{1}{\rho-1}},\\ \label{q06} G(s)=\frac{1}{\rho+1}|s|^\rho s. \end{gather} Recall that $G(s)=\int_0^s |\tau|^\rho d \tau$. With the above assumptions, we have the following result. \begin{theorem}\label{teorema1} Assume hypotheses {\rm (H1), (H3), (H4)} and \eqref{q4}. Consider \begin{equation}\label{q15b} u^0 \in V \cap H^2(\Omega),\quad u^1 \in V, f\in L^1(0, \infty; L^2(\Omega)), \quad f'\in L^1_{\rm loc}(0, \infty; L^2(\Omega)) \end{equation} satisfying \eqref{2.1} and \begin{equation}\label{0q16} \begin{gathered} \|u^0\| < \lambda^{*}_1,\\ \big(\frac{2}{\mu_0} \big)^{1/2}\Big[ (2N)^{1/2} + \int_0^{\infty}|f(t)|dt \Big]\exp\Big( \frac{2}{\mu_0}\int_0^{\infty}|\mu'(t)|dt \Big) < \lambda_1^{*}, \end{gathered} \end{equation} where \begin{equation}\label{q17} N=\frac{1}{2}|u^1|^2 + \frac{1}{2}\mu(0)\|u^0\|^2 + \frac{k_0^{\rho+1}}{\rho+1}\|u^0\|^{\rho +1}. \end{equation} and the real number $\lambda^{*}_1$ defined in \eqref{00q16}. Then there exists a function $u$ with \begin{equation}\label{q18} \begin{gathered} u\in L^{\infty}(0, \infty; V),\quad u'\in L^{\infty}(0, \infty; L^2(\Omega))\cap L^{\infty}_{\rm loc}(0, \infty; V) \\ u'' \in L^{\infty}_{\rm loc}(0, \infty; L^2(\Omega)),\quad u'\in L^{\infty}(0, \infty; L^2(\Gamma_1)); \\ u''\in L^{\infty}_{\rm loc}(0, \infty; L^2(\Gamma_1)) \end{gathered} \end{equation} satisfying \begin{gather}\label{q19} u'' - \mu \Delta u + |u|^\rho= f\quad\text{in } L^2_{\rm loc}(0, \infty; L^2(\Omega)),\\ \label{q20} \frac{\partial u}{\partial \nu} + \delta h(u')=0\quad\text{in } L^{2}_{\rm loc}(0, \infty; H^{1/2}(\Gamma_1)), \\ \label{0q20} \frac{\partial u'}{\partial \nu} + \delta h'(u')u''=0\quad\text{in } L^2_{\rm loc}(0, \infty; L^2(\Gamma_1)), \\ \label{q21} u(0)=u^0,\quad u'(0)=u^1. \end{gather} \end{theorem} \subsection*{Proof of Theorem \ref{teorema1}} By Lemma \ref{pr3.3}, we obtain sequences $(u_l^0), (u_l^1)$ of vectors of $V \cap H^2(\Omega) $ satisfying \begin{equation}\label{equacao3.9} \begin{gathered} \lim_{l \to \infty}u_l^0=u^0 \quad \text{in } V\cap H^2(\Omega) \\ \lim_{l \to \infty}u_l^1=u^1 \quad \text{in } V \\ \frac{\partial u_l^0}{\partial \nu} + \delta h(u_l^1)=0 \quad \text{on } \Gamma_1,\; \forall l \in \mathbb{N}. \end{gathered} \end{equation} We construct a special basis of $V \cap H^2(\Omega)$ as follows: Fix $l \in \mathbb{N}$. Consider the basis $$ \{w_1^l, w_2^l, \dots , w_j^l,\dots \} , $$ of $V \cap H^2(\Omega)$ satisfying $u^0, u^1 \in [w_1^l, w_2^l]$, where $[w_1^l, w_2^l]$ denotes the subspace generated by $w_1^l, w_2^l$. With this basis determine approximate solutions $u_{lm}(t)$ of Problem \eqref{3.1}, that is, \begin{equation}\label{equacao3.10} \begin{gathered} u_{lm}(t)=\sum_{j=1}^mg_{jlm}(t)w_j^l, \\ \begin{aligned} &(u''_{lm}(t), v) + \mu(t)((u_{lm}(t), v)) + (|u_{lm}(t)|^\rho, v) \\ &+ \mu(t)\int_{\Gamma_1}\delta h(u'_{lm}(t))v d\Gamma =(f(t),v), \quad \forall v \in V_m^l, \end{aligned} \\ u_{lm}(0)=u_l^0, \quad u'_{lm}(0)=u_l^1, \end{gathered} \end{equation} where $V_m^l$ is the subspace generated by $w_1^l, w_2^l, \dots, w_m^l$. The above finite-dimensional system has a solution $u_{lm}$ defined in $[0, t_{lm})$. The following estimates allow us to extend this solution to the interval $[0, \infty)$ \subsection*{First Estimate} Set $v=u'_{lm}$ in \eqref{equacao3.10}$_1$. We have \begin{align*} &\frac{1}{2}\frac{d}{dt}|u_{lm}'(t)|^2 + \frac{1}{2}\frac{d}{dt} \big[ \mu(t) \|u_{lm}'(t)\|^2\big] +\frac{d}{dt}\int_{\Omega} G(u_{lm}(t))dx \\ &+ \mu(t) \int_{\Gamma_1}\delta h(u'_{lm}(t))u'_{lm}(t) d\Gamma \\ &=(f(t), u'_{lm}(t)) + \frac{1}{2}\mu'(t) \|u'_{lm}(t)\|^2. \end{align*} Integrating on $[0,t]$, $0< t < t_{lm}$, we obtain \begin{equation}\label{q3.4} \begin{aligned} &\frac{1}{2}|u_{lm}'(t)|^2 + \frac{\mu(t)}{2}\|u_{lm}'(t)\|^2 +\int_{\Omega} G(u_{lm}(t))dt \\ &+ \int_0^t \int_{\Gamma_1}\mu(t)h(u'_{lm}(\tau))u'_{lm}(\tau) d\Gamma d\tau \\ &=\int_0^t (f(\tau), u'_{lm}(\tau))d\tau + \frac{1}{2}\int_0^t\mu'(\tau) \|u'_{lm}(\tau)\|^2d \tau \\ &\quad + \frac{1}{2}|u_l^1|^2 + \frac{\mu(0)}{2}\|u^0_l\|^2 + \int_{\Omega}G(u^0_l)dx. \end{aligned} \end{equation} Using \eqref{q06}, it follows that \begin{gather*} \big| \int_{\Omega}G(u_{lm}(t))dx \big| \leq \frac{1}{\rho+1}k_0^{\rho+1}\|u_{lm}(t)\|^{\rho+1}, \\ \big| \int_{\Omega}G(u^0_l)dx \big|\leq \frac{1}{\rho+1}k_0^{\rho+1} \|u^0_l\|^{\rho+1}. \end{gather*} Taking into account the last two inequalities in \eqref{q3.4}, and using hypotheses \eqref{q4}$_2$ and the fact $h_l(s)s \geq d_0$, we find \begin{equation}\label{q3.5} \begin{aligned} &\frac{1}{2}|u_{lm}'(t)|^2 + \frac{\mu_0}{2}\|u_{lm}(t)\|^2- \frac{1}{\rho+1}k_0^{\rho+1}\|u_{lm}(t)\|^{\rho+1} \\ & \leq \frac{1}{2}|u_{lm}'(t)|^2 + \frac{\mu(t)}{2}\|u_{lm}(t)\|^2 + \int_{\Omega} G(u_{lm}(t))dx \\ &\quad + \mu_0d_0 \int_0^t \int_{\Gamma_1}[u'_{lm}(\tau)]^2 d\Gamma d\tau \\ &\leq \int_0^t |f(\tau)| |u'_{lm}(\tau)|d\tau + \frac{1}{2}\int_0^t|\mu'(\tau)| \|u'_{lm}(\tau)\|^2d \tau + N_{1l} \end{aligned} \end{equation} where \begin{equation}\label{q3.6} N_{l}= \frac{1}{2}|u_l^1|^2 +\frac{\mu(0)}{2}\|u^0\|^2 +\frac{1}{\rho+1}k_0^{\rho +1}\|u^0\|^{\rho +1}. \end{equation} Motivated by the expression $$ \frac{\mu_0}{2}\|u_{lm}(t)\|^2-\frac{1}{\rho+1}k_0^{\rho+1}\|u_{lm}(t)\|^{\rho+1} $$ we introduce the function \begin{equation}\label{q3.7} J(\lambda)= \frac{1}{4}\mu_0\lambda^2 - \frac{3}{2}\frac{k_0^{\rho +1}}{\rho+1}\lambda^{\rho +1},\quad \lambda \geq 0. \end{equation} That is, $$ J'(\lambda)=\frac{1}{2}\mu_0\lambda -\frac{3}{2}k_0^{\rho +1}\lambda^{\rho}. $$ We are interested in $\lambda \geq 0$ such that $J'(\lambda)\geq 0$, that is, \begin{equation}\label{q3.9} \frac{3}{2}k_0^{\rho +1}\lambda^{\rho-1} \leq \frac{1}{2}\mu_0 \end{equation} or \begin{equation}\label{q3.10} 0 \leq \lambda^{\rho-1} \leq \frac{\mu_0}{3k_0^{\rho+1}}. \end{equation} This inequality is equivalent to $0\leq \lambda \leq \lambda^{*}_1$, where $\lambda_1^{*}$ was defined in \eqref{q16}. Thus \begin{equation}\label{q3.11} J(\lambda) \geq 0\quad \text{for }\lambda \in [0, \lambda_1^{*}]. \end{equation} As consequence of \eqref{q3.11} and hypothesis \eqref{q16}$_1$, we obtain \begin{equation}\label{q3.12} \frac{\mu_0}{4}\|u_{lm}(t)\|^2- \frac{3}{2}\frac{k_0^{\rho+1}}{\rho+1}\|u_{lm}(t)\|^{\rho+1} \geq 0, \end{equation} for $\|u_{lm}(t)\|<\lambda_1^{*},~t\in [0, t_{lm})$. Inequality \eqref{q3.12} implies \begin{equation*} %\label{q3.13} \frac{1}{4}\mu_0\|u_{lm}(t)\|^2+ \frac{1}{2}\frac{k_0^{\rho+1}}{\rho+1}\|u_{lm}(t)\|^{\rho+1} \leq \frac{1}{2}\mu_0\|u_{lm}(t)\|^2- \frac{k_0^{\rho+1}}{\rho+1}\|u_{lm}(t)\|^{\rho+1}. \end{equation*} Taking into account this inequality and \eqref{q3.12}, we have \begin{equation}\label{q3.14} \begin{aligned} &\frac{1}{2}|u'_{lm}(t)|^2 + \frac{1}{4}\mu_0\|u_{lm}(t)\|^2+ \frac{1}{2}\frac{k_0^{\rho+1}}{\rho+1}\|u_{lm}(t)\|^{\rho+1} \\ &\leq \frac{1}{2}|u'_{lm}(t)|^2 + \frac{\mu(t)}{2}\|u_{lm}(t)\|^2 + \int_{\Omega} G(u_{lm}(t))dx \\ &\quad + \mu_0d_0 \int_0^t \int_{\Gamma_1}[u'_{lm}(\tau)]^2 d\Gamma d\tau \\ &\leq \int_0^t |f(\tau)||u'_{lm}(\tau)|d\tau + \frac{1}{2}\int_0^t |\mu'(\tau)|\|u_{lm}(\tau)\|^2d\tau + N_{l}. \end{aligned} \end{equation} Note that \begin{equation}\label{q3.15} N_{l} < N \quad \text{for all } l\geq l_0 \end{equation} where $N$ was introduced in \eqref{q17}. We set $$ \varphi(t)=|u'_{lm}(t)|^2 + \frac{1}{2}\mu_0\|u_{lm}(t)\|^2+ \frac{k_0^{\rho+1}}{\rho+1}\|u_{lm}(t)\|^{\rho+1}. $$ Then taking into account \eqref{q3.15} in \eqref{q3.14} and noting that $\frac{1}{\mu_1} \leq \frac{1}{\mu_0}$, we obtain $$ \varphi^2(t) \leq \frac{[(2N)^{1/2}]^2}{2} + \int_0^t |f(\tau)||\varphi(\tau)|d\tau + \int_0^t 2\frac{|\mu'(\tau)|}{\mu_0}\varphi^2(\tau)d\tau. $$ Then by Lemma \ref{lema1}, we obtain \begin{equation}\label{q3.16} \varphi(t) \leq \Big[ (2N)^{1/2}+ \int_0^{\infty} |f(t)|dt \Big] \exp\Big( \frac{2}{\mu_0}\int_0^{\infty}|\mu'(t)|dt \Big)=P. \end{equation} So \begin{equation}\label{q3.17} |u'_{lm}(t)| \leq P\quad \text{and}\quad \|u_{lm}(t)\|\leq \big( \frac{2}{\mu_0} \big)^{1/2}P \end{equation} for each $t\in [0, t_{lm})$ and $\|u_{lm}(t)\|<\lambda_1^{*}$. The following result ensures that inequalities \eqref{q3.17} hold for all $t\in [0, \infty)$. \begin{lemma}\label{lema3} Let $[0, t_{lm})$ be an interval of existence of the solution $u_{lm}(t)$ of \eqref{equacao3.10}. Then $$ \|u_{lm}(t)\|<\lambda_1^{*},\quad \forall t\in [0, \infty),\; \forall l \geq l_0,\; \forall m. $$ \end{lemma} \begin{proof} First, we note that by hypothesis \eqref{q16}, we have $$ \|u_{lm}(0)\|=\|u^0_l\|< \lambda_1^{*},\quad \forall l \geq l_0,\; \forall m. $$ Reasoning by contradiction, we assume that there exists $t_1 \in (0, t_{lm})$ such that $\|u_{lm}(t_1)\|=\lambda_1^{*}$. Let $$ t^{*}=\inf\{t_1 \in (0, t_{lm}): \|u_{lm}(t_1)\|=\lambda_1^{*}\}. $$ By the continuity of $\|u_{lm}(t)\|$, we obtain $\|u_{lm}(t^{*})\|=\lambda_1^{*}$. Note that $0< t^{*} < t_{lm}$. Consider $t\in [0, t^{*})$. Then $\|u_{lm}(t)\|<\lambda_1^{*}$. So inequality \eqref{q3.17} provides $$ \|u_{lm}(t)\| \leq \Big( \frac{2}{\mu_0} \Big)^{1/2}P,~\forall t \in [0,t^{*}) $$ that implies $$ \lambda_1^{*} = \|u_{lm}(t^{*})\|\leq \Big( \frac{2}{\mu_0} \Big)^{1/2}P $$ But this is a contradiction because by hypothesis \eqref{q16}$_2$, $\big( \frac{2}{\mu_0} \big)^{1/2}P< \lambda_1^{*}$. This concludes the proof. \end{proof} Lemma \ref{lema3} provides the estimates \begin{equation}\label{q3.18} |u'_{lm}(t)| \leq P,\quad \|u_{lm}(t)\|\leq \Big( \frac{2}{\mu_0} \Big)^{1/2}P, \quad \forall t \in [0,\infty),\; \forall l \geq l_0,\; \forall m. \end{equation} Also inequalities \eqref{q3.16}, \eqref{q3.18} and \eqref{q3.5} gives us \begin{equation}\label{q3.19} \int_0^{\infty}\|u'_{lm}(t)\|_{L^2(\Gamma_1)}dt\leq K, \quad \forall t \in [0,\infty),\; \forall l \geq l_0, \; \forall m. \end{equation} \subsection*{Second Estimate.} In this part, to facilitate the notation we do not write the variable $t$ and the subscripts $l$ and $m$. Differentiating with respect to $t$ equation \eqref{equacao3.10}$_1$ and then setting $w=u''$, we obtain \begin{align*} &\frac{1}{2}\frac{d}{dt}|u''|^2 + \frac{1}{2}\frac{d}{dt}[ \mu\|u'\|^2] + \mu'((u,u'')) + (\rho |u|^{\rho-2} uu',u'' ) \\ & + \mu \int_{\Gamma_1}\delta h'(u')[u'']^2d\Gamma + \mu' \int_{\Gamma_1}h(u')u''d\Gamma \\ &= (f', u'') + \frac{1}{2}\mu'\|u'\|^2. \end{align*} Considering $w=\frac{\mu'}{\mu}u''$ in approximate equation \eqref{equacao3.10}$_1$, we find $$ \mu'((u,u'') + \mu' \int_{\Gamma_1}h(u')u''d\Gamma = \big(f', \frac{\mu'}{\mu}u'' \big) -\big(u'',\frac{\mu'}{\mu}u'' \big) -\big(|u|^\rho,\frac{\mu'}{\mu}u'' \big). $$ Combining the last two equalities, we have \begin{equation}\label{q3.20} \begin{aligned} &\frac{1}{2}\frac{d}{dt}|u''|^2 + \frac{1}{2}\frac{d}{dt}\left[ \mu\|u'\|^2 \right] + \mu \int_{\Gamma_1}\delta h'(u)[u'']^2d\Gamma \\ &= (f', u'')+\frac{1}{2}\mu'\|u\|^2 -\big(f,\frac{\mu'}{\mu}u'' \big) + \big(u'',\frac{\mu'}{\mu}u'' \big) \\ &\quad + \big(|u|^{\rho},\frac{\mu'}{\mu}u'' \big) -(\rho |u|^{\rho-2} uu',u''). \end{aligned} \end{equation} Fix a real number $T>0$. We bound the last terms of the second member of \eqref{q3.20}. By $C=C(T)>0$ is denoted a generic constant which is independent of $l$ and $m$. By \eqref{q06}, \eqref{q12}$_1$ and estimate \eqref{q3.20}, we obtain \[ \big| \big(|u|^{\rho},\frac{\mu'}{\mu}u'' \big) \big| \leq k_2^{\rho}\|u\|^{\rho}\frac{|\mu'|}{\mu_0}|u''| \leq C\frac{|\mu'|}{\mu_0}|u''|. \] By \eqref{q12}$_2$, \eqref{q12}$_3$, estimates \eqref{q3.18} and noting that $\frac{1}{n} + \frac{1}{p^{*}}+ \frac{1}{2}=1$ ($p^{*}$ introduced in \eqref{q8}), we find \[ | (\rho |u|^{\rho-2} u u',u'' ) | \leq \rho k_3^{\rho-1}k_4\|u'\||u''| \leq C\|u'\||u''| \leq \frac{C}{2}\|u'\|^2 + \frac{C}{2}|u''|^2. \] Taking into account the last two inequalities \eqref{q3.20} and integrating on $[0,t]$, we obtain \begin{equation}\label{q3.21} \begin{aligned} &\frac{1}{2}|u_{lm}''(t)|^2 + \frac{1}{2} \mu(t)\|u_{lm}'(t)\|^2 + {\mu_0d_0} \int_0^t \int_{\Gamma_1}[u_{lm}''(\tau)]^2d\Gamma d\tau \\ &\leq \int_0^t\left[|f'(\tau)|+ \frac{|\mu'(\tau)|}{\mu_0}|f(\tau)| + \frac{C|\mu'(\tau)|}{\mu_0} \right]|u_{lm}''(\tau)| d\tau \\ &\quad + \int_0^t \frac{C}{2}|u_{lm}''(\tau)|^2d\tau + \int_0^t \frac{C}{2}\|u_{lm}'(\tau)\|^2d\tau \\ &\quad + \frac{1}{2}\int_0^t \frac{|\mu'(\tau)|}{\mu_0} \mu(\tau) \|u(\tau)\|^2d\tau + \frac{1}{2}|u''_{lm}(0)|^2 + \frac{\mu(0)}{2}\|u_l^1\|^2. \end{aligned} \end{equation} For this inequality provides an estimate, we need to bound $|u''_{lm}(0)|$. This is possible thanks to the choice of the special basis of $V \cap H^2(\Omega)$ and \eqref{equacao3.9}$_3$. We bound $|u''_{lm}(0)|$. Set $t=0$ in approximate equation \eqref{equacao3.10}$_1$ and then take $v=u''_{lm}(0)$. The Gauss theorem and \eqref{equacao3.9}$_3$ gives us $$ |u''_{lm}(0)|^2 + \mu(0)(-\Delta u^0_l, u''_{lm}(0)) + (|u^0_l|^\rho, u''_{lm}(0))= (f(0), u''_{lm}(0)). $$ This equality and \eqref{equacao3.9} gives us $$ |u''_{lm}(0)|^2 \leq K_1. $$ Taking into account this inequality in \eqref{q3.21} and using Lemma \ref{lema3}, follows that \begin{equation}\label{q3.22} \begin{gathered} \|u_{lm}'(t)\| \leq C,\quad \forall t \in [0,T],\; \forall l \geq l_0,\;\forall m \\ |u_{lm}''(t)| \leq C,\quad \forall t \in [0,T],\; \forall l \geq l_0,\; \forall m \\ \int_0^{t}\|u_{lm}''(t)\|_{L^2(\Gamma_1)} \leq C,\quad \forall t \in [0,T],\; \forall l \geq l_0,\; \forall m \end{gathered} \end{equation} \subsection*{Passage to the Limit in $m$.} Estimates \eqref{q3.18}, \eqref{q3.19}, \eqref{q3.22} and diagonal process allows to find a function $u_k$ and a subsequence of $(u_{lm})$, still denoted by $(u_{lm})$, such that \begin{equation}\label{q3.23} \begin{gathered} u_{lm} \to u_l \quad\text{weak star in }L^{\infty}(0, \infty, V); \\ u'_{lm} \to u_l' \quad\text{weak star in }L^{\infty}(0, \infty, L^2(\Omega)) \cap L^{\infty}_{\rm loc}(0, \infty, V); \\ u''_{lm} \to u_l''\quad\text{weak star in }L^{\infty}_{\rm loc}(0, \infty, L^2(\Omega)); \\ u'_{lm} \to u_l' \quad\text{weak star in }L^{\infty}(0, \infty, L^2(\Gamma_1)); \\ u''_{lm} \to u_l''\quad\text{weak star in }L^{\infty}_{\rm loc}(0, \infty, L^2(\Gamma_1)). \\ \end{gathered} \end{equation} Estimates \eqref{q3.23}$_1$, \eqref{q3.23}$_2$ and Aubin-Lions Theorem provides us $$ u_{lm}(x,t) \to u_l(x,t)\quad\text{a.e. in } Q=\Omega \times (0,T). $$ Then \begin{equation}\label{q3.24} |u_{lm}(x,t)|^\rho \to |u_{l}(x,t)|^\rho\quad\text{a.e. in } Q=\Omega \times (0,T). \end{equation} By \eqref{q06}, \eqref{q12}$_2$ and \eqref{q3.18}, we find \begin{equation}\label{q3.25} \int_{\Omega}|u_{lm}|^{2\rho} dx \leq k_2^{2\rho}\|u_{lm}\|^{2\rho} \leq C. \end{equation} Expressions \eqref{q3.24}, \eqref{q3.25}, Lions Lema \cite{Lions} and diagonal process provide \begin{equation}\label{q3.26} |u_{lm}|^\rho~\to |u_{l}|^\rho \quad \text{weak star in } L^{\infty}_{\rm loc}(0, \infty; L^2(\Omega)). \end{equation} Estimate \eqref{q3.23}$_3$ yields $$ u'_{lm}\to u'_{l} \quad \text{weak star in }L^{\infty}(0, \infty; H^{1/2}(\Gamma_1)). $$ This, convergence \eqref{q3.23}$_5$ and Aubin-Lions Theorem and fact $h$ Lipchitizian function gives us $$ h(u'_{lm}(x,t))\to h(u'_{l}(x,t)) \quad\text{a.e. in }Q $$ and by trace theorem and \eqref{q3.23}, we obtain \[ (h(u'_{lm})) ~\text{bounded in}~L^{\infty}_{\rm loc}(0, \infty; H^{1/2}(\Gamma_1)). \] Therefore, by Lions Lemma, we conclude that \begin{equation}\label{q3.27} h(u'_{lm})\to h(u'_{l}) ~~\text{weak star in}~L^{\infty}_{\rm loc} (0, \infty; H^{1/2}(\Gamma_1)). \end{equation} Convergences \eqref{q3.23}, \eqref{q3.26}-\eqref{q3.27} allows us to pass to the limit in approximate equation \eqref{equacao3.10}$_1$. Then by density of $V\cap H^2(\Omega) $ in $V$, we obtain \begin{equation}\label{q3.29} \begin{aligned} &\int_0^{\infty}(u_{l}''(t), v)\theta(t)dt + \mu \int_0^{\infty}((u_{l}(t), v))\theta(t)dt +\int_0^{\infty}(|u_{l}(t)|^\rho, v )\theta(t)dt \\ &+ \int_0^{\infty}\int_{\Gamma_1} \mu(t)\delta h(u'_{l}(t))v\theta(t) d\Gamma dt\\ & =\int_0^{\infty}(f(t), v)\theta(t)dt, \quad v \in V,\; \forall \theta \in C_0^{\infty}(\Omega). \end{aligned} \end{equation} Taking $v \in {\mathcal{D}}(\Omega)$ in \eqref{q3.29}, and observing the regularities of $u_{l}'', |u_{l}|^\rho$ and $f$, follows that \begin{equation}\label{q3.30} u_{l}''- \mu \Delta u_{l} + |u_{l}|^\rho=f\quad \text{in }L^2_{\rm loc}(0, \infty; L^2(\Omega)). \end{equation} This equation provides $\Delta u_{l} \in L^{\infty}(0, \infty; L^2(\Omega))$ and \eqref{q3.23}$_1$, $u_{l} \in L^{\infty}(0, \infty; V)$. Then \begin{equation}\label{q3.31} \frac{\partial u_{l}}{\partial \nu} \in L^{\infty}_{\rm loc} (0, \infty; H^{1/2}(\Gamma_1)). \end{equation} Multiply both sides of \eqref{q3.30} by $v \theta, v\in V$ and $\theta \in C_0^{\infty}(0,\infty)$, and integrate on $\Omega \times (0, \infty)$. Using regularity \eqref{q3.31} of $\frac{\partial u}{\partial \nu}$, we conclude \begin{align*} &\int_0^{\infty}(u_{l}''(t), v)\theta(t)dt + \mu \int_0^{\infty}((u_{l}(t), v))\theta(t)dt -\int_0^{\infty}\mu(t)\langle \frac{\partial u_{l}}{\partial \nu}, v \rangle \theta(t)dt \\ &+ \int_0^{\infty}(|u_{l}(t)|^\rho, v )\theta(t)dt \\ &=\int_0^{\infty}(f(t), v)\theta(t)dt, \quad v \in V,\; \forall \theta \in C_0^{\infty}(\Omega). \end{align*} where $\langle \cdot, \cdot \rangle$ denotes the duality paring between $H^{-\frac{1}{2}}(\Gamma_1)$ and $H^{1/2}(\Gamma_1)$. Comparing this equality with \eqref{q3.29} and observing the regularity of $h(u'_l)$, we find (see \cite{milla02}) \begin{equation}\label{q3.32} \frac{\partial u_{l}}{\partial \nu} +\delta h(u'_{l})=0 \quad\text{in } L^{2}_{\rm loc}(0, \infty; H^{1/2}(\Gamma_1)). \end{equation} \subsection*{Passage to the Limit in $l$} Estimates \eqref{q3.18}, \eqref{q3.19}, \eqref{q3.22} and convergence \eqref{q3.23} provide \begin{equation}\label{q3.33} \begin{gathered} |u_{l}'(t)| \leq P,\|u_{l}(t)\|\leq \Big( \frac{2}{\mu_0} \Big)^{1/2} \quad \forall t \in [0,\infty),\; \forall l \geq l_0, \\ \int_0^{\infty}\|u_{l}''(t)\|_{L^2(\Gamma_1)}^2dt \leq C,\quad \forall t \in [0,\infty),\; \forall l \geq l_0; \\ \|u_{l}'(t)\| \leq C,\quad |u_{l}''(t)| \leq C\quad \forall t \in [0,T],]; \forall l \geq l_0, \\ \int_0^{t}\|u_{l}''(\tau)\|_{L^2(\Gamma_1)}^2d\tau \leq C,\quad \forall t \in [0,T],\; \forall l \geq l_0. \end{gathered} \end{equation} These estimates allows to obtain similar convergence to those obtained in \eqref{q3.23}. So there exists a function $u$ and subsequence of $(u_{l})$, still denoted by $(u_{l})$, such that \begin{equation}\label{q3.34} \begin{gathered} u_{l} \to u \quad\text{weak star in }L^{\infty}(0, \infty, V); \\ u'_{l} \to u' \quad\text{weak star in }L^{\infty}(0, \infty, L^2(\Omega)) \cap L_{\rm loc}^{\infty}(0, \infty; V); \\ u''_{l} \to u'' \quad\text{weak star in }L^{\infty}_{\rm loc}(0, \infty, L^2(\Omega)); \\ u'_{l} \to u' \quad\text{weak in } L^{2}(0, \infty, L^2(\Gamma_1)); \\ u''_{l} \to u'' \quad\text{weak in } L^{2}_{\rm loc}(0, \infty, L^2(\Gamma_1)). \end{gathered} \end{equation} By arguments similar to those used for \eqref{q3.26}, we find \begin{equation} \label{0q3.34} |u_{l}|^{\rho} \to |u|^{\rho} \quad \text{weak in } L^2_{\rm loc}(0, \infty; L^2(\Omega)). \end{equation} This convergence, \eqref{q3.34}$_3$ and \eqref{q3.30} provide \begin{equation}\label{q3.35} \Delta u_{l} \to \Delta u\quad \text{weak in } L^2_{\rm loc}(0, \infty; L^2(\Omega)) \end{equation} and therefore \begin{equation}\label{q3.36} u'' -\mu\Delta u + |u|^{\rho}=f\quad \text{in } L^2_{\rm loc} (0, \infty; L^2(\Omega)). \end{equation} Also convergences \eqref{q3.34}$_1$ and \eqref{q3.35} provide us with \begin{equation}\label{q3.37} \frac{\partial u_{l}}{\partial \nu}\to \frac{\partial u}{\partial \nu}\quad \text{weak in } L^2_{\rm loc}(0, \infty; H^{-\frac{1}{2}}(\Gamma_1)). \end{equation} As done in \eqref{q3.27}, we find \begin{equation}\label{0q3.27} \delta h(u'_{l})\to \delta h(u') \quad\text{weak star in } L^{\infty}_{\rm loc}(0, \infty; H^{1/2}(\Gamma_1)). \end{equation} So these two convergences and \eqref{q3.32}, we met \begin{equation}\label{0q3.32} \frac{\partial u}{\partial \nu} + \delta h(u')=0 \quad\text{in } L^{2}_{\rm loc}(0, \infty; H^{1/2}(\Gamma_1)). \end{equation} From the regularity $$ u \in L_{\rm loc}^{\infty}(0, \infty; V),\quad \Delta u \in L_{\rm loc}^{\infty}(0, \infty; L^2(\Omega)), \quad \frac{\partial u}{\partial \nu} \in L_{\rm loc}^{\infty} (0, \infty; H^{1/2}(\Gamma_1)) $$ and by Proposition \ref{pr3.1}, we obtain \begin{equation}\label{equacao3.29} u \in L^{\infty}_{\rm loc}(0,\infty; V \cap H^{2}(\Omega)). \end{equation} Also, by estimate \eqref{q3.34}$_4$ and noting that $h$ is a Lipschitz continuous function we find \begin{equation}\label{equacao3.300} \frac{\partial u'}{\partial \nu}+ \delta h'(u')u''=0 \quad \text{in } L^{2}_{\rm loc}(0,\infty; L^{2}(\Gamma_1)). \end{equation} The verification of initial conditions follows in the usual way. In what follows, we prove the uniqueness of solutions. Let $u$ and $v$ two functions in class \eqref{q18} which satisfy equations \eqref{q19}, \eqref{q20} and initial conditions \eqref{q21}. Consider $w=u-v$. Then \begin{equation}\label{E3.28} \begin{gathered} w'' -\mu \Delta w + |u|^\rho-|v|^\rho= 0\quad \text{in } L^{\infty}(0,T; L^2(\Omega)), \\ \frac{\partial w}{\partial \nu} + \delta [h(u')-h(v')]=0 \quad \text{in } L^{\infty}(0,T; H^{1/2}(\Gamma_1)), \\ w(0)=0,\quad w'(0)=0 \end{gathered} \end{equation} Multiplying both sides of \eqref{E3.28}$_1$ by $w'$ integrating on $\Omega$ and using Gauss Theorem, we obtain \begin{equation}\label{E3.29} \begin{aligned} &\frac{1}{2}\frac{d}{dt}|w'(t)|^2+ \frac{1}{2}\|w(t)\|^2 + \int_{\Gamma_1}\delta[h(u'(t))-h(v'(t))]d\Gamma \\ &= -(|u(t)|^\rho- |v(t)|^\rho,w'(t)). \end{aligned} \end{equation} We have $$ |u(x,t)|^\rho -|v(x,t)|^\rho =\rho|\xi|^{\rho-2}\xi w(x,t) $$ where $\xi$ is between $u(x,t)$ and $v(x,t)$. Then $$ ||u(x,t)|^\rho -|v(x,t)|^\rho| = \rho |\xi|^{\rho-1}| w(x,t)| $$ that provides \begin{equation}\label{E3.30} \begin{aligned} ||u(t)|^\rho- |v(t)|^\rho| &\leq \rho [|u(x,t)| + |v(x,t)|]^{\rho-1}|w(x,t)| \\ &\leq C(\rho)[|u(x,t)|^{\rho-1}|w(x,t)| + |v(x,t)|^{\rho-1}|w(x,t)|]. \end{aligned} \end{equation} We obtain \begin{align*} \int_{\Omega}|u(x,t)|^{\rho-1}|w(x,t)||w'(x,t)|dx &\leq \|u(t)\|_{L^{(\rho-1)n}(\Omega)}^{\rho-1}\|w(t)\|_{L^{p^{*}}(\Omega)}|w'(t)| \\ &\leq k_3k_4\|u(t)\|^{\rho-1}\|w(t)\||w'(t)|. \end{align*} Thus $$ |(|u(t)|^\rho- |v(t)|^\rho, w'(t))| \leq C\|w(t)\||w'(t)| \leq \frac{C}{2}\|w(t)\|^2 + \frac{C}{2}|w'(t)|^2. $$ This inequality, \eqref{E3.29} and property of monotony of $h$, imply $$ \frac{1}{2}\frac{d}{dt}|w'(t)|^2 + \frac{1}{2}\frac{d}{dt}\|w(t)\|^2 + \delta_0d_0\int_{\Gamma_1}w'(t)^2d\Gamma \leq \frac{C}{2}\|w(t)\|^2 + \frac{C}{2}|w'(t)|. $$ Then the Gronwall inequality provides $w'(t)=0$ and $w(t)=0$. This concludes the proof of Theorem \ref{teorema1}. \subsection{Proof of Theorem \ref{teo2.1}} We introduce some notation to apply the Banach Fixed-Point Theorem. Consider a real number $R>0$ such that \begin{equation}\label{3.30} R> M_0 \end{equation} where $M_0=\max\{M_1,M_2\}$ is defined in \eqref{local5}, $M_1$, $M_2$ are defined by \eqref{local1} and \eqref{local3} respectively. Let \begin{gather} R_1^2=N_1^2 =|u^1|^2+ M(0,\|u^0\|^{2})\|u^0\|^2 + \frac{1}{\rho+1}k_0\|u_0\|^{\rho +1}, \label{3.31} \\ R_2^2=M(0,\|u^0\|^{2})\|u^1\|^2 + M(0,\|u^0\|^{2})|\Delta u^0| +|u^0|^\rho + |f(0)|. \label{3.32} \end{gather} We define $B_{R,T_0}$ as the set of vectors \begin{align*} B_{R,T_0}=\Big\{& {u}: {u} \in L^{\infty}(0, T_0; V), \; {u'} \in L^{\infty}(0, T_0; V)\cap C^0([0, T_0]; L^2(\Omega)), \\ &\|u\|_{L^{\infty}(0, T_0; V)}+ \|u'\|_{L^{\infty}(0, T_0; V)}\leq R, \\ &u(0)=u^0,\; u'(0)=u^1.\Big\} \end{align*} The real number $T_0$ with $0< T_0 \leq 1$ will be determined later. We equipped $B_{R,T_0} $ with the metric $$ d(u,v)=\|u-v\|_{L^{\infty}(0,T_0;V)}+\|u'-v'\|_{C^{0}([0,T_0]; L^2(\Omega))} $$ where $u$ and $v$ belong to $B_{R,T_0}$. In \cite{Jutuca} is proved that $(B_{R,T_0}, d(u,v))$ is a complete metric space. Consider the map $S: B_{R,T_0}\to {\mathcal{H}},z\mapsto S(z)=\varphi$, where $\mathcal{H}$ denotes the set of solutions $\varphi$, of the problem \begin{equation}\label{3.34} \begin{gathered} \varphi'' - M(\cdot,\|z\|^{2})\Delta \varphi + |\varphi|^\rho=f \quad \text{in }\Omega \times (0, T_0) \\ \varphi=0 \quad \text{on } \Gamma_0 \times (0, T_0) \\ \frac{\partial \varphi}{\partial \nu} + \delta h(\varphi')=0 \quad \text{on } \Gamma_1 \times (0, T_0) \\ \varphi(0)=u^0,\quad \varphi'(0)=u^1 \quad \text{in } \Omega \end{gathered} \end{equation} We prove that the map $S$ is well defined. Set \begin{equation}\label{3.35} K=\max\Big\{\big|\frac{\partial M}{\partial t}(t, \lambda) \big|,\big|\frac{\partial M}{\partial \lambda}(t, \lambda) \big|; t \in[0,1],\; \lambda \in [0,R^2]\Big\}. \end{equation} Consider \begin{equation}\label{3.36} \mu(t)=M(t, \|z(t)\|^2),\quad t \in [0, T_0]. \end{equation} We have that $\mu \in W^{1,\infty}(0,T_0)$. In fact, \[ \mu'(t)=\frac{\partial M}{\partial t}(t,\|z(t)\|^{2}) + \frac{\partial M}{\partial\lambda}(t,\|z(t)\|^{2})\frac{d}{dt}\|z(t)\|^2. \] As $z\in B_{R,T_0}$, we find that \begin{equation}\label{3.37} |\mu'(t)|\leq K(1 + 4R^{2}),\quad\text{a.e. }t \in ]0,T_0[. \end{equation} Thus, $\mu \in W^{1,\infty}(0,T_0)$ with $\mu_0 =m_0$. Theorem \ref{teorema1} says that there exists a unique solution ${\varphi}$ of system \eqref{3.34} and this solution has the regularity of the vectors of $B_{R, T_0}$. Our objective now is to show that $S(B_{R,T_0})$ is contained $B_{R,T_0}$ and that $S$ is a strict contraction. Let ${\varphi}$ be a solution of the problem \eqref{3.34} given by the Theorem \ref{teorema1} with $\mu(t)$ defined in \eqref{3.36}. Let $\varphi_{lm}$ be the approximate solution given in the proof of Theorem \ref{teorema1}. Then by first a priori estimate given the proof of Theorem \ref{teorema1}, we obtain $$ \|\varphi_{lm}(t)\|^2\leq M_1\exp\Big( \frac{2}{m_0}\int_0^t|\mu'(\tau)|d\tau \Big),\quad 0\leq t \leq T_0, $$ where \begin{equation}\label{local1} M_1=(2R_1)^{1/2} + \int_0^{T_0}|f(t)|dt. \end{equation} This and \eqref{3.37} gives \begin{equation}\label{3.38} \|\varphi_{lm}(t)\|\leq M_1 \exp({\mathcal{K}}_1T_0),\quad 0\leq t \leq T_0, \text{ for } m \geq 2 \text{ and } l\geq l_0(1). \end{equation} where \begin{equation}\label{local2} \mathcal{K}_1=\frac{2K(1+ R^2)}{m_0}. \end{equation} The second priori estimates Theorem \ref{teo2.1} gives us \begin{equation}\label{3.40} \|\varphi'_{lm}(t)\|\leq M_2\exp\left({\mathcal{K}}_2T_0 \right),\quad 0\leq t \leq T_0,\text{ for } m \geq 2 \text{ and } l \geq l_0(1). \end{equation} where \begin{equation}\label{local3} \begin{aligned} M_2&=2R_2^{1/2} + \int_0^{T_0}\left[|f'(t)| + \frac{|\mu'(t)|}{m_0}|f(t)| + \frac{C}{m_0}|\mu'(t)|\right]dt \\ &\leq 2R_2^{1/2} + \int_0^{T_0}\big[|f'(t)| + \frac{K(1 + 4R^{2})}{m_0}|f(t)| + \frac{C}{m_0}K(1 + 4R^{2})\big]dt \end{aligned} \end{equation} and \begin{equation}\label{local4} \mathcal{K}_2=\frac{(2+m_0)K(1 + 4R^{2})}{2m_0} + \frac{3C}{2}. \end{equation} Consider \begin{equation}\label{local5} M_0=\max\{M_1, M_2\}, \quad \mathcal{K}=\max\{\mathcal{K}_1, \mathcal{K}_2\}. \end{equation} From \eqref{3.38}, \eqref{3.40} and \eqref{local5} and taking the maximum on $[0,T_0]$ of both of members the \eqref{3.38} and \eqref{3.40} and then the limit inferior, first with respect to $m $ and later with respect to $l$, we obtain \begin{equation}\label{3.41} \begin{array}{l} \|\varphi\|_{L^{\infty}(0,T_0; V)}+ \|\varphi'\|_{L^{\infty}(0,T_0; V)} \leq M_0exp({\mathcal{K}}T_0). \end{array} \end{equation} We will choose $T_0>0$ so that the second member of the preceding inequality be less than or equal to $R$. In fact, set $$ q(t)=M_0e^{{\mathcal{K}}t}, \quad t\geq 0. $$ Then $q$ is continuous, increasing, $q(t) \to \infty$ when $t\to \infty$ and $q(0)=M_00$ such that $q(T_1^{*})=R$, that is, \begin{equation}\label{3.42} T_1^{*}=\frac{1}{\mathcal{{K}}}\ln\Big(\frac{R}{M_0} \Big). \end{equation} We choose \begin{equation}\label{3.43} 0< T_0 \leq \min\{1, T_1^{*} \}. \end{equation} Then expression \eqref{3.41} with $T_0$ given by \eqref{3.43} satisfies $$ \|\varphi\|_{L^{\infty}(0,T_0; V)}+ \|\varphi'\|_{L^{\infty}(0,T_0; V)} \leq R. $$ Therefore $\varphi$ belongs to $B_{R,T_0}$. Thus $S(B_{R,T_0})$ is contained in $ B_{R,T_0}$. In the sequel we prove that $S$ is a strict contraction. Set $r_1,y_1\in B_{R,T_0}$ and $S(r_1)=r$, $S(y_1)={y}$. Introduce the notation \begin{equation}\label{3.44} \varphi = r-y. \end{equation} We have \begin{equation}\label{3.45} \begin{gathered} \varphi''-M(\cdot,\|r_1\|^2) \Delta r + M(\cdot,\|y_1 \|^2)\Delta y + |r|^\rho-|y|^\rho=0 \quad\text{in } \Omega \times ]0,T_0[, \\ \varphi=0, \quad \psi =0\quad \text{on } \Gamma_0 \times ]0, T_0[, \\ \frac{\partial \varphi}{\partial \nu} + \delta [h(r')-h(y')]=0 \quad \text{on }\Gamma_1 \times ]0, T_0[, \\ \varphi(0)=0,\quad \varphi'(0)=0 \quad \text{in }\Omega. \end{gathered} \end{equation} Taking the scalar product in $L^2(\Omega)$ of \eqref{3.45}$_1$ with $\varphi'(t)$ we obtain \begin{equation}\label{3.46} \begin{aligned} &\frac{1}{2}\frac{d}{d}|\varphi'(t)|^2-M(t,\|r_1(t)\|^2)(\Delta r(t),\varphi'(t))\\ & + M(t,\|y_1(t)\|^2)(\Delta y(t),\varphi'(t)) + (|r|^\rho-|y|^\rho,\varphi'(t))=0. \end{aligned} \end{equation} We modify \eqref{3.46}, to obtain \begin{align*} &\frac{1}{2}\frac{d}{d}|\varphi'(t)|^2-M(t,\|r_1(t)\|^2) (\Delta \varphi(t),\varphi'(t)) \\ &= [ M(t,\|r_1(t)\|^2)-M(t,\|y_1(t)\|^2 )](\Delta y(t),\varphi'(t)) - (|y|^\rho-|r|^\rho, \varphi'(t)). \end{align*} We abbreviate the notation and write this expression in the form \begin{equation}\label{3.47} \frac{1}{2}\frac{d}{dt}|\varphi'(t)|^2 + A(t)=B(t). \end{equation} \smallskip \noindent$\bullet$ Analysis of $A(t)$. Using the Green's Theorem and the boundary condition in \eqref{3.45}$_3$, we find that \begin{align*} A(t)&=M(t,\|r_1(t)\|^2)\frac{1}{2}\frac{d}{dt}\|\varphi(t)\|^2 \\ &\quad + M(t,\|r_1(t)\|^2)\int_{\Gamma_1} \delta [h(r'(t))-h(y'(t))] \varphi'(t)d\Gamma. \end{align*} Note that, $\delta(x)\geq \delta_0>0$ and $\varphi'(t)=r'(t)-y'(t)$ then by the strong monotonicity of $h$, follows that $$ \int_{\Gamma_1} \delta [h(r'(t))-h(y'(t))]\varphi'(t)d\Gamma \geq 0. $$ Combining the last two expressions we conclude that \begin{equation}\label{3.48} A(t)\geq M(t,\|r_1(t)\|^2)\frac{1}{2}\frac{d}{dt}\|\varphi(t)\|^2~a.e.~t \in ]0, T_0[. \\ \end{equation} \smallskip \noindent$\bullet$ Analysis of $B(t)$. To facilitate the notation in this part we do not write the variable $t$. We have \begin{equation}\label{3.49} B= \left[ M(\cdot,\|r_1\|^2)-( M(\cdot ,\|y_1\|^2)\right] (\Delta y(t),\varphi'(t))-(|y|^\rho-|r|^\rho, \varphi'(t)). \end{equation} \begin{itemize} \item As $M \in C^1$ we have $$ \big|M(\cdot, \|r_1\|^2)-M(\cdot, \|y_1\|^2)\big|\leq 2{K}M_0\|r_1-y_1\| , $$ where ${K}$ and $M_0$ were defined in \eqref{3.35} and \eqref{local5}, respectively. \item Analysis of $(|y(t)|^\rho-|r(t)|^\rho, \varphi'(t))$. We have $$ |y(x,t)|^\rho -|r(x,t)|^\rho =\rho|\xi|^{\rho-2}\xi \varphi(x,t) $$ where $\xi$ is between $y(x,t)$ and $r(x,t)$. Then $$ ||y(x,t)|^\rho -|r(x,t)|^\rho| \leq \rho |\xi|^{\rho-1}| \varphi(x,t)| $$ which implies $$ ||y(x,t)|^\rho-|r(x,t)|^\rho|\leq C [|y(x,t)|^{\rho-1}+ |r(x,t)|^{\rho-1}]|\varphi(x,t)|. $$ Thus $$ | (|y(t)|^\rho -|r(t)|^\rho,\varphi'(t)) |\leq C \|y(t)\|_{L^{(\rho-1)n}(\Omega)}^{\rho-1}\|\varphi(t)\|_{L^{p^{*}}(\Omega)} |\varphi'(t)|. $$ By \eqref{q12}, we find that \begin{gather*} \|y(t)\|_{L^{(\rho-1)n}(\Omega)}^{\rho-1} \leq k_3^{\rho-1} \|y(t)\|^{\rho-1}\leq C,\quad \forall t \in [0,T_0],\\ \|r(t)\|_{L^{(\rho-1)n}(\Omega)}^{\rho-1} \leq C,~\forall t \in [0,T_0]. \end{gather*} Combining the last tree inequalities, we obtain $$ |(|y(t)|^\rho- |r(t)|^\rho, \varphi'(t))| \leq C\|\varphi(t)\||\varphi'(t)|\leq \frac{C}{2}\|\varphi(t)\|^2 + \frac{C}{2}|\varphi'(t)|^2. $$ \end{itemize} Taking into account the last two inequalities in \eqref{3.49}, we obtain \begin{equation}\label{3.50} |B(t)| \leq C|\Delta y(t)| |\varphi'(t)|d(r_1,y_1) + \frac{C}{2}\|\varphi(t)\|^2 + \frac{C}{2}|\varphi'(t)|^2. \end{equation} Next we find a bound for $|\Delta y(t)|$. We have $$ \varphi'' - M(\cdot, \|z\|^2)\Delta \varphi + |\varphi|^\rho=f \quad \text{in } L^{\infty}(0, T_0; L^2(\Omega)). $$ By estimates \eqref{3.38}, \eqref{3.40} and following the same reasoning used for \eqref{3.40}, we obtain \begin{equation}\label{03.50} |y''(t)|\leq M_0\exp({{\mathcal{K}}}T_0)\quad \text{a.e. }t \in ]0, T_0[. \end{equation} Hence, \begin{equation}\label{local6} \begin{aligned} |M(t,\|z(t))\|| |\Delta \varphi (t)| &\leq |f(t)| + |u(t)|^\rho + |\varphi'(t)| \\ &\leq \Big(\frac{C_1 + C_2}{m_0} \Big) + \frac{M_0}{m_0}\exp(\mathcal{K}T_0). \end{aligned} \end{equation} These last two expressions give \begin{equation}\label{3.51} |\Delta y(t)| \leq M_3+ M_3exp({{\mathcal{K}}}T_0)\quad \text{a.e. }t \in ]0, T_0[, \end{equation} where $$ M_3=\max\big\{\frac{C_1 + C_2}{m_0}, \frac{M_0}{m_0} \big\}. $$ Note that $e^{\mathcal{K}T_0}>1$, therefore $M_3 \leq M_3e^{\mathcal{K}T_0}$. Hence Combining \eqref{3.50}$ and \eqref{3.51}$ we derive \begin{equation}\label{3.52} |B(t)|\leq P_0[exp({\mathcal{K}}T_0)]|\varphi'(t)|d(r_1,y_1) \quad \text{a.e. }t \in ]0, T_0[ \end{equation} where \begin{equation}\label{3.54} P_0=4 {K} M_0 M_3. \end{equation} Combining \eqref{3.48} and \eqref{3.52} with \eqref{3.47}, we obtain \begin{equation}\label{3.53} \begin{aligned} &\frac{1}{2}\frac{d}{dt}|\varphi'(t)|^2 + M(t, \|r_1(t)\|^2) \frac{1}{2}\frac{d}{dt}\|\varphi(t)\|^2 \\ &\leq P_0 [exp({{\mathcal{K}}}T_0)]^2d^2(r_1,y_1) + |\varphi'(t)|^2 \quad \text{a.e } t \in ]0, T_0[. \end{aligned} \end{equation} We have \begin{align*} &M(\cdot ,\|r_1\|^2)\frac{1}{2}\frac{d}{dt}\|\varphi\|^2 &=\frac{1}{2}\frac{d}{dt} [M(\cdot ,\|r_1\|^2)\|\varphi(t)\|^2 ] \\ &\quad -\frac{1}{2} \Big[\frac{\partial M}{\partial t}(\cdot ,\|r_1\|^2) + \frac{\partial M}{\partial \lambda}(\cdot ,\|r_1\|^2)\frac{d}{dt}\|r_1\|^{2}\Big]\|\varphi\|^2. \end{align*} Substituting this equality in \eqref{3.53}, and using boundedness \eqref{3.35} and \eqref{3.32}, we find \begin{align*} &\frac{1}{2}\frac{d}{d}\left[|\varphi'(t)|^2 + M(t,\|r_1(t)\|^2) \|\varphi(t)\|^2 \right] \\ &\leq \frac{{K}(1+2R^2)}{2}\|\varphi(t)\|^2 + P_0^2[\exp({{\mathcal{K}}}T_0)]^2 d^2(r_1,y_1) + |\varphi'(t)|^2 \quad\text{a.e. }t \in ]0, T_0[. \end{align*} Integrating on $[0, t]$, $00$ such that $\psi(T_2^{*})<1$. Take $$ T_0=\min\{1, T_1^{*}, T_2^{*}\}>0, $$ where $T_1^{*}$ was defined in \eqref{3.42}. Then $T_0$ satisfies \eqref{3.43} and $$ 2b_1T_0\exp(b_2 T_0)=\alpha_0<1. $$ Substituting this constant in \eqref{3.59}, we conclude that $$ d(S(r_1),S(y_1)) \leq \alpha_0 d(r_1,y_1),\quad \forall r_1, y_1 \in B_{R, T_0}. $$ Thus $d$ is a strict contraction. By the Banach Fixed-Point Theorem there exists a unique point $u\in B_{R, T_0}$ such that $S(u)=u$. This fixed point satisfies all conditions required in the theorem. The uniqueness of solutions follows as in \cite{Jutuca}. The existence of global solutions to problem \eqref{2.3} and their asymptotic behavior with small data will be published in a future article. \begin{thebibliography}{00} \bibitem{Brezis} H. Brezis; \emph{Op\'erateurs Maximaux Monotones et Semi-groupes de Contraction dans les Espaces de Hilbert}, Nort-Holland, Amsterdam, 1973. \bibitem{cazenave} H. Brezis, T. 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