\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 112, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/112\hfil Propagation of perturbations] {Propagation of perturbations for a sixth-order thin film equation} \author[Z. Li, C. Liu\hfil EJDE-2012/112\hfilneg] {Zhenbang Li, Changchun Liu} % in alphabetical order \address{Zhenbang Li \newline Department of Mathematics, Jilin University, Changchun 130012, China} \email{jamesbom23@yahoo.com.cn} \address{Changchun Liu \newline Department of Mathematics, Jilin University, Changchun 130012, China} \email{liucc@jlu.edu.cn} \thanks{Submitted January 31, 2012. Published July 3, 2012.} \thanks{Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, \hfill\break\indent State Education Ministry} \thanks{Changchun Liu is the corresponding author} \subjclass[2000]{35D05, 35G25, 35Q99, 76A20} \keywords{Sixth-order thin film equation; radial solution; existence; \hfill\break\indent finite speed of propagation} \begin{abstract} We consider an initial-boundary problem for a sixth-order thin film equation, which arises in the industrial application of the isolation oxidation of silicon. Relying on some necessary uniform estimates of the approximate solutions, we prove the existence of radial symmetric solutions to this problem in the two-dimensional space. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} This article is devoted to the radial symmetric solutions for a sixth-order thin film equation $$\frac{\partial u}{\partial t}=\operatorname{div}[|u|^n\nabla\Delta^2 u], \quad x\in B,$$ with the boundary value conditions $$\frac{\partial u}{\partial\boldsymbol{\nu}}\big|_{\partial B}= \frac {\partial\Delta u}{\partial\boldsymbol{\nu}}\big|_{\partial B} =\frac {\partial\Delta^2 u}{\partial\boldsymbol{\nu}}\big|_{\partial B}=0,$$ and the initial value condition $$u\big|_{t=0}=u_0(x),$$ where $B$ is the unit ball in ${\mathbb R}^2$, $n>0$ is a constant, and $\boldsymbol{\nu}$ is the outward unit normal to $\partial B$. The equation is a typical higher order equation, which has a sharp physical background and a rich theoretical connotation. It was first introduced in \cite{King, King1} in the case $n = 3$ $$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x} \Big(u^3\frac{\partial^5u}{\partial x^5}\Big).$$ It describes the spreading of a thin viscous fluid under the driving force of an elastica (or light plate). During the past years, only a few works have been devoted to the sixth-order thin film equation \cite{bn3, FJC, Ga, King, King1}. Bernis and Friedman \cite{bn3} have studied the initial boundary value problems to the thin film equation $$\frac {\partial u}{\partial t}+(-1)^{m-1}\frac{\partial}{\partial x} \Big(f(u)\frac{\partial^{2m+1}u}{\partial x^{2m+1}}\Big)=0,$$ where $f(u)=|u|^nf_0(u)$, $f_0(u)>0$, $n\geq1$ and proved existence of weak solutions preserving nonnegativity. Barrett, Langdon and Nuernberg \cite{JW} considered the above equation with $m=2$. A finite element method is presented which proves to be well posed and convergent. Numerical experiments illustrate the theory. Recently, J\"{u}ngel and Mili\v{s}i\'{c} \cite{AJ} studied the sixth-order nonlinear parabolic equation $$\frac {\partial u}{\partial t} =\Big[u\Big(\frac1u(u(\ln u)_{xx})_{xx}+\frac12((\ln u)_{xx})^2 \Big)_x\Big]_x.$$ They proved the global-in-time existence of weak nonnegative solutions in one space dimension with periodic boundary conditions. Evans, Galaktionov and King \cite{Ev, Ev2} considered the sixth-order thin film equation containing an unstable (backward parabolic) second-order term $$\frac {\partial u}{\partial t}=\operatorname{div}\big[|u|^n\nabla\Delta^2 u\big] -\Delta(|u|^{p-1}u), n>0,\quad p>1.$$ By a formal matched expansion technique, they show that, for the first critical exponent $p = p_0 = n + 1 +\frac4N$ for $n \in( 0,\frac54)$, where $N$ is the space dimension, the free-boundary problem with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions $u_k(x,t)=(T-t)^{-\frac N{nN+6}}f_k(y)$, $k=1,2,\cdots$, $y=\frac x{(T-t)^{\frac1{nN+6}}}$, where $T > 0$ is the blow-up time. We also refer the following relevant equation $$\frac {\partial u}{\partial t}=-\frac{\partial}{\partial x} \Big(u^n\frac{\partial^3u}{\partial x^3}\Big),$$ which has been extensively studied. Bernis and Friedman \cite{bn3} studied the initial boundary value problems to the thin film equation $n>0$ and proved existence of weak solutions preserving nonnegativity (see also \cite{b2, L, L1, Y}). They proved that if $n\geq 2$ the support of the solutions $u(\cdot,t)$ is nondecreasing with respect to $t$. Our purpose in this paper is to study the radial symmetric solutions for the equation. We will study the problem in two-dimensional case, which has particular physical derivation of modeling the oil film spreading over a solid surface, see \cite{T1}. After introducing the radial variable $r=|x|$, we see that the radial symmetric solution satisfies \begin{gather} \frac{\partial(ru)}{\partial t} =\frac{\partial}{\partial r}\big\{r|u|^n\frac{\partial W}{\partial r}\big\},\quad rW=\frac{\partial}{\partial r}\Big(r\frac{\partial V}{\partial r}\Big),\quad rV=\frac{\partial}{\partial r}\Big(r\frac{\partial u}{\partial r}\Big), \label{1-1} \\ \frac{\partial u}{\partial r}\big|_{r=0}=\frac{\partial u}{\partial r}\big|_{r=1}=\frac{\partial V}{\partial r}\big|_{r=0} =\frac{\partial V}{\partial r}\big|_{r=1}=\frac{\partial W}{\partial r}\big|_{r=0}=\frac{\partial W}{\partial r}\big|_{r=1}=0, \label{1-2} \\ u\big|_{t=0}=u_0(r). \label{1-3} \end{gather} It should be noticed that the equation \eqref{1-1} is degenerate at the points where $r=0$ or $u=0$, and hence the arguments for one-dimensional problem can not be applied directly. Because of the degeneracy, the problem does not admit classical solutions in general. So, we introduce the weak solutions in the following sense \begin{definition} \label{def1.1}\rm A function $u$ is said to be a weak solution of the problem \eqref{1-1}--\eqref{1-3}, if the following conditions are fulfilled: \begin{itemize} \item[(1)] $ru(r,t)$ is continuous in $\overline Q_T$, where $Q_T=(0,1)\times(0,T)$; \item[(2)] $\sqrt{r}|u|^{n/2}\frac{\partial W}{\partial r}\in L^2(Q_T)$; \item[(3)] For any $\varphi\in C^1(\overline Q_T)$, the following integral equality holds \begin{align*} &\int_0^1ru(r,T)\varphi(r,T)\,dr-\int_0^1ru_0(r)\varphi(r,0)\,dr\\ &-\iint_{Q_T} ru\frac{\partial \varphi}{\partial t} \,dr\,dt +\iint_{Q_T} r |u|^n\frac{\partial W}{\partial r} \frac{\partial \varphi}{\partial r} \,dr\,dt=0. \end{align*} \end{itemize} \end{definition} Our interest lies in the existence of weak solutions. Because of the degeneracy, we will first consider the regularized problem. Based on the uniform estimates for the approximate solutions, we obtain the existence. Owing to the background, we are much interested in the nonnegativity of the weak solutions and the solutions with the property of finite speed of propagation of perturbations. Using weighted Nirenberg's inequality and Hardy's inequality, we proved these properties. This paper is arranged as follows. We shall prove several preliminary lemmas and obtain some a priori estimates on the solutions of regularized problem in Section 2, and then establish the existence in Section 3. Subsequently, we discuss the nonnegativity of weak solutions in Section 4 and the finite speed of propagation in Section 5. \section{Regularized problem} \label{sec2} Bernis and Friedman \cite{bn3} obtained several uniform estimations for the regularized solutions of fourth order thin film equation with the initial boundary value problems. To discuss the existence of weak solutions of problem \eqref{1-1}-\eqref{1-3}, we adopt the method of parabolic regularization, namely, the desired solution will be obtained as the limit of some subsequence of solutions of the following regularized problem \begin{gather} \frac{\partial(r_\varepsilon u)}{\partial t}=\frac{\partial}{\partial r} \big\{r_\varepsilon m_\varepsilon(u)\frac {\partial W}{\partial r}\big\},\quad r_\varepsilon W=\frac{\partial}{\partial r} \Big(r_\varepsilon\frac{\partial V}{\partial r}\Big),\quad r_\varepsilon V=\frac{\partial}{\partial r} \Big(r_\varepsilon\frac {\partial u} {\partial r}\Big), \label{2-1} \\ \frac {\partial u}{\partial r}\big|_{r=0}=\frac {\partial u}{\partial r}\big|_{r=1}= \frac{\partial V}{\partial r}\big|_{r=0}=\frac{\partial V}{\partial r}\big|_{r=1}=\frac{\partial W}{\partial r}\big|_{r=0} =\frac{\partial W}{\partial r}\big|_{r=1}=0, \label{2-2} \\ u\big|_{t=0}=u_{0\varepsilon}(r),\label{2-3} \end{gather} where $r_\varepsilon=r+\varepsilon$, $m_\varepsilon(u)=(|u|^2+\varepsilon)^{n/2}$ and $u_{0\varepsilon}(r)$ is a smooth approximation of the initial data $u_0(r)$. From the classical approach \cite{bn3}, it is not difficult to conclude that the problem \eqref{2-1}--\eqref{2-3} admits a global classical solution. We need some uniform estimates on the classical solutions. We first introduce some notation. Let $I=(0,1)$ and for any fixed $\varepsilon\geq0$ denote by $W^{1,2}_{*,\varepsilon}(I)$ the class of all functions satisfying $$\|u\|_{*,\varepsilon}=\Big(\int_0^1(r+\varepsilon)|u'(r)|^2dr\Big)^{1/2} +\Big(\int_0^1(r+\varepsilon)|u(r)|^2dr\Big)^{1/2}<+\infty.$$ It is obvious that $W^{1,2}(I)\subset W^{1,2}_{*,0}(I)$, but the class $W^{1,2}_{*,0}(I)$ is quite different from $W^{1,2}(I)$. In particular, we notice that the functions in $W^{1,2}_{*,0}(I)$ may not be bounded. For $\varepsilon>0$ the spaces $W^{1,2}(I)$ and $W^{1,2}_{*,\varepsilon}(I)$ coincide. However, it is not difficult to prove that for $u\in W^{1,2}_{*,\varepsilon}(I)$, the following properties hold: \begin{lemma}\label{lem2.1} If $0<\alpha\leq1$, and $u\in W^{1,2}_{*,\varepsilon}(I)$. Then \sup_{00, there is a constant M independent of \varepsilon such that \begin{gather} r_\varepsilon^\alpha|u(r,t)|\leq M,\quad \|u\|_{*,\varepsilon}\leq M , \label{2-10}\\ |r_\varepsilon u(r,t_2)-r_\varepsilon u(r,t_1)|\leq M|t_2-t_1|^{1/{16}} \label{2-11} \end{gather} for all r\in (0,1), t_1,t_2\in(0,T). \end{lemma} \begin{proof} The first two estimates have already been seen from the arguments in Lemma \ref{lem2.1}. Now, we begin to show \eqref{2-11}. Without loss of generality, we assume that t_10 be fixed and set P_\delta=\{(r,t):r|u|^n>\delta\}. We choose \varepsilon_0(\delta)>0, such that $$r_\varepsilon (|u_\varepsilon|^2+\varepsilon)^{n/2} \geq \frac{\delta}2,\quad (r,t)\in P_\delta,\;0<\varepsilon<\varepsilon_0(\delta). \label{3-1}$$ Then from \eqref{2-5} $$\iint_{P_\delta}\Big(\frac{\partial W_\varepsilon}{\partial r}\Big)^2 \,dr\,dt \leq \frac C\delta. \label{3-2}$$ To prove the integral equality in the definition of solutions, it suffices to pass the limit as \varepsilon\to0 in \begin{align*} &\int_0^1r_\varepsilon u_\varepsilon(r,T)\varphi(r,T)\,dr -\int_0^1r_\varepsilon u_{0\varepsilon}\varphi(r,0)\,dr -\iint_{Q_T}r_\varepsilon u_\varepsilon\frac{\partial \varphi}{\partial t}\,dr\,dt \\ &+\iint_{Q_T} r_\varepsilon (u_\varepsilon^2+\varepsilon)^{n/2}\frac{\partial W_\varepsilon}{\partial r}\frac{\partial\varphi}{\partial r}\,dr\,dt=0. \end{align*} The limits \begin{gather*} \lim_{\varepsilon\to0}\int_0^1r_\varepsilon u_\varepsilon(r,T)\varphi(r,T)=\int_0^1ru(r,T)\varphi(r,T)\,dr, \\ \lim_{\varepsilon\to0}\int_0^1r_\varepsilon u_{0\varepsilon}(r)\varphi(r,0)\,dr=\int_0^1u_0(r)\varphi(0,r)\,dr, \\ \lim_{\varepsilon\to0}\iint_{Q_T} r_\varepsilon u_\varepsilon\frac{\partial\varphi}{\partial t}\,dr\,dt=\iint_{Q_T}ru\frac{\partial\varphi}{\partial t}\,dr\,dt, \end{gather*} are obvious. It remains to show that $$\lim_{\varepsilon\to0}\iint_{Q_T} r_\varepsilon (u_\varepsilon^2+\varepsilon)^{n/2}\frac{\partial W_\varepsilon}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt =\iint_{Q_T} r |u|^n\frac{\partial W}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt. \label{3-3}$$ In fact, for any fixed \delta>0, \begin{align*} &\big|\iint_{Q_T} r_\varepsilon (u_\varepsilon^2+\varepsilon)^{n/2}\frac{\partial W_\varepsilon}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt -\iint_{Q_T} r |u|^n\frac{\partial W}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt\big| \\ &\leq \big|\iint_{P_\delta} r_\varepsilon (u_\varepsilon^2+\varepsilon)^{n/2}\frac{\partial W_\varepsilon}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt -\iint_{P_\delta} r|u|^n\frac{\partial W}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt\big| \\ &\quad +\big|\iint_{{Q_T}\backslash{P_\delta}} r_\varepsilon (u_\varepsilon^2+\varepsilon)^{n/2}\frac{\partial W_\varepsilon}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt\big| +\big|\iint_{{Q_T}\backslash{P_\delta}} r|u|^n\frac{\partial W}{\partial r} \frac{\partial \varphi}{\partial r}\,dr\,dt\big|. \end{align*} From the estimates \eqref{2-5}, we have \begin{gather*} \Big|\iint_{{Q_T}\backslash {P_\delta}} r_\varepsilon (u_\varepsilon^2+\varepsilon)^{n/2}\frac{\partial W_\varepsilon}{\partial r} \frac{\partial \varphi}{\partial r}\,dr\,dt\Big| \leq C\delta\sup\big|\frac{\partial \varphi}{\partial r}\big|,\quad 0<\varepsilon<\varepsilon_0(\delta), \\ \Big|\iint_{{Q_T}\backslash {P_\delta}} r|u|^n\frac{\partial W}{\partial r} \frac{\partial \varphi}{\partial r}\,dr\,dt\Big| \leq C\delta\sup\big|\frac{\partial \varphi}{\partial r}\big|, \\ \begin{aligned} &\Big|\iint_{P_\delta} r_\varepsilon (u_\varepsilon^2+\varepsilon)^{n/2}\frac{\partial W_\varepsilon}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt -\iint_{P_\delta} r|u|^n\frac{\partial W}{\partial r} \frac{\partial \varphi}{\partial r}\,dr\,dt\Big| \\ &\leq \iint_{P_\delta} \big|r_\varepsilon(u_\varepsilon^2+\varepsilon)^{n/2} -r|u|^n\big| \big|\frac{\partial W_\varepsilon}{\partial r}\big|\big| \frac{\partial \varphi}{\partial r}\big|\,dr\,dt\\ &\quad +\Big|\iint_{P_\delta} r|u|^n \Big(\frac{\partial W_\varepsilon}{\partial r}-\frac{\partial W}{\partial r}\Big) \frac{\partial \varphi}{\partial r}\,dr\,dt\Big| \\ &\leq \sup|r_\varepsilon(u_\varepsilon^2+\varepsilon)^{n/2} -r|u|^n| \big|\frac{\partial \varphi}{\partial r}\big|\frac C{\sqrt{\delta}} +\Big|\iint_{P_\delta}r|u|^n\Big(\frac{\partial W_\varepsilon}{\partial r} -\frac{\partial W}{\partial r}\Big) \frac{\partial \varphi}{\partial r}\,dr\,dt\Big| \end{aligned} \end{gather*} and hence \limsup_{\varepsilon\to0} \Big|\iint_{Q_T} r_\varepsilon \frac{\partial W_\varepsilon}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt -\iint_{Q_T} r\frac{\partial W}{\partial r}\frac{\partial \varphi}{\partial r}\,dr\,dt \Big| \leq C\delta\sup\big|\frac{\partial \varphi}{\partial r}\big|. $$By the arbitrariness of \delta, we see that the limit \eqref{3-3} holds. The proof is complete. \end{proof} \section{Nonnegativity}\label{sec4} Just as mentioned by several authors, it is much interesting to discuss the physical solutions. For the two-dimensional problem \eqref{1-1}--\eqref{1-3}, a very typical example is the modeling of oil films spreading over a solid surface, where the unknown function u denotes the height from the surface of the oil film to the solid surface. Motivated by this idea, we devote this section to the discussion of the nonnegativity of solutions. \begin{theorem}\label{th4.1} The weak solution u obtained in Section 3 satisfies u(r,t)\geq0, if u_0(r)\geq0. \end{theorem} \begin{proof} Suppose the contrary, that is, the set $$E=\{(r,t)\in\overline Q_T; u(r,t)<0\} \label{4-1}$$ is nonempty. For any fixed \delta>0, choose a C^\infty function H_\delta(s) such that H_\delta(s)=-\delta for s\geq-\delta, H_\delta(s)=-1, for s\leq-2\delta and that H_\delta(s) is nondecreasing for -2\delta0 in (-1,1), and \int_{-1}^1\alpha(s)ds=1. For any fixed k>0,\delta>0, define \begin{gather*} u^h(r,t)=\int_{\mathbb R}\bar{u}(s,r)\alpha_h(t-s)ds, \\ \beta_\delta(t)=\int_t^{+\infty}\alpha\Big(\frac{s-\frac T2}{\frac T2-\delta}\Big)\frac1{\frac T2-\delta}ds, \end{gather*} where \alpha_h(s)=\frac1h\alpha(s/h). The function$$ \varphi_\delta^h(r,t)\equiv[\beta_\delta(t)H_\delta(u^h)]^h is clearly an admissible test function, that is, the following integral equality holds \begin{aligned} &\int_0^1ru(r,T)\varphi^h_\delta(T,r)\,dr-\int_0^1ru_0(r)\varphi^h_\delta(r,0)\,dr\\ &-\iint_{Q_T} ru\frac {\partial\varphi^h_\delta} {\partial t} \,dr\,dt +\iint_{Q_T} r|u|^n\frac{\partial W}{\partial r} \frac {\partial\varphi^h_\delta}{\partial r} \,dr\,dt=0. \end{aligned}\label{4-2} To proceed further, we give an analysis on the properties of the test function \varphi^h_\delta(r,t). The definition of \beta_\delta(t) implies that $$\varphi_\delta^h(r,t)=0,\quad t\geq T-\frac \delta2, \quad h<\frac\delta2. \label{4-3}$$ Since \bar{u}(r,t) is continuous, for fixed \delta, there exists \eta_1(\delta)>0, such that $$u^h(r,t)\geq-\frac\delta2,\quad t\leq \eta_1(\delta),\quad 0\leq r\leq1,\quad h<\eta_1(\delta), \label{4-4}$$ which together with the definition of \beta_\delta(t) and H_\delta(s) imply $$H_\delta(u^h(r,t))=-\delta,\quad t\leq \eta_1(\delta),\quad 0\leq r\leq1,\quad h<\eta_1(\delta) \label{4-5}$$ and hence $$\varphi_\delta^h=-\delta,\quad t\leq \frac12\eta_1(\delta),\quad 0\leq r\leq1,\quad h<\frac12\eta_1(\delta). \label{4-6}$$ We note also that for any functions f(t),g(t)\in L^2(R), \begin{align*} \int_{\mathbb R}f(t)g^h(t)dt&=\int_{\mathbb R}f(t)dt\int_{\mathbb R}g(s)\alpha_n(t-s)ds =\int_{\mathbb R}f(t)\int_{\mathbb R}g(s)\alpha_n(s-t)ds \\ &=\int_{\mathbb R}g(s)ds\int_{\mathbb R}f(t)\alpha_n(s-t)dt=\int_{\mathbb R}f^h(t)g(t)dt. \end{align*} Taking this into account and using \eqref{4-3}, \eqref{4-5}, \eqref{4-6}, we have \begin{align*} \iint_{Q_T}ru\frac{\partial}{\partial t}\varphi_\delta^h\,dr\,dt &=\int_{-\infty}^{+\infty}dt\int_0^1ru\big[\frac{\partial}{\partial t}\big(\beta_\delta(t) H_\delta(u^h)\big)\big]^hdr \\ &= \iint_{Q_T}(ru)^h\frac{\partial}{\partial t}\big(\beta_\delta(t)H_\delta(u^h)\big)\,dr\,dt \end{align*} and hence by integrating by parts \begin{align*} &\iint_{Q_T}(ru)^h\frac{\partial}{\partial t}\left(\beta_\delta(t)H_\delta(u^h)\right)\,dr\,dt \\ &= \int_0^1(ru)^h(r,T)\beta_\delta(T)H_\delta(u^h(r,T))\,dr -\int_0^1(ru)^h(r,0)\beta_\delta(0)H_\delta(u^h(r,0))\,dr \\ &\quad -\iint_{Q_T}\beta_\delta(t)H_\delta(u^h)\frac{\partial(ru)^h}{\partial t}\,dr\,dt \\ &= \delta\int_0^1(ru)^h(r,0)\,dr -\iint_{Q_T}r\beta_\delta(t)\frac{\partial}{\partial t}F_\delta(u^h)\,dr\,dt, \end{align*} where F_\delta(s)=\int_0^sH_\delta(\sigma)d\sigma. Again by \eqref{4-5} \begin{align*} F_\delta(u^h(r,0))&=\int_0^{u^h(r,0)}H_\delta(\sigma)d\sigma =\int_0^1H_\delta(\lambda u^h(r,0))d\lambda\cdot u^h(r,0)\\ &=-\delta u^h(r,0) \end{align*} and hence \begin{aligned} &\iint_{Q_T}(ru)^h\frac{\partial}{\partial t}(\beta_\delta(t)H_\delta(u^h))\,dr\,dt \\ &= \delta\int_0^1(ru)^h(r,0)\,dr +\int_0^1r\beta_\delta(0)F_\delta(u^h(r,0))\,dr +\iint_{Q_T}rF_\delta(u^h)\beta'_\delta(t)\,dr\,dt \\ &= -\frac1{\frac T2-\delta}\iint_{Q_T}rF_\delta(u^h)\alpha \Big(\frac{t-\frac T2}{\frac T2-\delta}\Big)\,dr\,dt. \end{aligned} \label{4-7} From \eqref{4-3}, \eqref{4-6} it is clear that \begin{gather} \int_0^1ru(r,T)\varphi^h_\delta(T,r)\,dr=0,\quad 00, such that $$u(r,t)\geq-\frac\delta2\quad\forall (r,t)\in P^\delta, \label{4-11}$$ where P^\delta=\{(r,t);\operatorname{dist}((r,t),P)<\eta_2(\delta)\}. Here we have used the fact that u(r,t)>0 in P. Thus H_\delta(u^h(r,t))=-\delta,\quad\forall (r,t)\in P^{\delta/2},\quad 00$for$00$and$n\geq 6$, then the weak solution$u$satisfies$u(r,t)>0$. \end{theorem} \begin{proof} Since we have proved that$u(r,t)\geq0$, if the conclusion were false, then there would exist a point$(r_0,t_0)\in Q_T$, such that$u(r_0, t_0)=0$. From the H\"{o}lder continuity of$ru$, we see that $$ru(r, t_0)\leq C|r-r_0|^{1/4}.$$ Since$n\geq 6$, we have $$\int_0^1ru(r,t_0)^{2-n}dr\geq C\int_0^1|r-r_0|^{(2-n)/4}dr=\infty.$$ On the other hand, by Lemma \ref{lem4.1}, $$\int_0^1ru(r,t_0)^{2-n}dr\leq C,$$ which is a contradiction. The proof is complete. \end{proof} \section{Finite speed of propagation of perturbations} As is well known, one of the important properties of solutions of the porous medium equation is the finite speed of propagation of perturbations. So from the point of view of physical background, it seems to be natural to investigate this property for thin film equation. Bernis and Friedman \cite{bn3}, Bernis \cite{bn4} considered this property for thin film equation. On the other hand, the mathematical description of this property is that if$\operatorname{supp} u_0$is bounded, then for any$t>0$,$\operatorname{supp} u(\cdot,t)$is also bounded. So from the point of view of mathematics, this problem seems to be quite interesting. We adopt the weighted energy method and the main technical tools are weighted Nirenberg's inequality and Hardy's inequality. \begin{theorem}\label{th5.1} Assume$00$, we have $$\operatorname{supp}u(x,\cdot)\subset[r_1(t),r_2(t)]\cap[0,1],$$ where$r_1(t)=r_1-C_1t^\gamma$,$r_2(t)=r_2+C_2t^\gamma$,$C_1, C_2, \gamma>0$. \end{theorem} We need some uniform estimates on such approximate solutions$u_\varepsilon$. \begin{lemma}\label{lem5.1} Let$u$be the limit function of the approximate solutions obtained above. Then for any$y\in\mathbb R^+, the following integral inequality holds \begin{align*} &\int_0^1r(r-y)_+^\alpha u^{2-n}dr +\frac 12\iint_{Q_t}r(r-y)_+^\alpha\Big(\frac{\partial^3 u}{\partial r^3}\Big)^2\,dr\,ds \\ &\leq C\iint_{Q_t}r(r-y)_+^{\alpha-4}\Big(\frac{\partial u}{\partial r}\Big)^2\,dr\,ds +C\iint_{Q_t}r(x-y)_+^{\alpha-2}\Big(\frac{\partial^2 u}{\partial r^2}\Big)^2\,dr\,ds \\ &\quad+C\int_0^1r(r-y)_+^\alpha|u_0|^{2-n}dr +C\Big(\iint_{Q_t}r|u_0|^{2-n}\,dr\,ds\Big)^{1/2}, \end{align*} whereC$depends only on$n,u_0$and$\alpha\ge 2p-1$, where$(r-y)_+$denotes the positive part of$r-y$. \end{lemma} \begin{proof} Let$g_\varepsilon(u)$and$G_\varepsilon(u)$be defined as in the proof of Lemma \ref{lem4.1}. Let$u_\varepsilonbe the approximate solutions derived from the problem \eqref{2-1}--\eqref{2-3}. Then, using the equation \eqref{2-1} and integrating by parts, we obtain \begin{align*} &\int_0^1r(r-y)_+^\alpha G_\varepsilon(u_\varepsilon)\,dr -\int_0^1r(r-y)_+^\alpha G_\varepsilon(u_0)\,dr \\ &= -\iint_{Q_t}r_\varepsilon(|u_\varepsilon|^2+\varepsilon)^{n/2} \frac{\partial W}{\partial r} \frac{\partial}{\partial r}\left[(r-y)_+^\alpha g_\varepsilon(u_\varepsilon)\right]\,dr\,ds \\ &= -\iint_{Q_t}r_\varepsilon \frac{\partial W}{\partial r} (r-y)_+^\alpha \frac{\partial u_\varepsilon}{\partial r} \,dr\,ds \\ &\quad-\iint_{Q_t}r_\varepsilon(|u_\varepsilon|^2+\varepsilon)^{n/2} \frac{\partial W}{\partial r} \left[\alpha(r-y)_+^{\alpha-1} g_\varepsilon(u_\varepsilon)\right]\,dr\,ds \\ &\equiv I_1+I_2. \end{align*} As forI_1, integrating by parts, we have %% \begin{align*} &I_1\\ &= -\iint_{Q_t}r_\varepsilon \frac{\partial W}{\partial r} (r-y)_+^\alpha \frac{\partial u_\varepsilon}{\partial r} \,dr\,ds \\ &= \iint_{Q_t}Wr_\varepsilon\frac{\partial u_\varepsilon}{\partial r}\alpha(r-y)_+^{\alpha-1}\,dr\,ds +\iint_{Q_t}W\frac{\partial}{\partial r} \Big(r_\varepsilon\frac{\partial u_\varepsilon}{\partial r}\Big)(r-y)_+^\alpha \,dr\,ds \\ &= -\iint_{Q_t}r_\varepsilon\frac{\partial V}{\partial r}\frac{\partial^2 u_\varepsilon}{\partial r^2}\alpha(r-y)_+^{\alpha-1}\,dr\,ds -\iint_{Q_t}r_\varepsilon\frac{\partial V}{\partial r}\frac{\partial u_\varepsilon}{\partial r}\alpha(\alpha-1)(r-y)_+^{\alpha-2}\,dr\,ds \\ &\quad -\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial V}{\partial r}\Big)^2\,dr\,ds -\iint_{Q_t}r_\varepsilon\alpha(r-y)_+^{\alpha-1}\frac{\partial V}{\partial r}V \,dr\,ds. \end{align*} Therefore, \begin{align*} &\int_0^1r_\varepsilon(r-y)_+^\alpha G_\varepsilon(u_\varepsilon)\,dr -\int_0^1r_\varepsilon(r-y)_+^\alpha G_\varepsilon(u_0)\,dr +\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha\Big(\frac{\partial V}{\partial r}\Big)^2\,dr\,ds \\ &= -\iint_{Q_t}r_\varepsilon\frac{\partial V}{\partial r}\frac{\partial^2 u_\varepsilon}{\partial r^2}\alpha(r-y)_+^{\alpha-1}\,dr\,ds -\iint_{Q_t}r_\varepsilon\frac{\partial V}{\partial r}\frac{\partial u_\varepsilon}{\partial r}\alpha(\alpha-1)(r-y)_+^{\alpha-2}\,dr\,ds \\ &\quad -\iint_{Q_t}r_\varepsilon\alpha(r-y)_+^{\alpha-1}\frac{\partial V}{\partial r} V\,dr\,ds \\ &\quad -\iint_{Q_t}r_\varepsilon(|u_\varepsilon|^2+\varepsilon)^{n/2} \frac{\partial W}{\partial r} \left[\alpha(r-y)_+^{\alpha-1} g_\varepsilon(u_\varepsilon)\right]\,dr\,ds \\ &\equiv I_a+I_b+I_c+I_d. \end{align*} H\"{o}lder's inequality yields \begin{gather*} |I_a|\leq \frac 18\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial V}{\partial r}\Big)^2\,dr\,ds +C\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-2} \Big(\frac{\partial^2 u_\varepsilon}{\partial r^2}\Big)^2\,dr\,ds, \\ |I_b| \leq \frac 18\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha\left(\frac{\partial V}{\partial r}\right)^2\,dr\,ds +C_1\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-4} \Big(\frac{\partial u_\varepsilon}{\partial r}\Big)^2\,dr\,ds, \\ |I_c| \leq\frac 14\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial V}{\partial r}\Big)^2\,dr\,ds +C_2\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-2}V^2\,dr\,ds. \end{gather*} Noticing that $$(|u_\varepsilon|^2+\varepsilon)^{n/2}|g_\varepsilon(u_\varepsilon)| \leq\frac2{1-n}|u_\varepsilon|,$$ using \eqref{2-5}, we have \begin{align*} |I_d|&\leq C\Big(\iint_{Q_t}r_\varepsilon(|u_\varepsilon|^2+\varepsilon)^{n/2} \Big(\frac{\partial W}{\partial r}\Big)^2\,dr\,ds\Big)^{1/2}\\ &\quad\times \Big(\iint_{Q_t}r_\varepsilon(|u_\varepsilon|^2 +\varepsilon)^{n/2}g_\varepsilon(u_\varepsilon)^2\,dr\,ds\Big)^{1/2} \\ &\leq C\Big(\iint_{Q_t}r_\varepsilon|u_\varepsilon|^{2-n}\,dr\,ds\Big)^{1/2}. \end{align*} Summing up, we have \begin{align*} &\int_0^1r_\varepsilon(r-y)_+^\alpha G_\varepsilon(u_\varepsilon)\,dr -\int_0^1r_\varepsilon(r-y)_+^\alpha G_\varepsilon(u_0)\,dr\\ &+\frac12\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial V}{\partial r}\Big)^2\,dr\,ds \\ &\leq C\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-2} \Big(\frac{\partial^2 u_\varepsilon}{\partial r^2}\Big)^2\,dr\,ds +C_1\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-4} \Big(\frac{\partial u_\varepsilon}{\partial r}\Big)^2\,dr\,ds \\ &\quad +C_2\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-2}V^2\,dr\,ds +C_3\Big(\iint_{Q_t}r_\varepsilon|u_\varepsilon|^{2-n}\,dr\,ds\Big)^{1/2}. \end{align*} A simple calculation shows that \begin{align*} &\frac12\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial V}{\partial r}\Big)^2\,dr\,ds \\ &= \frac12\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \bigg(\Big(\frac{\partial^3u_\varepsilon}{\partial r^3}\Big)^2+ 2\frac{\partial^3u_\varepsilon}{\partial r^3}\frac{\partial}{\partial r} \Big(\frac1{r_\varepsilon}\frac{\partial u_\varepsilon}{\partial r}\Big) +\Big(\frac{\partial}{\partial r}\Big(\frac1{r_\varepsilon} \frac{\partial u_\varepsilon}{\partial r}\Big)\Big)^2\bigg)\,dr\,ds \\ &= \frac12\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial^3u_\varepsilon}{\partial r^3}\Big)^2\,dr\,ds +\frac12\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial}{\partial r}\Big(\frac1{r_\varepsilon} \frac{\partial u_\varepsilon}{\partial r}\Big)\Big)^2\,dr\,ds \\ &\quad +\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \frac{\partial^3u_\varepsilon}{\partial r^3}\frac{\partial}{\partial r} \Big(\frac1{r_\varepsilon}\frac{\partial u_\varepsilon}{\partial r}\Big)\,dr\,ds. \end{align*} Note that \begin{align*} &\Big|\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \frac{\partial^3u_\varepsilon}{\partial r^3}\frac{\partial}{\partial r} \Big(\frac1{r_\varepsilon}\frac{\partial u_\varepsilon}{\partial r}\Big)\,dr\,ds\Big| \\ &= \Big|\iint_{Q_t}(r-y)_+^\alpha\frac{\partial^3u_\varepsilon}{\partial r^3}\frac{\partial^2u_\varepsilon}{\partial r^2}\,dr\,ds -\iint_{Q_t}\frac1{r_\varepsilon}(r-y)_+^\alpha \frac{\partial^3u_\varepsilon}{\partial r^3} \frac{\partial u_\varepsilon}{\partial r}\,dr\,ds\Big| \\ &\leq \frac18\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial^3u_\varepsilon}{\partial r^3}\Big)^2\,dr\,ds +C\iint_{Q_t}\frac{r_\varepsilon}{r_\varepsilon^2}(r-y)_+^\alpha \Big(\frac{\partial^2u_\varepsilon}{\partial r^2}\Big)^2\,dr\,ds \\ &\quad +\frac18\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial^3u_\varepsilon}{\partial r^3}\Big)^2\,dr\,ds +C\iint_{Q_t}\frac{r_\varepsilon^2}{r_\varepsilon^4}(r-y)_+^\alpha \Big(\frac{\partial u_\varepsilon}{\partial r}\Big)^2\,dr\,ds \\ &\leq \frac14\iint_{Q_t}r_\varepsilon(r-y)_+^\alpha \Big(\frac{\partial^3u_\varepsilon}{\partial r^3}\Big)^2\,dr\,ds +C\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-2} \Big(\frac{\partial^2u_\varepsilon}{\partial r^2}\Big)^2\,dr\,ds \\ &\quad +C\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-4} \Big(\frac{\partial u_\varepsilon}{\partial r}\Big)^2\,dr\,ds. \end{align*} On the other hand, we have \begin{align*} &\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-2}V^2\,dr\,ds \\ &\leq 2\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-2} \Big(\frac{\partial^2u_\varepsilon}{\partial r^2}\Big)^2\,dr\,ds +2\iint_{Q_t}r_\varepsilon(r-y)_+^{\alpha-4} \Big(\frac{\partial u_\varepsilon}{\partial r}\Big)^2\,dr\,ds. \end{align*} Letting\varepsilon\to0$, we immediately get the desired conclusion and complete the proof of the lemma. \end{proof} \begin{proof}[Proof of Theorem \ref{th5.1}] For any$y\geq r_2$, Lemma \ref{lem5.1} and Hardy's inequality \cite{HG} imply that for any$t\in [0,T], \begin{aligned} &\int_0^1r(r-y)_+^\alpha u^{2-n}dr+\frac 12\iint_{Q_t}r(r-y)_+^\alpha \big|\frac{\partial^3u}{\partial r^3}\big|^2\,dr\,ds \\ &\leq C\iint_{Q_t}r(r-y)_+^{\alpha-4}\big|\frac{\partial u}{\partial r}\big|^2\,dr\,ds +C\iint_{Q_t}r(r-y)_+^{\alpha-2}\big|\frac{\partial^2u}{\partial r^2}\big|^2\,dr\,ds \\ &\leq C\iint_{Q_t}(r-y)_+^{\alpha-2}\big|\frac{\partial^2u}{\partial r^2}\big|^2\,dr\,ds. \end{aligned}\label{5-2} For any positive numberm$, define $f_m(y)=\int_0^t\int_0^1r(r-y)_+^m\big|\frac{\partial^3u}{\partial r^3}\big|^2\,dr\,ds, \quad f_0(y)=\int_0^t\int_y^1r\big|\frac{\partial^3u}{\partial r^3}\big|^2\,dr\,ds.$ By$y>r_2>0, then, weighted Nirenberg's inequality \cite{BF} and estimate \eqref{5-2} imply that \begin{align*} &f_{2p+1}(y)\\ &\leq C\iint_{Q_t}(r-y)_+^{2p-1}\big|\frac{\partial^2u}{\partial r^2}\big|^2\,dr\,ds \\ &\leq C_1\int_0^t \Big(\int_0^1(r-y)_+^{2p-1}\big|\frac{\partial^3u}{\partial r^3}\big|^2dr\Big)^a \Big(\int_0^1 (r-y)_+^{2p-1}| u|^qdr\Big)^{2(1-a)/q}ds \\ &\leq C\sup_{00, $1<\mu$. Applying H\"{o}lder's inequality, we have %% \begin{align*} f_{2p+1}(y) &\leq C t^\lambda\Big[\iint_{Q_t}(r-y)_+^{2p-1} \big|\frac{\partial^3u}{\partial r^3}\big|^2\,dr\,ds\Big]^\mu \\ &\leq C t^\lambda\Big[\iint_{Q_t}r(r-y)_+^{2p-2} \big|\frac{\partial^3u}{\partial r^3}\big|^2\,dr\,ds\Big]^\mu \\ &\leq C t^\lambda\Big[\iint_{Q_t}r(r-y)_+^{2p+1}\big| \frac{\partial^3u}{\partial r^3}\big|^2\,dr\,ds\Big]^{(2p-2)\mu/(2p+1)}\\ &\quad\times \Big[\int_0^t\int_y^1r\big|\frac{\partial^3u}{\partial r^3}\big|^2\,dr\,ds \Big]^{3\mu/(2p+1)} \\ &\leq C t^\lambda[f_{2p+1}(y)]^{(2p-2)\mu/(2p+1)} [f_0(y)]^{3\mu/(2p+1)}. \end{align*} Therefore, $$f_{2p+1}(y)\leq Ct^{\lambda/\sigma}[f_0(y)]^{3\mu/(2p+1)\sigma}, \quad \sigma=1-\frac{2p-2}{2p+1}\mu>0.$$ Using H\"{o}lder's inequality again, we obtain $$f_1(y)\leq[f_0(y)]^{2p/2p+1}[f_{2p+1}(y)]^{1/2p+1}\leq Ct^\gamma[f_0(y)]^{1+\theta},$$ where $$\gamma=\frac\lambda{(2p+1)\sigma},\quad \theta=\frac{2\mu}{(2p+1)^2\sigma}-\frac1{2p+1}>0.$$ Noticing that $f_1'(y)=-f_0(y)$, we obtain $$f_1'(y)\leq -Ct^{-\gamma/(\theta+1)}[f_1(y)]^{1/(\theta+1)}.$$ If $f_1(r_2)=0$, then $\hbox{supp}u\subset[0,r_2]$. If $f_1(r_2)>0$, then there exists a maximal interval $(r_2,r_2^*)$ in which $f_1(y)>0$ and $$\left[f_1(y)^{\theta/(\theta+1)}\right]'=\frac\theta{\theta+1}\frac{f_1'(y)} {[f_1(y)]^{1/(\theta+1)}}\leq-Ct^{-\gamma/(\theta+1)}.$$ Integrating the above inequality over $(r_2,r_2^*)$, we have $$f_1(r_2^*)^{\theta/(\theta+1)}-f_1(r_2)^{\theta/(\theta+1)} \leq -Ct^{-\gamma/(\theta+1)}(r_2^*-r_2),$$ which implies $$r_2^*\leq r_2+Ct^\gamma(f_0(r_2))^\theta.$$ Therefore. $$\sup \operatorname{supp} u(\cdot,t)\le r_2+Ct^\gamma\equiv r_2(t).$$ We have thus completed the proof of Theorem \ref{th5.1}. \end{proof} \subsection*{Acknowledgment} The authors would like to express their gratitude to the anonymous referees for their valuable suggestions which lead to improvements in this article. \begin{thebibliography}{00} \bibitem{BF} F. Bernis; \emph{Qualitative properties for some nonlinear higher order degenerate parabolic equations}, Houston J. Math., 14(3)(1988), 319-352. \bibitem{JW} J. W. 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