\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 17, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/17\hfil Life span of blow-up solutions] {Life span of blow-up solutions for higher-order semilinear parabolic equations} \author[F. Sun \hfil EJDE-2010/17\hfilneg] {Fuqin Sun} \address{Fuqin Sun \newline School of Science, Tianjin University of Technology and Education, Tianjin 300222, China} \email{sfqwell@163.com} \thanks{Submitted August 17, 2009. Published January 27, 2010.} \thanks{Supported by grant 10701024 from the National Natural Science Foundation of China} \subjclass[2000]{35K30, 35K65} \keywords{Higher-order parabolic equation; critical exponent; life span; \hfill\break\indent test function method} \begin{abstract} In this article, we study the higher-order semilinear parabolic equation \begin{gather*} u_t+(-\Delta)^m u=|u|^p, \quad (t,x)\in \mathbb{R}^1_+\times \mathbb{R}^N,\\ u(0,x)= u_0(x),\quad x\in \mathbb{R}^N. \end{gather*} Using the test function method, we derive the blow-up critical exponent. And then based on integral inequalities, we estimate the life span of blow-up solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \section{Introduction} This article concerns the cauchy problem for the higher-order semilinear parabolic equation $$\begin{gathered} u_t+(-\Delta)^m u=|u|^p, \quad (t,x)\in \mathbb{R}^1_+\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\quad x\in \mathbb{R}^N, \end{gathered} \label{1.1}$$ where $m, p>1$. Higher-order semilinear and quasilinear heat equations appear in numerous applications such as thin film theory, flame propagation, bi-stable phase transition and higher-order diffusion. For examples of these mathematical models, we refer the reader to the monograph \cite{PT}. For studies of higher-order heat equations we refer also to \cite{GC, SC, EVVP1, EVVP2, GP, WSP} and the references therein. In \cite{GP}, under the assumption that $u_0\in L^1(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$, $u_0\not\equiv0$ and $$\int_{\mathbb{R}^N}u_0(x){\rm d}x\geq0, \label{1.2}$$ Galaktionov and Pohozaev studied the Fujita critical exponent of problem \eqref{1.1} and showed that $p_F=1+2m/N$. The critical exponents $p_F$ is calculated from both sides: \begin{itemize} \item[(i)] blow-up of any solutions with \eqref{1.2} for $1p_F$. \end{itemize} Egorov et al \cite{EVVP2} studied the asymptotic behavior of global solutions with suitable initial data in the supercritical Fujita range $p>p_F$ by constructing self-similar solutions of higher-order parabolic operators and through a stability analysis of the autonomous dynamical system. For other studies of the problem, we refer to \cite{EVVP1} where global non-existence was proved for $p\in(1,p_F]$ by using the test function approach, and \cite{GC} where a general situation was discussed with nonlinear function $h(u)$ in place of $|u|^p$. In a recent paper \cite{WSP}, we discussed the system $$\begin{gathered} u_t+(-\Delta)^m u=|v|^p, \quad (t,x)\in \mathbb{R}^1_+\times \mathbb{R}^N,\\ v_t+(-\Delta)^m v=|u|^q, \quad (t,x)\in \mathbb{R}^1_+\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \ v(0, x)=v_0(x), \quad x\in \mathbb{R}^N. \end{gathered} \label{1.3}$$ It is proved that if $N/(2m>\max \big\{\frac{1+p}{pq-1}, \frac{1+q}{pq-1}\big\}$ then solutions of \eqref{1.3} with small initial data exist globally in time. Moreover the decay estimates $\|u(t)\|_\infty\leq C(1+t)^{-\sigma_1}$ and $\|v(t)\|_\infty\leq C(1+t)^{-\sigma_2}$ with $\sigma_1>0$ and $\sigma_2>0$ are also satisfied. On the other hand, under the assumption that $\int_{\mathbb{R}^N}u_0(x){\rm d}x>0, \quad \int_{\mathbb{R}^N}v_0(x){\rm d}x>0,$ if $N/(2m)\leq\max\big\{\frac{1+p}{pq-1},\frac{1+q}{pq-1}\big\}$ then every solution of \eqref{1.3} blows up in finite time. In our present work, exploiting the test function method, we shall give the life span of blow-up solution for some special initial data. The main idea comes from \cite{HK} for discussing cauchy problem of the second order equation $$\begin{gathered} \rho(x)u_t-\Delta u^m=h(x,t)u^{1+p}, \quad (t,x)\in \mathbb{R}^1_+\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\quad x\in \mathbb{R}^N. \end{gathered} \label{1.5}$$ Using the test function method, the author gave the blow-up type critical exponent and the estimates for life span $[0,T)$ like that in \cite{LN}. For the construction of a test function, the author mainly based on the eigenfunction $\Phi$ corresponding to the principle eigenvalue $\lambda_1$ of the Dirichlet problem on unit ball $B_1$, \begin{gather*} -\Delta w(x)=\lambda_1 w(x), \quad x\in B_1,\\ w(x)=0,\quad x\in \partial B_1. \end{gather*} However, for the operator $(-\Delta)^m$, the eigenfunction $\Phi$ corresponding to the principal eigenvalue $\lambda_1$ of the Dirichlet problem may change sign (see \cite{EV}). We will use a non-negative smooth function $\Phi$ constructed in \cite{GC} and \cite{GP}. The organization of this paper is as follows. In section 2, by the test function method, we derive some integral inequalities and reacquire the Fujita critical exponent $p_F$ obtained in the paper \cite{GP}. Section 3 is for the estimate of life span of blow-up solution. \section{Fujita critical exponent} In this section, we shall use the test function method to derive the Fujita critical exponent and some useful inequalities. From the reference \cite{GP}, we know that if $u_0\in L^1(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$, then the solution $u(t, \cdot)\in C^1([0,T]; L^1(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N))$ for some $T>0$. Therefore, without loss of generality, we may consider $u_0(x)$ concentrated around the origin and bounded below by a positive constant in some neighborhood of origin. Further, $u_0(x)\to 0$ as $|x|\to\infty$. With these choices, the solution $u$ and its spatial derivatives vanish as $|x|\to\infty$ for $t>0$. First we construct a test function. For this aim, we shall use a non-negative smooth function $\Phi$ which was constructed in the papers \cite{GC} and \cite{GP}. Let $\Phi(x)=\Phi(|x|)>0,\quad \Phi(0)=1;\quad 0<\Phi(r)\leq1\quad \mbox{for } r>0,$ where $\Phi(r)$ is decreasing and $\Phi(r)\to 0$ as $r\to \infty$ sufficiently fast. Moreover, there exists a constant $\lambda_1>0$ such that $$|\Delta^m \Phi|\leq \lambda_1\Phi,\quad x \in \mathbb{R}^N,\label{2.1}$$ and such that $$\|\Phi\|_1=\int_{\mathbb{R}^N}\Phi(x){\rm d}x=1.$$ This can be done by letting $\Phi(r)=e^{-r^\nu}$ for $r\gg1$ with $\nu\in (0,1]$, and then extending $\Phi$ to $[0, \infty)$ by a smooth approximation. Take $\theta>p/(p-1)$, and define $\phi(t)=\begin{cases} 0,& t>T,\\ (1-(t-S)/(T-S))^\theta, & 0\leq t\leq T,\\ 1, & t0.  Suppose that u exists in [0, t_*)\times \mathbb{R}^N. For TR^{2m}0.\label{dd} Let u be a global solution with u_0 satisfying \eqref{dd}, then \[ \int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm d}x{\rm d}t>0.$ Suppose $s<0$. Letting $R$ tend to infinity in \eqref{2.7} to obtain $$\int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm d}x{\rm d}t +\int_{\mathbb{R}^N}u_0(x){\rm d}x=0.$$ Hence $u\equiv0$, a contradiction. Suppose $s=0$. We first show $J\geq 0$ for all $R>0$. In fact, from the assumptions on initial datum, there exists $\varepsilon_0>0$ such that $u_0(x)\geq\delta>0$ for $|x|\leq \varepsilon_0$. Set \begin{align*} J&= \int_{|x|\leq \varepsilon_0}u_0(x)\Phi(x/R){\rm d}x +\int_{|x|> \varepsilon_0}u_0(x)\Phi(x/R){\rm d}x \\ &> \delta\int_{|x|\leq \varepsilon_0}\Phi(x/R){\rm d}x +\int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm d}x\\ &= \delta R^N\int_{|\eta|\leq \varepsilon_0/R}\Phi(\eta){\rm d}\eta +\int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm d}x\\ &\geq \int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm d}x. \end{align*} By the choice of $\Phi$, we have $\lim_{R\to0}\int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm d}x=0.$ And so there exists $R_0>0$ such that $J\geq 0$ for all $00$ such that $$\int_{|x|\leq R_0M}u_0(x){\rm d}x >\int_{|x|>R_0M}|u_0(x)|{\rm d}x.$$ In addition, by a slight modification of $\Phi$, we may set $\Phi(x)\equiv 1$ in $\{x:\ |x|\leq M \}$. Note that since $0\leq\Phi\leq1$ we have, for $R\geq R_0$, \begin{align*} J&= \int_{|x|\leq R_0M}u_0(x)\Phi(x/R){\rm d}x +\int_{|x|>R_0M}u_0(x)\Phi(x/R){\rm d}x\\ &\geq \int_{|x|\leq R_0M}u_0(x){\rm d}x -\int_{|x|>R_0 M}|u_0(x)|\Phi(x/R){\rm d}x\\ &\geq \int_{|x|\leq R_0M}u_0(x){\rm d}x -\int_{|x|>R_0 M}|u_0(x)|{\rm d}x>0. \end{align*} Now we are in the position to complete the proof of case $s=0$. Since $$A(S, T)=\frac{\theta(T-S)^{-1/p}}{[\theta-1/(p-1)]^{(p-1)/p}},\quad B(T)=\Big[S+\frac{T-S}{\theta+1}\Big]^{(p-1)/p},$$ we may choose $S$ small and $\theta$ large, $T-S$ bounded, such that $$B(T)\leq \int_{\mathbb{R}^N}u_0(x){\rm d}x/\Big[2\lambda_1\Big(\int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm d}x{\rm d}t\Big)^{1/p} \Big].\label{2.8}$$ Moreover, note that $J\geq0$, from \eqref{2.7} we get that $I(0,T)$ is uniformly bounded for all $R>0$. Then, keeping $T-S$ bounded, $$\lim_{R\to \infty}I(S,T)^{1/p}A(S, T)=0.\label{2.9}$$ Letting $R\to\infty$, \eqref{2.7}--\eqref{2.9} give $$\int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm d}x{\rm d}t+\frac12\int_{\mathbb{R}^N}u_0(x){\rm d}x=0,$$ which also implies $u\equiv0$. \end{proof} Let $\sigma$ be an arbitrary positive number. For $x\in [0, \infty)$ and $0<\omega<1$, define $$\Psi(\omega; \sigma):=\max_x(\sigma x^{\omega}-x).$$ It is easy to check that $\Psi(\omega; \sigma)=(1-\omega)\omega^{\frac{\omega}{1-\omega}}\sigma^{\frac1{1-\omega}}$. Set $$A(T)=A(0,T),\quad S(T)=A(T)+\lambda_1B(T).$$ We have the following result. \begin{theorem}\label{thm2} If $u$ is a solution of \eqref{1.1} defined on $[0, t_*)\times \mathbb{R}^N$. Then, for $R>0$ and $0\leq\tau \leq t_*R^{-2m}$, we have $$\int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x\leq \Psi\Big(\frac1p;\ S(T)R^s\Big).\label{2.10}$$ Moreover, if $u$ is a global solution of {\rm\eqref{1.1}}, then $$\lim_{R\to \infty}\sup R^{-\hat{s}}\int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x\leq \lambda_1^{1/(p-1)},\label{2.11}$$ where $\hat{s}=sp/(p-1)$. \end{theorem} \begin{proof} Denote $I(T)=I(0, T)$. Firstly, by the definition of $\Psi$, from \eqref{2.7} we know that $$J\leq I(T)^{1/p}S(T)R^s-I(T) \leq\Psi\Big(\frac1p;\ S(T)R^s\Big).$$ This is exactly \eqref{2.10}. By means of \eqref{2.10}, we deduce that \begin{aligned} \int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x &\leq \Psi\Big(\frac1p;\ S(T)R^s\Big)\\ &= (1-1/p)(1/p)^{\frac{1/p}{1-1/p}}[S(T)R^s]^{\frac1{1-1/p}}\\ &= (p-1)p^{p/(1-p)}R^{sp/(p-1)}S(T)^{\frac{p}{p-1}}, \end{aligned} \label{2.12} which leads to $$\lim_{R\to \infty}\sup R^{-\hat{s}}\int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x\leq (p-1)p^{p/(1-p)}[\inf_T S(T)]^{\frac{p}{p-1}}.\label{2.13}$$ To estimate $S(T)$, we need estimate $A(T)$ and $B(T)$ respectively. Denote $a_p=\frac{\theta}{[\theta-1/(p-1)]^{(p-1)/p}},\quad b_p=\frac{\lambda_1}{(\theta+1)^{(p-1)/p}}.$ We obtain $$S(T)=a_pT^{-1/p}+b_pT^{(p-1)/p}.$$ Since \begin{align*} \min_T S(T)&= p[a_p/(p-1)]^{(p-1)/p}b_p^{1/p}\\ &= \frac{p(p-1)^{-(p-1)/p}\lambda_1^{1/p}\theta^{(p-1)/p}} {[\theta-1/(p-1)]^{(p-1)^2/p^2}(1+\theta)^{(p-1)/p^2}}, \end{align*} we have $$\lim_{\theta\to\infty}\min_T S_p(T)=p(p-1)^{-(p-1)/p}\lambda_1^{1/p}.\label{2.14}$$ Combining \eqref{2.13} and \eqref{2.14}, we obtain \eqref{2.11}. The proof is complete. \end{proof} \section{Life span of blow-up solutions} In this section, we shall estimate the life span of the blow-up solution with some special initial datum. To this aim, we assume that $u_0$ satisfies \begin{itemize} \item[(H)] There exist positive constants $C_0, L$ such that $u_0(x) \geq \begin{cases} \delta, & |x|\leq \varepsilon_0,\\ C_0|x|^{-\kappa}, & |x|>\varepsilon_0, \end{cases}$ where $\delta$ and $\varepsilon_0$ are as in the proof of Theorem \ref{thm1}, and $N< \kappa< 2m/(p-1)$ if $p<1+2m/N$; $0<\kappa0$. Denote $[0,T_\varepsilon)$ be the life span of $u_\varepsilon$. Then there exists a positive constant $C$ such that $T_\varepsilon\leq C\varepsilon^{1/\hat{\beta}}$, where $\hat{\beta}=\frac{\kappa}{2m}-\frac1{p-1}<0.$ \end{theorem} \begin{remark}\label{rmk3} \rm When $p=1+2m/N$, note that $\hat{\beta}=(\kappa-N)/(2m)$. \end{remark} \begin{proof} Choose $R$ such that $R\geq R^0>0$. By the definition of $J$ and the assumptions of initial data, we have \begin{aligned} J&= \varepsilon\int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x\\ &\geq \varepsilon \int_{|x|>\varepsilon_0} u_0(x)\Phi(x/R){\rm d}x\\ &= \varepsilon R^{N}\int_{|\eta|>\varepsilon_0/R }u_0(R\eta)\Phi(\eta){\rm d}\eta\\ &\geq \varepsilon C_0R^{N-\kappa}\int_{|\eta|>\varepsilon_0/R}|\eta|^{-\kappa}\Phi(\eta){\rm d}\eta\\ &\geq \varepsilon C_0R^{N-\kappa}\int_{|\eta|>\varepsilon_0/R^0}|\eta|^{-\kappa}\Phi(\eta){\rm d}\eta\\ &= \widetilde{C}R^{N-\kappa}. \end{aligned}\label{4.1} Using \eqref{2.12}, we know from \eqref{4.1} that, for $0<\tau0$. Now we derive some estimates on $H(\tau, R)$. If we can find a function $G(\tau)$ such that $$H(\tau, R)\geq G(\tau),\quad \forall \ \tau>0,$$ and for each value of $R\geq R^0$ there exists a value of $\tau_R$ such that $H(\tau_R, R)=G(\tau_R)$, then \eqref{4.2} holds for all $R\geq R^0$ if and only if $$\varepsilon\leq \widetilde{C}^{-1}(p-1)p^{p/(1-p)}G(\tau). \label{4.3}$$ Set $$y=R^{\alpha_1+\alpha_2}=R^{2m},\quad \beta_1=\alpha_2/(\alpha_1+\alpha_2)=\alpha_2/(2m).$$ Then $$H(\tau,R)=\tau^{-1/(p-1)}h(\tau, y)^{p/(p-1)}$$ with $h(\tau, y)=a_py^{1-\beta_1}+b_p y^{-\beta_1}\tau$. Denote $$\sigma=a_pb_p^{-1}(1-\beta_1)\beta_1^{-1}y,\quad G(\tau)=\tau^{-1/(p-1)}g(\tau)^{p/(p-1)},$$ where $$g(\tau)=[a_py^{1-\beta_1}\sigma^{\beta_1-1} +b_py^{-\beta_1}\sigma^{\beta_1}] \tau^{1-\beta_1}.$$ It is easy to check that $0<\beta_1<1$. Then, $\zeta=g(\tau)$ ia a concave curve. Furthermore, $\zeta=h(\tau, y)$ is a tangent line of $\zeta=g(\tau)$ at the point of $( \sigma, g(\sigma))$. Therefore, we get that $h(\tau,y)\geq g(\tau)$, for all $\tau>0$. Hence $H(\tau, R)\geq G(\tau)$, for all $\tau>0$. Moreover, $H(\tau,R_{\tau})=G(\tau)$ with $$\tau_R=a_pb_p^{-1}(1-\beta_1)\beta_1^{-1}R^{2m}.$$ By computations, $$G(\tau)=\tau^{-1/(p-1)}g(\tau)^{p/(p-1)}= C_1\tau^{\hat{\beta}}. \label{4.4}$$ for some positive constant $C$, where $\hat{\beta}=\frac \kappa{2m}-\frac{1}{p-1}.$ The choice of $\kappa$ implies that $\hat{\beta}<0$. Combining \eqref{4.3} and \eqref{4.4}, we find that $$\varepsilon\leq K\tau^{\hat{\beta}} \label{4.5}$$ for some $K>0$. From \eqref{4.5}, it follows that $\tau\leq C\varepsilon^{1/\hat{\beta}}$ for some $C>0$. The proof is complete. \end{proof} \begin{thebibliography}{00} \bibitem{GC} M. Chaves, V. A. Galaktionov, \newblock{Regional blow-up for a higher-order semilinear parabolic equation}, \emph{Europ J. Appl. Math.}, \textbf{12} (2001), 601--623. \bibitem{SC} S. B. 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