\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 228, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/228\hfil Asymptotic behavior of solutions] {Asymptotic behavior of solutions to parabolic problems with nonlinear nonlocal terms} \author[M. Loayza \hfil EJDE-2013/228\hfilneg] {Miguel Loayza} % in alphabetical order \address{Miguel Loayza \newline Departamento de Matem\'atica, Universidade Federal de Pernambuco, 50740-540, Recife, PE, Brazil} \email{miguel@dmat.ufpe.br} \thanks{Submitted August 9, 2012. Published October 16, 2013.} \subjclass[2000]{35K15, 35B40, 35E15} \keywords{Nonlocal parabolic equation; global solution; self-similar solution} \begin{abstract} We study the existence and asymptotic behavior of self-similar solutions to the parabolic problem $$ u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s)ds\quad\text{on } (0,\infty)\times \mathbb{R}^N, $$ with $p>1$ and $u(0,\cdot) \in C_0(\mathbb{R}^N)$. \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}\label{In} In this work we study the existence and asymptotic behavior of global solutions of the semilinear parabolic problem \begin{equation} \begin{gathered} u_t-\Delta u = \int_0^tk(t,s)|u|^{p-1}u(s)ds \quad \text{in } (0,\infty)\times \mathbb{R}^N,\\ u(0,x)=\psi(x) \quad \text{in }\mathbb{R}^N, \end{gathered}\label{In.uno} \end{equation} where $p>1$ and $k: \mathcal{R} \to \mathbb{R}$ satisfies \begin{itemize} \item[(K1)] $k$ is a continuous function on the region $\mathcal{R}=\{(t,s)\in \mathbb{R}^2;00$ and some $\gamma \in \mathbb{R}$, \item[(K3)] $k(1, \cdot ) \in L^1(0,1)$, \item[(K4)] $\limsup_{\eta \to 0^+} \eta^l|k(1, \eta)|<\infty$ for some $l \in \mathbb{R}$. \end{itemize} Problem \eqref{In.uno} models diffusion phenomena with memory effects and has been considered by several authors for some values of the function $k$ (see \cite{Be,CDE,Fino,FinoKi,L,S2} and the references therein). When $k(t,s)=(t-s)^{-\gamma}$, $\gamma \in [0,1)$ and $\psi \in C_0(\mathbb{R}^N)$, it was shown in \cite{CDE} that if $$ p>p_*=\max\{1/\gamma, 1+(4-2\gamma)/[(N-2+2\gamma)^+]\} \in (0,\infty], $$ then the solution of \eqref{In.uno} is global, for $\| \psi\|_{r^*}$ small enough, where $r^*=N(p-1)/[2(2-\gamma)]$. The value $p_*$ is the Fujita critical exponent and is not given by a scaling argument. Similar results were obtained in \cite{Fino} replacing the operator $-\Delta$ by the operator $(-\Delta)^{\beta/2}$ with $0<\beta\leq 2$. When the function $k$ is nonnegative and satisfies conditions (K1)--(K4), with $\gamma<2$ and $l<1$, it was shown in \cite{L} that if $$ p(2-\gamma)/(p-1)0$, the function $u_\lambda(t,x)=\lambda^\alpha u(\lambda^2 t, \lambda x)$ satisfies \begin{equation} \begin{gathered} u_t-\Delta u = \lambda^{2[\alpha(1-p)+2-\gamma]}\int_0^tk(t,s)|u|^{p-1}u(s)ds \quad \text{in } (0,\infty)\times \mathbb{R}^N,\\ u(0,x)=\lambda^{2 \alpha} \psi(\lambda x) \quad \text{in }\mathbb{R}^N. \end{gathered}\label{Inr.uno} \end{equation} In particular, if $\alpha=(2-\gamma)/(p-1)$, then $u_\lambda$ is also a solution of problem \eqref{In.uno}. A solution satisfying $u=u_\lambda$ for all $\lambda>0$ is called a self-similar solution of problem \eqref{In.uno}. Note that, in this case, $\psi(x)=\lambda^{2\alpha} \psi(\lambda x)$; that is, the function $\psi$ is a homogeneous function of degree $-2\alpha$. Our objective is to determine the asymptotic behavior of global solutions of \eqref{In.uno} in terms of the self-similar solution $w$ corresponding to the cases (see Theorem \ref{Th.sel} for details): \begin{itemize} \item[(i)] $\alpha(p-1)=2-\gamma$. \begin{gather*} w_t-\Delta w= \int_0^tk(t,s)|w|^{p-1}w(s)ds \quad \text{in } (0,\infty)\times \mathbb{R}^N,\\ w(0,x)=|x|^{-2\alpha} \quad\text{in }\mathbb{R}^N , \end{gather*} \item[(ii)] $\alpha(p-1)>2-\gamma$. \begin{gather*} w_t-\Delta w =0 \quad \text{in } (0,\infty)\times \mathbb{R}^N,\\ w(0,x) =|x|^{-2\alpha} \quad\text{in }\mathbb{R}^N. \end{gather*} \end{itemize} For $\alpha(p-1)<2-\gamma$, we show that there is no nonnegative global solution of \eqref{In.uno}, if $w(0,x) \sim |x|^{-2\alpha}$ for $|x|$ large enough (see Theorem \ref{Th.nex} for details). To show the existence of global solutions to \eqref{In.uno} we use a contraction mapping argument on the associated integral equation \begin{equation} u(t)= e^{t\Delta}\psi + \int_0^t e^{(t-s)\Delta}\int_0^s k(s,\sigma) |u|^{p-1}u(\sigma)d\sigma ds, \label{Eq.int} \end{equation} where $(e^{t\Delta})_{t\geq 0}$ is the heat semigroup. Precisely, this contraction mapping argument is done on a given Banach space equipped with a norm chosen so that we obtain directly the global character of the solution. Our approach works for unbounded and sign changing initial data. On the other hand, the self-similar solutions constructed in this work may be not radially symmetric. In fact,we adapt a method introduced by Fujita and Kato \cite{FuKa,KaFu} and used later in \cite{Cann,CannPl,CazWie,ST,W1}. Since the homogeneous function $\psi= | \cdot |^{-2\alpha}$, does not belong to any $L^p(\mathbb{R}^N)$ space, we consider initial data so that $\sup_{t>0}t^{\alpha-N/(2r_1)}\| e^{t\Delta}\psi \|_{r_1}<\infty$, for some $r_1\geq 1$. Hence, it is necessary to consider that $\alphaN/(2\alpha)>1$, $\beta_1=\alpha-N/(2r_1)$ and let $\varphi_h$ be a tempered distribution homogeneous of degree $-2\alpha$ such that $\varphi_h(x)=\mu(x)|x|^{-2\alpha}$, where $\mu \in L^{r_1}(S^{N-1})$ is a function homogeneous of degree $0$. Assume that $\eta$ is a cut-off function, that is, identically 1 near the origin and of compact support. Then \begin{itemize} \item[(i)] $\sup_{t>0}t^{\beta_1}\| e^{t\Delta} \varphi_h\|_{r_1}<\infty$; \item[(ii)] $\sup_{t>0}t^{\beta_1+\delta }\| e^{t\Delta} (\eta \varphi_h)\|_{r_1}<\infty$ for $0<\delta0} t^{\beta_1}\| e^{t\Delta} (1-\eta )\varphi_h\|_{r_1}<\infty$. \end{itemize} \label{Pr.h} \end{proposition} Our first result is technical. It will be used to formulate the global existence and asymptotic behavior results. \begin{proposition} Let $l<1,\gamma<2$ and set $a=\min\{1-l,2-\gamma\}$. Assume that $\alpha \in (0, N/2)$ satisfies \begin{gather} 2-\gamma+\alpha < \frac{N}{2}+ a, \label{In.sei} \\ (2-\gamma +\alpha)\frac{1-\gamma}{2-\gamma}\frac{N}{2\alpha}(2-\gamma), r_1 > \frac{2-\gamma}{\alpha}+1$ and $r_1> \frac{N}{2\alpha}$. \item[(ii)] $(2-\gamma+\alpha)(1-\frac{N}{2r_1\alpha}) < a$. \end{itemize} \label{pr.a} \end{proposition} We now give the following existence result for problem \eqref{In.uno} shows the existence of global solutions and its continuous dependence. \begin{theorem} Let $p>1$ and $k$ satisfying conditions $K1)-K4)$ with $\gamma<2$ and $l<1$. Assume \begin{equation} p>1+2(2-\gamma)/N \label{In.dos} \end{equation} and $\alpha \in (0,N/2)$ satisfying \eqref{In.sei}, \eqref{In.sie} and \begin{equation} \frac{2-\gamma}{p-1}\leq \alpha <\frac{N}{2}. \label{In.tre} \end{equation} Fix $\tilde \alpha>0$ such that \begin{equation} \tilde \alpha \leq \frac{2-\gamma}{p-1}. \label{Inr.tre} \end{equation} Let $r_1>1$ be given by Proposition \ref{pr.a}, and let $r_2>1$ be defined by $r_2=\alpha r_1/\tilde \alpha$. For every $\varphi \in \mathcal{S}'(\mathbb{R}^N)$ define $\mathcal{N}$ by \begin{equation} \mathcal{N}(\varphi)=\sup_{t>0}\{t^{\beta_1}\| e^{t\Delta}\varphi \|_{r_1}, t^{\beta_2}\| e^{t\Delta}\varphi \|_{r_2}\}, \label{In.och} \end{equation} where $\beta_1=\alpha -N/(2r_1)$ and $\beta_2=\tilde \alpha -N/(2r_2)$. Let $M>0$ be such that $C=C(M)<1$, where $C$ is a positive constant given by \eqref{Gl.die}. Choose $R>0$ such that $R+C M \leq M$. If $\varphi$ is a tempered distribution such that \begin{equation} \mathcal{N}(\varphi)\leq R, \label{In.siec} \end{equation} then there exits a unique global solution $u$ of \eqref{In.uno} satisfying $$ \sup_{t>0}\{t^{\beta_1} \| u(t)\|_{r_1}, t^{\beta_2} \| u(t)\|_{r_2}\}\leq M. $$ In addition, if $\varphi, \psi$ satisfy \eqref{In.siec} and if $u_\varphi$ and $u_\psi$ respectively are the solutions of \eqref{Eq.int} with initial data $\varphi,\psi$, then \begin{equation} \sup_{t>0}[t^{\beta_1}\| u_\varphi(t)-u_\psi(t)\|_{r_1},t^{\beta_2}\| u_\varphi(t)-u_\psi(t)\|_{r_2}]\leq (1-C)^{-1}\mathcal{N}(\varphi-\psi). \label{In.conk} \end{equation} Moreover, if $\varphi, \psi$ are such that \begin{equation} \mathcal{N}_\delta(\varphi-\psi)=\sup_{t>0}\{t^{\beta_1+\delta}\| e^{t\Delta}(\varphi-\psi)\|_{r_1}, t^{\beta_2+\delta}\| e^{t\Delta}(\varphi-\psi)\|_{r_2}\}<\infty, \label{In.conka} \end{equation} for some $\delta \in (0,\delta_0)$, where $\delta_0=1-l-(2-\gamma+\alpha)[1-N/(2r_1\alpha)]>0$. Then \begin{equation} \sup_{t>0} \{t^{\beta_1+\delta}\| u_\varphi-u_\psi\|_{r_1}, t^{\beta_2+\delta}\| u_\varphi-u_\psi\|_{r_2}\} \leq (1-C_\delta)^{-1}\mathcal{N}_\delta(\varphi-\psi), \label{In.conkb} \end{equation} where $C_\delta$ is given by \eqref{Gl.d} below and the constant $M>0$ is chosen small enough so that $C_\delta<1$. \label{Th.sol} \end{theorem} \begin{remark}\label{Rem.ex} \rm Suppose that $\alpha (p-1) =2-\gamma$ in Theorem \ref{Th.sol}. (i) From \eqref{In.tre} and \eqref{Inr.tre}, we see that it is possible to choose $\tilde \alpha=\alpha $. It follows that $r_1=r_2$, $\beta_1=\beta_2$. Therefore, Theorem \ref{Th.sol} holds replacing the norm $\mathcal{N}$ of \eqref{In.och} by $ \mathcal{N}_s(\varphi):=\sup_{t>0}\{t^{\beta_1} \| e^{t\Delta}\varphi\|_{r_1}\}$. (ii) Assume that $k(t,s)=(t-s)^{-\gamma}$ with $\gamma \in (0,1)$. Then $k$ satisfies $K1)-K4)$ with $l=0$, and therefore $a=\min\{1-l,2-\gamma\}=1$. From conditions \eqref{In.sei}-\eqref{In.tre} we have that $p(N-2+2\gamma)>N+2$, $p\gamma>1$ and $p>1+2(2-\gamma)/N$ respectively. Since $p>1+(4-2\gamma)/[(N-2+2\gamma)^+] >1+2(2-\gamma)/N$, we conclude that $p>p^*=\max\{1/\gamma,1+(4-2\gamma)/[(N-2+2\gamma)^+]\}$ which coincides with the condition encountered in \cite{CDE}. (iii) Conditions \eqref{In.sei}-\eqref{In.dos} become $2(2-\gamma)p<(N+2a)(p-1)$, $p(1-\gamma)1+2(2-\gamma)/N$ respectively. The last inequality is obtained from the first one, since $2(2-\gamma)p<(N+2a)(p-1)\leq [N+2(1-l)](p-1)$ and $\gamma<2$. Indeed, $p>1+2(2-\gamma)/[N-2+2(\gamma-l)^+]>1+2(2-\gamma)/N$. These conditions were used in \cite{L} to show global existence of \eqref{In.uno}. \end{remark} We now state the following asymptotic behavior result for some global solution of problem \eqref{In.uno} with small initial data with respect to the norm $\mathcal{N}$ given by \eqref{In.och}. \begin{theorem}[Asymptotically self-similar solutions] Let $p>1$ satisfying \eqref{In.dos} and $k$ be a function satisfying conditions $K1)-K4)$ with $\gamma<2$ and $l<1$. Let $\alpha \in (0,N/2)$ be satisfying \eqref{In.sei}, \eqref{In.sie} and \eqref{In.tre}, $\tilde \alpha >0$ satisfying \eqref{Inr.tre}, $r_1$ given by Proposition 2 and $r_2=\alpha r_1/ \tilde \alpha$. Set $\varphi_h(x)=\mu(x)|x|^{-2\alpha}$, where $\mu$ is homogeneous of degree $0$ and $\mu \in L^{r_1}(S^{N-1})$. Suppose that $\varphi \in \mathcal{S}'(\mathbb{R}^N)$ satisfies \eqref{In.siec}, $u$ is the corresponding solution of \eqref{In.uno} given by Theorem \ref{Th.sol}, and \begin{equation} \sup_{t>0}t^{\beta_1+\delta} \| e^{t\Delta}(\varphi-\varphi_h) \|_{r_1}<\infty \label{In.cat} \end{equation} for some $\delta \in (0,\delta_0)$, where $\delta_0=1-l-p(2-\gamma)/(p-1)+Np/(2r_1)$ when $\alpha(p-1)=2-\gamma$ and given by Lemma \ref{Lem.sub} when $\alpha(p-1)>2-\gamma$. We have the following: \begin{itemize} \item[(i)] If $\alpha(p-1)>2-\gamma$, then $\sup_{t>0}t^{\beta_1+\delta}\| u(t)-e^{t\Delta}\varphi_h\|_{r_1}\leq C_\delta$, for some constant $C_\delta>0$. \item[(ii)] If $\alpha(p-1)=2-\gamma$ and $w$ is the solution of \eqref{In.uno} given by Theorem \ref{Th.sol} with initial data $\varphi_h$(we multiplied $\varphi_h$ by a small constant so that \eqref{In.siec} is satisfied), then $w$ is self-similar and $\sup_{t>0} t^{\beta_1+\delta}\| u(t)-w(t)\|_{r_1}\leq C_\delta$ for some constant $C_\delta>0$. \end{itemize} \label{Th.sel} \end{theorem} \begin{remark} \rm The class of functions $\varphi$ satisfying the condition \eqref{In.cat} is nonempty. Indeed, from Proposition \ref{Pr.h}(2), condition \eqref{In.cat} is satisfied for $\varphi= (1-\eta)\varphi_h$. \end{remark} In the following result, we analyze the non existence of global solutions of problem \eqref{In.uno}, under the assumption \begin{itemize} \item[(K5)] There exist $T>0$ and a nonnegative, non-increasing continuous function $\phi \in C([0, \infty))$ with integrable derivative such that $\phi(0)=1$ and $\phi(t)=0$ for $t\geq T$ satisfying $k(\cdot, t) \phi(\cdot) \in L^1(t, T)$ for $t>0$ and \begin{equation}\label{K5} \int_0^T \phi(t)^{p'}\Big( \int_t^T k(s,t) \phi(s)ds\Big)^{-p'/p}dt <\infty, \end{equation} where $p'$ is the conjugate of $p$. \end{itemize} \begin{theorem} Let $p>1$ and let $k$ be a nonnegative function satisfying conditions {\rm (K1)--(K3), (K5)}. If $\psi \in C_0(\mathbb{R}^N), \psi \geq 0$ satisfies $\liminf_{|x|\to \infty} |x|^{2(2-\gamma)/(p-1)}\psi (x)=\infty$ and $u$ is a corresponding nonnegative solution of problem \eqref{In.uno}, then $u$ is not a global solution. \label{Th.nex} \end{theorem} \begin{remark}\rm Regarding Theorem \ref{Th.nex} we have the following statements: (i) Under conditions (K1)--(K3), existence of local solutions for \eqref{In.uno} in the class $C([0,T), C_0(\mathbb{R}^N))$ and initial data $\psi \in C_0(\mathbb{R}^N)$, were studied in \cite{L}. In particular, we know that if $k$ and $\psi$ are nonnegative, then the solution of \eqref{In.uno} is nonnegative. (ii) Let $k(t,s)=(t-s)^{-\gamma_1} s^{-\gamma_2}$ for $01/(p-1)$. Indeed, since $\phi \leq 1$, we have for $t>0$ $$ \int_t^1k(s,t)\phi(s)ds=t^{-\gamma_2} \int_t^1 (s-t)^{-\gamma_1} \phi(s)ds \leq \frac{t^{-\gamma_2}}{1-\gamma_1}(1-t)^{1-\gamma_1}<\infty. $$ On the other hand, $$ \int_t^1 k(s,t)\phi(s)ds =t^{-\gamma_2}\int_t^1 (s-t)^{-\gamma_1} \phi(s) ds \geq t^{-\gamma_2} \int_t^1 \phi(s)ds =\frac{t^{-\gamma_2}}{1+q} (1-t)^{1+q}. $$ Therefore, $$ \int_0^1 \phi(t)^{p'}\Big( \int_t^1 k(s,t) \phi(s)ds\Big)^{-p'/p}dt =(1+q)^{p'/p}\int_0^1 (1-t)^{p'(q-\frac{1+q}{p})}t^{\frac{\gamma_2 p'}{p}}dt, $$ which is finite, since $$ 1+p'(q-\frac{1+q}{p})=\frac{p'}{p}[p-2+q(p-1)]>\frac{p'}{p}(p-1)>0. $$ \end{remark} \section{Existence of global solutions} \subsection*{Proof of Proposition \ref{pr.a}} Let $A=\frac{2\alpha}{N}(1-\frac{a}{2-\gamma+\alpha})$. Since $a>0$ we conclude that $A<2\alpha/N<1$. From \eqref{In.sie} and \eqref{In.sei} we have $A<2\alpha/[N(2-\gamma)]$ and $A<\alpha/(2-\gamma+\alpha)$, respectively. Now, it is sufficient to choose $r_1> 1$ satisfying $A<\frac{1}{r_1}<\min\{\frac{2\alpha}{N(2-\gamma)}, \frac{\alpha}{2-\gamma+\alpha},\frac{2\alpha}{N}\}$. \begin{lemma} Assume the conditions \eqref{In.sei}-\eqref{Inr.tre}. Let $r_2=\frac{\alpha r_1}{\tilde \alpha}$, $\beta_1=\alpha-\frac{N}{2r_1}$, $\beta_2=\tilde \alpha-\frac{N}{2r_2}$, $\frac{1}{\eta_1}=\frac{1}{pr_1}(\frac{2-\gamma}{\alpha} +1)$, $\frac{1}{\eta_2}=\frac{1}{pr_1}(\frac{2-\gamma}{\alpha} +\frac{\tilde \alpha}{\alpha})$, $\theta_1=\frac{2-\gamma+\alpha-p\tilde \alpha}{p(\alpha-\tilde \alpha)}$, and $\theta_2=\frac{2-\gamma+(1-p)\tilde \alpha}{p(\alpha-\tilde \alpha)}$. For $i=1,2$ we have \begin{itemize} \item[(i)] $\eta_i \in [r_1,r_2]$ and $\eta_i \in (p,r_ip)$. \item[(ii)] $ \frac{p}{\eta_1}-\frac{1}{r_1} =\frac{p}{\eta_2}-\frac{1}{r_2}=\frac{2-\gamma}{r_1\alpha}<\frac{2}{N}$. \item[(iii)] $\theta_i \in [0,1]$, $ \frac{1}{\eta_i} =\frac{\theta_i}{r_1}+\frac{(1-\theta_i)}{r_2}$. \item[(iv)] $\frac{1}{p}a>\theta_i \beta_1+(1-\theta_i)\beta_2$, with \begin{gather*} \theta_1 \beta_1+(1-\theta_1)\beta_2 =\frac{1}{p}(2-\gamma+\alpha)(1-\frac{N}{2r_1\alpha}),\\ \theta_2 \beta_1+(1-\theta_2)\beta_2=\frac{1}{p} (2-\gamma+\tilde \alpha)(1-\frac{N}{2r_1\alpha}). \end{gather*} \item[(v)] $2-\gamma +\beta_i-\frac{N}{2} (\frac{p}{\eta_i}-\frac{1}{r_i})-p[\beta_1 \theta_i+\beta_2(1-\theta_i)]=0$. \end{itemize} \label{Lem.b} \end{lemma} \begin{proof} (i) From \eqref{In.tre}, we see that $\eta_1\geq r_1$ and $\eta_2\leq r_2$. Since $\tilde \alpha\leq \alpha$, it follows from \eqref{In.tre} and \eqref{Inr.tre} that \begin{equation}\label{Gl.un} 2-\gamma +\tilde \alpha\leq p\alpha, \ p \tilde \alpha \leq 2-\gamma+\alpha \end{equation} respectively. From here, $\eta_2\geq r_1$ and $\eta_1\leq r_2$. The condition $r_1> (2-\gamma)/\alpha+1$ of Proposition \ref{pr.a}(i) and $\gamma<2$ ensure that $\eta_1 \in (p,r_1p)$. Moreover, since $r_1> (2-\gamma)/\alpha+1\geq (2-\gamma+\tilde \alpha)/\alpha$ and $\gamma<2$, we conclude that $\eta_2 \in (p,r_2 p)$. Item (ii) follows from Proposition \ref{pr.a}(i). (iii) From \eqref{In.tre} and \eqref{Inr.tre} we get $\theta_1\leq 1$ and $\theta_2\geq 0$ respectively, and from \eqref{Gl.un} we see that $\theta_2 \leq 1$ and $\theta_1\geq 0$ respectively. We obtain (iv) from Proposition \ref{pr.a}(ii). \end{proof} \begin{proof}[Proof of Theorem \ref{Th.sol}] The proof is based on a contraction mapping argument. Let $E$ be the set of Bochner measurable functions $u: (0,\infty)\to L^{r_1}(\mathbb{R}^N)\cap L^{r_2}(\mathbb{R}^N)$, such that $\| u \|_E= \sup_{t>0}\{t^{\beta_1}\| u(t)\|_{r_1}, t^{\beta_2}\| u(t)\|_{r_2}\}<\infty$, where $\beta_1=\alpha-N/(2r_1), \beta_2=\tilde \alpha-N/(2r_2)$. The space $E$ is a Banach space. Let $M>0$ and $K$ be the closed ball of radius $M$ in $E$. Let $\Phi_\varphi:K \to E$ be the mapping defined by \begin{equation} \Phi_\varphi (u)(t)=e^{t\Delta}\varphi +\int_0^t e^{(t-s)\Delta}\int_0^sk(s,\sigma)|u|^{p-1}u(\sigma)d\sigma ds. \label{Ap} \end{equation} We will prove that $\Phi_\varphi$ is a strict contraction mapping on $K$. Let $\varphi, \psi$ satisfying \eqref{In.siec} and $u,v \in K$. We will use several times the smoothing effect for the heat semigroup: if $1\leq s\leq r\leq \infty$ and $\varphi \in L^r$, then $$ \| e^{t\Delta} \varphi \|_{r}\leq t^{-\frac{N}{2}(\frac{1}{s} -\frac{1}{r})}\| \varphi\|_s $$ for all $t>0$. From \eqref{Ap}, we deduce \begin{equation} \begin{aligned} &t^{\beta_1}\| \Phi_\varphi(u)(t)- \Phi_\psi(v)(t)\|_{r_1}\leq t^{\beta_1} \| e^{t\Delta}(\varphi-\psi)\|_{r_1} \\ &+ pt^{\beta_1}\int_0^t \| e^{(t-s)\Delta}\int_0^s |k(s,\sigma)|(|u|^{p-1}+|v|^{p-1})|u(\sigma)-v(\sigma)|\|_{r_1}d\sigma\\ &\leq t^{\beta_1}\| e^{t\Delta}(\varphi-\psi)\|_{r_1} + pt^{\beta_1}\int_0^t(t-s)^{-\frac{N}{2}(\frac{p}{\eta_1} -\frac{1}{r_1})}\\ &\quad\times \int_0^s |k(s,\sigma)|(\| u\|^{p-1}_{\eta_1} +\| v\|^{p-1}_{\eta_1})\| u(\sigma)-v(\sigma)\|_{\eta_1}d\sigma ds. \end{aligned} \label{Gl.tre} \end{equation} From Lemma \ref{Lem.b},(i) and (iii), and an interpolation inequality $$ \| u\|_{\eta_1}\leq \| u\|_{r_1}^{\theta_1}\| u\|_{r_2}^{1-\theta_1} $$ where $\frac{1}{\eta_1}=\frac{\theta_1}{r_1}+\frac{1-\theta_1}{r_2}$. Replacing this inequality into \eqref{Gl.tre} we obtain \begin{equation} \begin{aligned} &t^{\beta_1}\| \Phi_\varphi(u)(t)- \Phi_\psi(v)(t)\|_{r_1}\\ &\leq t^{\beta_1}\| e^{t\Delta}(\varphi-\psi)\|_{r_1} +2M^{p-1}p\| u-v\|_Et^{\beta_1}\\ &\quad\times \int_0^t(t-s)^{-\frac{N}{2}(\frac{p}{\eta_1} -\frac{1}{r_1})}\int_0^s |k(s,\sigma)|\sigma^{-p[\theta_1 \beta_1 +(1-\theta_1)\beta_2]}d\sigma ds. \end{aligned} \label{Gl.trea} \end{equation} From (K4), there exist $\eta_0,\nu>0$ such that $\eta^l |k(1,\eta)|<\nu$ for $\eta \in (0, \eta_0)$. Thus, if $\theta_1 \beta_1+ \beta_2(1-\theta_1)=\Theta_1$, we have \begin{equation} \begin{aligned} \int_0^s |k(s,\sigma)|\sigma^{-p\Theta_1}d\sigma &= s^{1-\gamma-p\Theta_1}\int_0^1|k(1,\sigma)|\sigma^{-p\Theta_1}d\sigma\\ &\leq s^{1-\gamma-p \Theta_1}\Big[\nu \int_0^{\eta_0}\sigma^{-l- p\Theta_1}d\sigma+ \eta_0^{-p\Theta_1}\int_{\eta_0}^1 |k(1,\sigma)|d\sigma\Big]\\ & =C_1s^{1-\gamma- p\Theta_1}, \end{aligned}\label{Gl.unoa} \end{equation} where \begin{equation} C_1=\nu \int_0^{\eta_0}\sigma^{-l- p\Theta_1}d\sigma+ \eta_0^{-p\Theta_1}\int_{\eta_0}^1 |k(1,\sigma)|d\sigma. \label{Gl.cuno} \end{equation} Since $p \Theta_1< a$ (see Lemma \ref{Lem.b}(iv)) and $k$ satisfies (K3), we conclude that $C_1<\infty$. From \eqref{Gl.trea}, \eqref{Gl.unoa} and properties (iv) and (v) of Lemma \ref{Lem.b}, \begin{equation} \begin{aligned} &t^{\beta_1}\| \Phi_\varphi u(t)-\Phi_\psi v(t)\|_{r_1}\\ & \leq t^{\beta_1}\| e^{t\Delta}(\varphi-\psi)\|_{r_1} \\ &\quad+ 2C_1M^{p-1}pt^{\beta_1} \| u-v\|_E\int_0^t (t-s) ^{-\frac{N}{2}(\frac{p}{\eta_1}-\frac{1}{r_1})}s^{1-\gamma-p\Theta_1}ds\\ & \leq t^{\beta_1}\| e^{t\Delta}(\varphi-\psi)\|_{r_1}+C_1'\| u-v\|_E, \end{aligned} \label{Gl.cua} \end{equation} where $C_1'=2C_1M^{p-1}p \int_0^1(1-s)^{-\frac{N}{2} (\frac{p}{\eta_1}-\frac{1}{r_1})}s^{1-\gamma-p\Theta_1}ds$. From Lemma \ref{Lem.b}, (ii) and (iv), we see that $C_1'<\infty$. Similarly, one can prove that \begin{equation} t^{\beta_2}\|\Phi_\varphi u(t)-\Phi_\psi v(t)\|_{r_2} \leq t^{\beta_2}\| e^{t\Delta}(\varphi-\psi)\|_{r_2}+C_2'\| u-v\|_E, \label{Gl.och} \end{equation} where \begin{gather*} C_2'= 2C_2 M^{p-1}p\int_0^1(1-s)^{-\frac{N}{2} (\frac{p}{\eta_2}-\frac{1}{r_2})}s^{1-\gamma-p \Theta_2}ds<\infty,\\ C_2=\nu \int_0^{\eta_0} \sigma^{-l-p\Theta_2}d\sigma +\eta_0^{-p\Theta_1}\int_{\eta_0}^1|k(1,\sigma)| d\sigma<\infty,\\ \Theta_2=\frac{1}{p}(1-\frac{N}{2r_1\alpha})(2-\gamma+\tilde \alpha). \end{gather*} From \eqref{Gl.cua} and \eqref{Gl.och} we obtain \begin{equation} \| \Phi_\varphi(u)(t)-\Phi_\psi(v)(t)\|_E \leq \mathcal{N}(\varphi-\psi) + C \| u-v\|_E, \label{Gl.sei} \end{equation} where \begin{equation} C=\max\{C_1',C_2'\}. \label{Gl.die} \end{equation} Setting $\psi=0, v=0$ in \eqref{Gl.sei} we get $ \|\Phi_\varphi(u)\|_E \leq \mathcal{N}(\varphi)+C\| u\|_E$. Since $\varphi$ satisfies \eqref{In.siec} and $R+CM \leq M$, we conclude that $\Phi_\varphi u \in K$. Moreover, since $C<1$ we conclude from \eqref{Gl.sei} that $\Phi_\varphi$ is a strict contraction from $K$ into itself, so $\Phi_\varphi$ has a unique fixed point in $K$. The continuous dependence \eqref{In.conk} follows clearly from \eqref{Gl.sei}. To show \eqref{In.conkb}, let \begin{equation} \| u-v\|_{E,\delta}=\sup_{t>0}\{t^{\beta_1+\delta}\| u(t)\|_{r_1}, t^{\beta_2+\delta}\| v(t)\|_{r_2}\}. \label{Nord} \end{equation} Proceeding as \eqref{Gl.tre} we obtain \begin{equation} \begin{aligned} &t^{\beta_1+\delta}\| u(t)-v(t)\|_{r_1}\\ &\leq t^{\beta_1+\delta}\| e^{t\Delta }( \varphi-\psi)\|_{r_1} + 2pM^{p-1}t^{\beta_1+\delta}\int_0^t(t-s)^{-\frac{N}{2} (\frac{p}{\eta_1}-\frac{1}{r_1})} \\ &\quad\times \int_0^s |k(s,\sigma)|\sigma^{-[\theta_1\beta_1 +(1-\theta_1)\beta_2](p-1)}\| u-v\|_{\eta_1}d\sigma ds \\ &\leq t^{\beta_1+\delta}\| e^{t\Delta } (\varphi-\psi)\|_{r_1} +2pM^{p-1} \sup_{\sigma \in (0,t)}\{\sigma^{\beta_1+\delta}\| u(\sigma)\|_{r_1}, \sigma^{\beta_2+\delta}\| v(\sigma)\|_{r_2}\} \\ &\quad\times t^{\beta_1+\delta}\int_0^t(t-s)^{-\frac{N}{2} (\frac{p}{\eta_1}-\frac{1}{r_1})}\int_0^s |k(s,\sigma) |\sigma^{-p[\theta_1 \beta_1+(1-\theta_1) \beta_2]-\delta}d\sigma dt \end{aligned} \label{Gl.a} \end{equation} For $0<\delta<1-l-p\Theta_1$, arguing as in \eqref{Gl.unoa}, we have \begin{align*} &\int_0^s |k(s,\sigma)|\sigma^{-p[\theta_1 \beta_1 +\theta_2 \beta_2]-\delta}d\sigma\\ &= s^{1-\gamma-p\Theta_1 -\delta}\int_0^1 |k(1,\sigma) |\sigma^{-p\Theta_1-\delta}\\ &\leq s^{1-\gamma-p\Theta_1 -\delta} \Big[ \nu \int_0^{\eta_0}\sigma^{-l-p\Theta_1-\delta}d\sigma + \eta_0^{-p\Theta_1-\delta}\int_{\eta_0}^1|k(1,\sigma)|d\sigma \Big]\\ &= C_{1,\delta} s^{1-\gamma-p\Theta_1 -\delta}, \end{align*} where $$ C_{1,\delta}= \nu \int_0^{\eta_0}\sigma^{-l-p\Theta_1-\delta}d\sigma + \eta_0^{-p\Theta_1-\delta}\int_{\eta_0}^1|k(1,\sigma)|d\sigma<\infty. $$ Therefore, from \eqref{Gl.a} we obtain \begin{equation} \begin{aligned} &t^{\beta_1+\delta}\| u(t)-v(t)\|_{r_1}\\ &\leq t^{\beta_1+\delta}\| e^{t\Delta }( \varphi-\psi)\|_{r_1} + C_{1,\delta}' \sup_{\sigma \in (0,t)}\{\sigma^{\beta_1+\delta} \| u(\sigma)\|_{r_1}, \sigma^{\beta_2+\delta}\| v(\sigma)\|_{r_2}\}, \end{aligned}\label{Gl.b} \end{equation} and \begin{equation} C_{1,\delta}'=2pM^{p-1} C_{1,\delta}. \label{Gl.e} \end{equation} Similarly, for $0<\delta<1-l-p\Theta_2$, one can to obtain \begin{equation} \begin{aligned} &t^{\beta_2+\delta}\| u(t)-v(t)\|_{r_2}\\ &\leq t^{\beta_2+\delta}\| e^{t\Delta } \varphi-\psi\|_{r_2} + C_{2,\delta}' \sup_{\sigma \in (0,t)}\{\sigma^{\beta_1+\delta} \| u(\sigma)\|_{r_1}, \sigma^{\beta_2+\delta}\| v(\sigma)\|_{r_2}\}, \end{aligned} \label{Gl.c} \end{equation} where $$ C_{2,\delta}= \nu \int_0^{\eta_0}\sigma^{-l-p\Theta_2-\delta}d\sigma + \eta_0^{-p\Theta_2-\delta}\int_{\eta_0}^1k(1,\sigma)d\sigma<\infty $$ and $ C_{2,\delta}'=2pM^{p-1}C_{2,\delta}$. From \eqref{Gl.b} and \eqref{Gl.c} it follows that $$ (1-C_\delta)\| u-v\|_{E,\delta}\leq \mathcal{N}_\delta(\varphi-\psi), $$ where \begin{equation} C_\delta=\max\{C_{1,\delta}',C_{2,\delta}'\}. \label{Gl.d} \end{equation} \end{proof} \section{Asymptotic behavior} The next result will be used in the proof of Theorem \ref{Th.sel}(1). \begin{lemma} Let $l<1$, $\gamma<2$, $p>1$ and $a=\min\{1-l,2-\gamma\}$. Assume \eqref{In.tre} and let $\alpha$ satisfying \eqref{In.sei}, \eqref{In.sie} and \eqref{In.tre}. Let $\tilde \eta$ satisfying \eqref{Inr.tre}. For $\delta >0$ we define $\eta'\geq 1$ by $ \frac{1}{\eta_1'}=\frac{1}{pr_1}\big(\frac{2-\gamma+\delta}{\alpha}+1\big)$, and $\theta_1'= \frac{2-\gamma+\delta+\alpha-p\tilde \alpha} {p(\alpha-\tilde \alpha)}$. If $\frac{2-\gamma}{p-1}<\alpha$, then there exists $\delta_0>0$ small such that for all $\delta \in (0,\delta_0]$: \begin{itemize} \item[(i)] $\eta'_1 \in [r_1,r_2]$ and $\eta_1'\in (p,r_1 p)$, where $r_2=(\alpha r_1)/\tilde \alpha$. \item[(ii)] $ \frac{N}{2}(\frac{p}{\eta'_1}-\frac{1}{r_1}) =\frac{N}{2r_1 \alpha}(2-\gamma+\delta)<1$. \item[(iii)] $\theta_1 \in [0,1]$, $ \frac{1}{\eta'_1} =\frac{\theta_1'}{r_1}+\frac{1-\theta_1'}{r_2}$. \item[(iv)] If $\beta_1=\alpha-\frac{N}{2r_1}$ and $\beta_2 =\tilde \alpha-\frac{N}{2r_2}$, then $$ a>p[\beta_1 \theta_1' + \beta_2(1-\theta_1')] =(2-\gamma+\alpha+\delta)(1-\frac{N}{2r_1\alpha}). $$ \item[(v)] $ 2-\gamma+\beta_1+\delta -\frac{N}{2}(\frac{p}{\eta_1'} -\frac{1}{r_1})-p[\beta_1 \theta_1'+\beta_2 (1-\theta_1')]=0$. \end{itemize} \label{Lem.sub} \end{lemma} \begin{proof} Since $\alpha>(2-\gamma)/(p-1)$, \eqref{In.sei} and \eqref{In.sie} hold, it follows from Proposition \ref{pr.a} that there exists $\delta_0>0$ small so that such that $\alpha>(2-\gamma+\delta_0)/(p-1)$, $r_1>\frac{N}{2\alpha}(2-\gamma+\delta_0)$, $r_1>(2-\gamma+\delta_0)/\alpha+1$ and $(2-\gamma+\alpha+\delta_0)(1-N/(2r_1\alpha))0}\{t^{\beta_1}\| u(t)\|_{r_1},t^{\beta_2}\| u(t)\|_{r_2}\}\leq M. $$ Arguing as in \eqref{Gl.a}, \eqref{Gl.unoa} and \eqref{Gl.cuno}, we conclude conclude that \begin{align*} t^{\beta_1+\delta}\| u(t)-e^{t\Delta} \varphi_h\|_{r_1} &\leq t^{\beta_1+\delta}\| e^{t\Delta } (\varphi-\varphi_h)\|_{r_1} + 2pM^{p} t^{\beta_1+\delta}\int_0^t(t-s)^{-\frac{N}{2}(\frac{p}{\eta_1'} -\frac{1}{r_1})} \\ &\quad\times \int_0^s |k(s,\sigma)|\sigma^{-p[\theta_1' \beta_1+(1-\theta_1') \beta_2]}d\sigma dt\\ &\leq t^{\beta_1+\delta}\| e^{t\Delta } (\varphi-\varphi_h)\|_{r_1}+ C_{\delta}', \end{align*} where $C_{\delta}'= 2pM^{p}C_\delta$, $C_\delta= \nu \int_0^{\eta_0}\sigma^{-l-p\Theta_\delta}d\sigma +\eta_0^{-p\Theta_\delta}\int_{\eta_0}^1|k(1,\sigma)|d\sigma$ and $\Theta_\delta=\frac{1}{p}(1-\frac{N}{2}\frac{P_1}{r_1})(2-\gamma +\frac{1}{P_1}+\delta)$. From the above result and \eqref{In.cat} we have the desired conclusion. (ii) For $\lambda>0$, we define $z(t,x)=\lambda^{(4-2\gamma)/(p-1)} w(\lambda^2 t, \lambda x)$ for all $t>0, x \in \mathbb{R}^N$. Clearly $z$ is a solution of \eqref{In.uno}. We claim that $\sup_{t>0}t^{\beta_1}\| z\|_{r_1}\leq M$. To see this, we observe that \begin{align*} t^{\beta_1}\| z\|_{r_1} &= t^{\beta_1}\lambda^{\frac{4-2\gamma}{p-1}}\| w(\lambda^2 t, \lambda \cdot)\|_{r_1}\\ &= t^{\beta_1}\lambda^{\frac{4-2\gamma}{p-1}-\frac{N}{r_1}}\| w(\lambda^2 t)\|_{r_1}\\ &=(\lambda^2 t)^{\beta_1}\| w(\lambda^2 t)\|_{r_1}. \end{align*} Since $z(0)=\varphi_h$, we have from \eqref{In.conk} that $w=z$; that is, $w$ is self-similar. The conclusion now follows from \eqref{In.conkb} and the Remark \ref{Rem.ex}(i). \end{proof} \section{Non existence of global solutions} \begin{proof}[Proof of Theorem \ref{Th.nex}] Let $B_R$ be the open ball in $\mathbb{R}^N$ with radius $R>0$. Let $\lambda_R>0$ and $\rho_R>0$ be the first eigenvalue and the first normalized (i.e. $\int_{B_R} \rho_R=1$) eigenfunction of $-\Delta$ on $B_R$ with zero Dirichlet boundary condition. Set $w_R(t)=\int_{B_R}u(t)\rho_R$. Then by Green's identity and Jensen's inequality we obtain \begin{equation} ( w_R )_t+ \lambda_R w_R \geq \int_0^t k(t,s)w_R^p(s)ds. \label{Nex.uno} \end{equation} Set $\phi_R(t)=\phi(t/R^2)$ for all $t\geq 0$. Multiplying \eqref{Nex.uno} by $\phi_R$ and integrating on $[0,TR^2]$, we have \begin{equation} \begin{aligned} -w_R(0) + \lambda_R \int_0^{TR^2} w_R(t) \phi_R(t)dt &\geq \int_0^{TR^2} \int_0^t k(t,s) w_R^p(s)ds \, \phi_R(t)dt\\ &= \int_0^{TR^2} I_R(s) w_R^p(s)ds, \end{aligned}\label{Nex.dos} \end{equation} where $$ I_R(s)=\int_s^{TR^2} k(t,s) \phi_R(t)dt. $$ On the other hand, by H\"older's inequality, \begin{equation}\label{Nex.tre} \begin{aligned} \int_0^{TR^2} w_R(t) \phi_R(t)dt &= \int_0^{TR^2} w_R(t) I_{R}(t)^{1/p} I_{R}(t)^{-1/p}\phi_R(t)dt\\ &\leq \Big\{ \int_0^{TR^2} w_R^p I_{R}(t)dt\Big\}^{1/p} \underbrace {\Big\{ \int_0^{TR^2} I_{R}(t)^{-p'/p} \phi^{p'}_R(t)dt \Big\}^{1/p'}}_{II}. \end{aligned} \end{equation} Since $$ I_R(R^2 s)=\int_{R^2s}^{TR^2}k(t,s)\phi(t/R^2)dt=(R^2)^{1-\gamma} \int_s^T k(t,s)\phi(t)dt=(R^2)^{1-\gamma} I_1(s), $$ we have \begin{equation}\label{Nex.cua} \begin{aligned} II^{p'}&=R^2\int_0^T I_R (R^2t)^{-p'/p} \phi_R^{p'}(R^2t)dt\\ &=(R^2)^{1-(p'/p)(1-\gamma)} \int_0^T I_1(t)^{-p'/p} \phi^{p'}(t)dt\\ &= C(T)(R^2)^{1-(p'/p)(1-\gamma)}, \end{aligned} \end{equation} where $C(T)=\int_0^T \phi^{p'}(t)I_1(t)^{-p'/p}dt < \infty $ by \eqref{K5}. From \eqref{Nex.dos}--\eqref{Nex.cua} it follows that \begin{align*} &\lambda_R \Big\{ \int_0^{TR^2} w_R^p(t) I_{R}(t)dt\Big\}^{1/p} C(T)^{1/p'} (R^2)^{\frac{1}{p'}-\frac{1-\gamma}{p}}\\ &\geq \int_0^{TR^2}I_{R}(s) w_R^p(s)ds +w_R(0), \end{align*} and by Young's inequality, $$ \frac{1}{p}\int_0^{TR^2} w_R^p(t)I_R(t)dt+\frac{1}{p'} \lambda_R^{p'}C(T)(R^2)^{1-\frac{(1-\gamma)p'}{p}}\geq w_R(0) + \int_0^{TR^2}I_{R}(t) w_R^p(t)dt. $$ Thus, $$ \frac{1}{p'}\lambda_R^{p'} C(T)(R^2)^{1-\frac{(1-\gamma)p'}{p}}\geq w_R(0). $$ Since $\lambda_R=\lambda_1/ R^{2}$ we concluded that \begin{equation}\label{Nex.cin} w_R(0) \leq C(T)(\frac{\lambda_1^{p'}}{p'})(R^2)^{-p'+1-\frac{(1-\gamma)p'}{p}} =C'(T)(R^2)^{-\frac{2-\gamma}{p-1}}, \end{equation} where $C'(T)=[C(T)\lambda_1^{p'}]/p'$. On the other hand, for $\epsilon \in (0,1)$ small \begin{align*} w_R(0)&=\int_{B_R} u_0(x)\rho_R(x)dx\\ &\geq \Big(\inf_{R\geq |x|\geq \epsilon R}u_0(x)\Big) \int_{\{\epsilon R\leq |x|\leq R\}} \rho_R(x)dx\\ &\geq \Big(\inf_{R\geq |x|\geq \epsilon R}u_0(x)\Big) \int_{\{\epsilon \leq |x|\leq 1\}} \rho_1(x)dx. \end{align*} Thus, from \eqref{Nex.cin}, it follows that $$ C'(T)\geq \Big(\inf_{R\geq |x|\geq \epsilon R}|x|^{2(2-\gamma)/(p-1)}u_0(x)\Big) \int_{\{\epsilon \leq |x|\leq 1\}} \rho_1(x)dx. $$ Putting, $\epsilon=\kappa/R>0$ and letting $R\to \infty$ we have $\inf_{|x\geq \kappa}|x|^{2(\frac{2-\gamma}{p-1})}u_0(x)\leq C'(T)$. Since $C'(T)<\infty$ and $\kappa$ is arbitrary the conclusion follows. \end{proof} \begin{thebibliography}{00} \bibitem{Be} H. Bellout; \emph{Blow-up of solution of parabolic equations with nonlinear memory}, J. Diff. Eq. 70, (1987), 42-68. \bibitem{Cann} M. Cannone; \emph{A generalization of a theorem by Kato on Navier-Stokes equations}, Mat. 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