\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 $$\begin{gathered} u_t-\Delta u = \int_0^tk(t,s)|u|^{p-1}u(s)ds \quad \text{in } (0,\infty)\times \mathbb{R}^N,\\ u(0,x)=\psi(x) \quad \text{in }\mathbb{R}^N, \end{gathered}\label{In.uno}$$ 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{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}$$ In particular, if \alpha=(2-\gamma)/(p-1), then u_\lambda is also a solution of problem \eqref{In.uno}. A solution satisfying u=u_\lambda for all \lambda>0 is called a self-similar solution of problem \eqref{In.uno}. Note that, in this case, \psi(x)=\lambda^{2\alpha} \psi(\lambda x); that is, the function \psi is a homogeneous function of degree -2\alpha. Our objective is to determine the asymptotic behavior of global solutions of \eqref{In.uno} in terms of the self-similar solution w corresponding to the cases (see Theorem \ref{Th.sel} for details): \begin{itemize} \item[(i)] \alpha(p-1)=2-\gamma. \begin{gather*} w_t-\Delta w= \int_0^tk(t,s)|w|^{p-1}w(s)ds \quad \text{in } (0,\infty)\times \mathbb{R}^N,\\ w(0,x)=|x|^{-2\alpha} \quad\text{in }\mathbb{R}^N , \end{gather*} \item[(ii)] \alpha(p-1)>2-\gamma. \begin{gather*} w_t-\Delta w =0 \quad \text{in } (0,\infty)\times \mathbb{R}^N,\\ w(0,x) =|x|^{-2\alpha} \quad\text{in }\mathbb{R}^N. \end{gather*} \end{itemize} For \alpha(p-1)<2-\gamma, we show that there is no nonnegative global solution of \eqref{In.uno}, if w(0,x) \sim |x|^{-2\alpha} for |x| large enough (see Theorem \ref{Th.nex} for details). To show the existence of global solutions to \eqref{In.uno} we use a contraction mapping argument on the associated integral equation $$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}$$ 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 $$p>1+2(2-\gamma)/N \label{In.dos}$$ and \alpha \in (0,N/2) satisfying \eqref{In.sei}, \eqref{In.sie} and $$\frac{2-\gamma}{p-1}\leq \alpha <\frac{N}{2}. \label{In.tre}$$ Fix \tilde \alpha>0 such that $$\tilde \alpha \leq \frac{2-\gamma}{p-1}. \label{Inr.tre}$$ Let r_1>1 be given by Proposition \ref{pr.a}, and let r_2>1 be defined by r_2=\alpha r_1/\tilde \alpha. For every \varphi \in \mathcal{S}'(\mathbb{R}^N) define \mathcal{N} by $$\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}$$ where \beta_1=\alpha -N/(2r_1) and \beta_2=\tilde \alpha -N/(2r_2). Let M>0 be such that C=C(M)<1, where C is a positive constant given by \eqref{Gl.die}. Choose R>0 such that R+C M \leq M. If \varphi is a tempered distribution such that $$\mathcal{N}(\varphi)\leq R, \label{In.siec}$$ then there exits a unique global solution u of \eqref{In.uno} satisfying$$ \sup_{t>0}\{t^{\beta_1} \| u(t)\|_{r_1}, t^{\beta_2} \| u(t)\|_{r_2}\}\leq M. $$In addition, if \varphi, \psi satisfy \eqref{In.siec} and if u_\varphi and u_\psi respectively are the solutions of \eqref{Eq.int} with initial data \varphi,\psi, then $$\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}$$ Moreover, if \varphi, \psi are such that $$\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}$$ for some \delta \in (0,\delta_0), where \delta_0=1-l-(2-\gamma+\alpha)[1-N/(2r_1\alpha)]>0. Then $$\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}$$ 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 $$\sup_{t>0}t^{\beta_1+\delta} \| e^{t\Delta}(\varphi-\varphi_h) \|_{r_1}<\infty \label{In.cat}$$ for some \delta \in (0,\delta_0), where \delta_0=1-l-p(2-\gamma)/(p-1)+Np/(2r_1) when \alpha(p-1)=2-\gamma and given by Lemma \ref{Lem.sub} when \alpha(p-1)>2-\gamma. We have the following: \begin{itemize} \item[(i)] If \alpha(p-1)>2-\gamma, then \sup_{t>0}t^{\beta_1+\delta}\| u(t)-e^{t\Delta}\varphi_h\|_{r_1}\leq C_\delta, for some constant C_\delta>0. \item[(ii)] If \alpha(p-1)=2-\gamma and w is the solution of \eqref{In.uno} given by Theorem \ref{Th.sol} with initial data \varphi_h(we multiplied \varphi_h by a small constant so that \eqref{In.siec} is satisfied), then w is self-similar and \sup_{t>0} t^{\beta_1+\delta}\| u(t)-w(t)\|_{r_1}\leq C_\delta for some constant C_\delta>0. \end{itemize} \label{Th.sel} \end{theorem} \begin{remark} \rm The class of functions \varphi satisfying the condition \eqref{In.cat} is nonempty. Indeed, from Proposition \ref{Pr.h}(2), condition \eqref{In.cat} is satisfied for \varphi= (1-\eta)\varphi_h. \end{remark} In the following result, we analyze the non existence of global solutions of problem \eqref{In.uno}, under the assumption \begin{itemize} \item[(K5)] There exist T>0 and a nonnegative, non-increasing continuous function \phi \in C([0, \infty)) with integrable derivative such that \phi(0)=1 and \phi(t)=0 for t\geq T satisfying k(\cdot, t) \phi(\cdot) \in L^1(t, T) for t>0 and $$\label{K5} \int_0^T \phi(t)^{p'}\Big( \int_t^T k(s,t) \phi(s)ds\Big)^{-p'/p}dt <\infty,$$ where p' is the conjugate of p. \end{itemize} \begin{theorem} Let p>1 and let k be a nonnegative function satisfying conditions {\rm (K1)--(K3), (K5)}. If \psi \in C_0(\mathbb{R}^N), \psi \geq 0 satisfies \liminf_{|x|\to \infty} |x|^{2(2-\gamma)/(p-1)}\psi (x)=\infty and u is a corresponding nonnegative solution of problem \eqref{In.uno}, then u is not a global solution. \label{Th.nex} \end{theorem} \begin{remark}\rm Regarding Theorem \ref{Th.nex} we have the following statements: (i) Under conditions (K1)--(K3), existence of local solutions for \eqref{In.uno} in the class C([0,T), C_0(\mathbb{R}^N)) and initial data \psi \in C_0(\mathbb{R}^N), were studied in \cite{L}. In particular, we know that if k and \psi are nonnegative, then the solution of \eqref{In.uno} is nonnegative. (ii) Let k(t,s)=(t-s)^{-\gamma_1} s^{-\gamma_2} for 01/(p-1). Indeed, since \phi \leq 1, we have for t>0$$ \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. $$\end{remark} \section{Existence of global solutions} \subsection*{Proof of Proposition \ref{pr.a}} Let A=\frac{2\alpha}{N}(1-\frac{a}{2-\gamma+\alpha}). Since a>0 we conclude that A<2\alpha/N<1. From \eqref{In.sie} and \eqref{In.sei} we have A<2\alpha/[N(2-\gamma)] and A<\alpha/(2-\gamma+\alpha), respectively. Now, it is sufficient to choose r_1> 1 satisfying A<\frac{1}{r_1}<\min\{\frac{2\alpha}{N(2-\gamma)}, \frac{\alpha}{2-\gamma+\alpha},\frac{2\alpha}{N}\}. \begin{lemma} Assume the conditions \eqref{In.sei}-\eqref{Inr.tre}. Let r_2=\frac{\alpha r_1}{\tilde \alpha}, \beta_1=\alpha-\frac{N}{2r_1}, \beta_2=\tilde \alpha-\frac{N}{2r_2}, \frac{1}{\eta_1}=\frac{1}{pr_1}(\frac{2-\gamma}{\alpha} +1), \frac{1}{\eta_2}=\frac{1}{pr_1}(\frac{2-\gamma}{\alpha} +\frac{\tilde \alpha}{\alpha}), \theta_1=\frac{2-\gamma+\alpha-p\tilde \alpha}{p(\alpha-\tilde \alpha)}, and \theta_2=\frac{2-\gamma+(1-p)\tilde \alpha}{p(\alpha-\tilde \alpha)}. For i=1,2 we have \begin{itemize} \item[(i)] \eta_i \in [r_1,r_2] and \eta_i \in (p,r_ip). \item[(ii)]  \frac{p}{\eta_1}-\frac{1}{r_1} =\frac{p}{\eta_2}-\frac{1}{r_2}=\frac{2-\gamma}{r_1\alpha}<\frac{2}{N}. \item[(iii)] \theta_i \in [0,1],  \frac{1}{\eta_i} =\frac{\theta_i}{r_1}+\frac{(1-\theta_i)}{r_2}. \item[(iv)] \frac{1}{p}a>\theta_i \beta_1+(1-\theta_i)\beta_2, with \begin{gather*} \theta_1 \beta_1+(1-\theta_1)\beta_2 =\frac{1}{p}(2-\gamma+\alpha)(1-\frac{N}{2r_1\alpha}),\\ \theta_2 \beta_1+(1-\theta_2)\beta_2=\frac{1}{p} (2-\gamma+\tilde \alpha)(1-\frac{N}{2r_1\alpha}). \end{gather*} \item[(v)] 2-\gamma +\beta_i-\frac{N}{2} (\frac{p}{\eta_i}-\frac{1}{r_i})-p[\beta_1 \theta_i+\beta_2(1-\theta_i)]=0. \end{itemize} \label{Lem.b} \end{lemma} \begin{proof} (i) From \eqref{In.tre}, we see that \eta_1\geq r_1 and \eta_2\leq r_2. Since \tilde \alpha\leq \alpha, it follows from \eqref{In.tre} and \eqref{Inr.tre} that $$\label{Gl.un} 2-\gamma +\tilde \alpha\leq p\alpha, \ p \tilde \alpha \leq 2-\gamma+\alpha$$ respectively. From here, \eta_2\geq r_1 and \eta_1\leq r_2. The condition r_1> (2-\gamma)/\alpha+1 of Proposition \ref{pr.a}(i) and \gamma<2 ensure that \eta_1 \in (p,r_1p). Moreover, since r_1> (2-\gamma)/\alpha+1\geq (2-\gamma+\tilde \alpha)/\alpha and \gamma<2, we conclude that \eta_2 \in (p,r_2 p). Item (ii) follows from Proposition \ref{pr.a}(i). (iii) From \eqref{In.tre} and \eqref{Inr.tre} we get \theta_1\leq 1 and \theta_2\geq 0 respectively, and from \eqref{Gl.un} we see that \theta_2 \leq 1 and \theta_1\geq 0 respectively. We obtain (iv) from Proposition \ref{pr.a}(ii). \end{proof} \begin{proof}[Proof of Theorem \ref{Th.sol}] The proof is based on a contraction mapping argument. Let E be the set of Bochner measurable functions u: (0,\infty)\to L^{r_1}(\mathbb{R}^N)\cap L^{r_2}(\mathbb{R}^N), such that \| u \|_E= \sup_{t>0}\{t^{\beta_1}\| u(t)\|_{r_1}, t^{\beta_2}\| u(t)\|_{r_2}\}<\infty, where \beta_1=\alpha-N/(2r_1), \beta_2=\tilde \alpha-N/(2r_2). The space E is a Banach space. Let M>0 and K be the closed ball of radius M in E. Let \Phi_\varphi:K \to E be the mapping defined by $$\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}$$ We will prove that \Phi_\varphi is a strict contraction mapping on K. Let \varphi, \psi satisfying \eqref{In.siec} and u,v \in K. We will use several times the smoothing effect for the heat semigroup: if 1\leq s\leq r\leq \infty and \varphi \in L^r, then$$ \| 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{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} 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{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} 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{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} where $$C_1=\nu \int_0^{\eta_0}\sigma^{-l- p\Theta_1}d\sigma+ \eta_0^{-p\Theta_1}\int_{\eta_0}^1 |k(1,\sigma)|d\sigma. \label{Gl.cuno}$$ 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{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} 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 $$t^{\beta_2}\|\Phi_\varphi u(t)-\Phi_\psi v(t)\|_{r_2} \leq t^{\beta_2}\| e^{t\Delta}(\varphi-\psi)\|_{r_2}+C_2'\| u-v\|_E, \label{Gl.och}$$ 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 $$\| \Phi_\varphi(u)(t)-\Phi_\psi(v)(t)\|_E \leq \mathcal{N}(\varphi-\psi) + C \| u-v\|_E, \label{Gl.sei}$$ where $$C=\max\{C_1',C_2'\}. \label{Gl.die}$$ 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 $$\| u-v\|_{E,\delta}=\sup_{t>0}\{t^{\beta_1+\delta}\| u(t)\|_{r_1}, t^{\beta_2+\delta}\| v(t)\|_{r_2}\}. \label{Nord}$$ Proceeding as \eqref{Gl.tre} we obtain \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} 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{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} and $$C_{1,\delta}'=2pM^{p-1} C_{1,\delta}. \label{Gl.e}$$ Similarly, for 0<\delta<1-l-p\Theta_2, one can to obtain \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} 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), $$where $$C_\delta=\max\{C_{1,\delta}',C_{2,\delta}'\}. \label{Gl.d}$$ \end{proof} \section{Asymptotic behavior} The next result will be used in the proof of Theorem \ref{Th.sel}(1). \begin{lemma} Let l<1, \gamma<2, p>1 and a=\min\{1-l,2-\gamma\}. Assume \eqref{In.tre} and let \alpha satisfying \eqref{In.sei}, \eqref{In.sie} and \eqref{In.tre}. Let \tilde \eta satisfying \eqref{Inr.tre}. For \delta >0 we define \eta'\geq 1 by  \frac{1}{\eta_1'}=\frac{1}{pr_1}\big(\frac{2-\gamma+\delta}{\alpha}+1\big), and \theta_1'= \frac{2-\gamma+\delta+\alpha-p\tilde \alpha} {p(\alpha-\tilde \alpha)}. If \frac{2-\gamma}{p-1}<\alpha, then there exists \delta_0>0 small such that for all \delta \in (0,\delta_0]: \begin{itemize} \item[(i)] \eta'_1 \in [r_1,r_2] and \eta_1'\in (p,r_1 p), where r_2=(\alpha r_1)/\tilde \alpha. \item[(ii)]  \frac{N}{2}(\frac{p}{\eta'_1}-\frac{1}{r_1}) =\frac{N}{2r_1 \alpha}(2-\gamma+\delta)<1. \item[(iii)] \theta_1 \in [0,1],  \frac{1}{\eta'_1} =\frac{\theta_1'}{r_1}+\frac{1-\theta_1'}{r_2}. \item[(iv)] If \beta_1=\alpha-\frac{N}{2r_1} and \beta_2 =\tilde \alpha-\frac{N}{2r_2}, then$$ 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 $$( w_R )_t+ \lambda_R w_R \geq \int_0^t k(t,s)w_R^p(s)ds. \label{Nex.uno}$$ 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{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} where $$I_R(s)=\int_s^{TR^2} k(t,s) \phi_R(t)dt.$$ On the other hand, by H\"older's inequality, \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} 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 \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} 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 $$\label{Nex.cin} w_R(0) \leq C(T)(\frac{\lambda_1^{p'}}{p'})(R^2)^{-p'+1-\frac{(1-\gamma)p'}{p}} =C'(T)(R^2)^{-\frac{2-\gamma}{p-1}},$$ 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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