\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 41, 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 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/41\hfil Blow up of mild solutions] {Blow up of mild solutions of a system of partial differential equations with distinct \\ fractional diffusions} \author[J. Villa-Morales \hfil EJDE-2014/41\hfilneg] {Jos\'e Villa-Morales} \address{Jos\'e Villa Morales \newline Universidad Aut\'onoma de Aguascalientes, Departamento de Matem\'aticas y F\'isica \newline Aguascalientes, Aguascalientes, M\'exico} \email{jvilla@correo.uaa.mx} \thanks{Submitted June 18, 2013. Published February 5, 2014.} \thanks{Supported by grants 118294 of CONACyT and PIM13-3N from UAA} \subjclass[2000]{35K55, 35K45, 35B40, 35K20} \keywords{Blow up; weakly coupled system; mild solution; fractal diffusion; \hfill\break\indent nonautonomous initial value problem} \begin{abstract} We give a sufficient condition for blow up of positive mild solutions to an initial value problem for a nonautonomous weakly coupled system with distinct fractional diffusions. The proof is based on the study of blow up of a particular system of ordinary differential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Let $i\in \{1,2\}$ and $j=3-i$. In this paper we study blow up of positive mild solutions of \begin{equation} \label{wcs} \begin{gathered} \frac{\partial u_i(t,x) }{\partial t} = g_i(t) \Delta _{\alpha _i}u_i(t,x) +h_i(t) u_{j}^{\beta _i}(t,x) ,\quad t>0,\; x\in \mathbb{R}^{d}, \\ u_i(0,x) = \varphi _i(x) ,\quad x\in \mathbb{R}^{d}, \end{gathered} \end{equation} where $\Delta _{\alpha _i}=-(-\Delta ) ^{\alpha _i/2}$, $0<\alpha _i\leq 2$, is the $\alpha _i$-Laplacian, $\beta _i\geq 1$ are constants, $\varphi _i$ are non negative, not identically zero, bounded continuous functions and $h_i,g_i:(0,\infty )\to [0,\infty) $ are continuous functions. If there exist a solution $(u_1,u_2) $ of \eqref{wcs} defined in $[0,\infty ) \times \mathbb{R}^{d}$, we say that $(u_1,u_2) $ is a global solution, on the other hand if there exists a number $t_e<\infty $ such that $(u_1,u_2) $ is unbounded in $[0,t] \times \mathbb{R}^{d}$, for each $t>t_e$, we say that $(u_1,u_2) $ blows up in finite time. The associated integral system of \eqref{wcs} is \begin{equation} \label{ecintegrl} \begin{aligned} u_i(t,x) &= \int_{\mathbb{R}^{d}}p_i(G_i(t) ,y-x) \varphi _i(y)dy \\ &\quad +\int_{0}^{t}\int_{\mathbb{R}^{d}}p_i(G_i(s,t) ,y-x) h_i(s) u_{j}^{\beta _i}(s,y) \,dy\,ds. \end{aligned} \end{equation} Here $p_i(t,x) $ denote the fundamental solution of $\frac{\partial }{\partial t}-\Delta _{\alpha _i}$ and \begin{equation*} G_i(s,t) =\int_{s}^{t}g_i(r) dr,\quad 0\leq s\leq t, \end{equation*} where $G_i(t)=G_i(0,t)$. We say that $(u_1,u_2) $ is a mild solution of \eqref{wcs} if $(u_1,u_2) $ is a solution of \eqref{ecintegrl}. Our main result reads as follows. \begin{theorem}\label{TeoPr} Assume that $\beta _i\beta _{j}>1$ and \begin{equation} \lim_{t\to \infty }G_i(t) =\infty . \label{condG} \end{equation} Let $a\in \{1,2\}$ such that \begin{equation} \alpha _{a}=\min \{\alpha _1,\alpha _2\}\text{ \ and \ }b=3-a. \label{defa} \end{equation} Define \begin{equation} f_i(t)=h_i(t) \Big(\frac{G_b(t) }{ (G_{j}(t) ^{\alpha _b/\alpha _{j}}+G_b(t) )^{\beta _i}}\Big) ^{d/\alpha _b},\quad t>0. \label{dfi} \end{equation} Then the positive solution of \eqref{ecintegrl} blows up in finite time if \begin{equation} \int_{\cdot }^{\infty }F(s)ds=\infty , \label{cenfexplo} \end{equation} where \begin{equation} F(t)=\Big(f_i(t)^{1/(\beta _i+1)}f_{j}(t)^{1/(\beta _{j}+1)}\Big) ^{(\beta _i+1)(\beta _{j}+1)/(\beta _i+\beta _{j}+2)}. \label{dfexpl} \end{equation} \end{theorem} It is well known that a classical solution is a mild solution. Therefore, if we give a sufficient condition for blow up of positive solutions to \eqref{ecintegrl} then we have a condition for blow up of classical solutions to \eqref{wcs}. \begin{corollary}\label{SegRe} Moreover, assume that $\rho _i>0$, $\sigma _i>-1$ and \begin{equation} \label{conexpledi} \begin{aligned} &\frac{d\rho _b}{\alpha _b}+\frac{\sigma _i(1+\beta _{j})+\sigma _{j}(1+\beta _i)}{\beta _i+\beta _{j}+2}+1\\ &\geq \frac{d}{\beta _i+\beta _{j}+2}\big[\beta _i(\beta _{j}+1)\max \{ \frac{\rho _{j}}{\alpha _{j}},\frac{\rho _b}{\alpha _b} \} +\beta _{j}(\beta _i+1)\max \{ \frac{\rho _i}{\alpha _i} ,\frac{\rho _b}{\alpha _b}\} \big] , \end{aligned} \end{equation} then each (classical) solution to \begin{equation} \label{Edparpart} \begin{gathered} \frac{\partial u_i(t,x) }{\partial t} = \rho _it^{\rho_i-1}\Delta _{\alpha _i}u_i(t,x) +t^{\sigma _i}u_{j}^{\beta _i}(t,x) ,\quad t>0,\; x\in\mathbb{R}^{d}, \\ u_i(0,x) = \varphi _i(x) ,\quad x\in \mathbb{R}^{d}. \end{gathered} \end{equation} blow up in finite time. \end{corollary} In applied mathematics it is well known the importance of the study of equations such as \eqref{wcs}. In fact, for example, they arise in fields like molecular biology, hydrodynamics and statistical physics \cite{S-Z}. Also, notice that generators of the form $g_i(t) \Delta _{\alpha_i}$ arise in models of anomalous growth of certain fractal interfaces \cite{M-W}. There are many related works. Here are some of them: $\bullet$ When $\alpha _1=\alpha _2=2$, $\rho _1=\rho _2=1$, $\sigma _1=\sigma _2=0$ and $\varphi _1=\varphi _2$ in \eqref{Edparpart}, Fujita \cite{Fujita} showed that if $d<\alpha _1/\beta _1$, then for any non-vanishing initial condition the solution of \eqref{Edparpart} is infinite for all $t$ large enough. $\bullet$ When $\alpha _1=\alpha _2$, $\rho _1=\rho _2$, $\sigma _1=\sigma _2$ and $\varphi _1=\varphi _2$ in \eqref{Edparpart}, P\'erez and Villa \cite{P-V} showed that if $\sigma _1+1\geq d\rho _1(\beta _1-1)/\alpha _1$, then the solutions of \eqref{Edparpart} blow up in finite time. $\bullet$ When $\alpha _1=\alpha _2=2$ and $\rho _1=\rho _2=1$ in \eqref{Edparpart}, Uda \cite{Uda} proved that all positive solutions of \eqref{Edparpart} blow up if $\max \{ \frac{(\sigma _2+1) \beta _1+\sigma _1+1}{\beta _1\beta _2-1}, \frac{(\sigma_1+1) \beta _2+\sigma _2+1}{\beta _1\beta _2-1}\} \geq \frac{d}{2}$. $\bullet$ When $\alpha _1=\alpha _2$, $g_1(t)=g_2(t)=t^{\rho -1}$, $\rho >0$, and $h_1(t)=h_2(t)=1$ in \eqref{wcs}, P\'{e}rez \cite{P} proved that every positive solution blows up in finite time if $\min \{ \frac{\alpha _1}{\rho (\beta _1-1) },\frac{\alpha _1}{\rho (\beta _2-1) }\} >d$. $\bullet$ When $\rho _1=\rho _2=1$ and the nonlinear terms in \eqref{Edparpart} are of the form $h(t,x) u^{\beta _i}$, $h(t,x) =O(t^{\sigma }| x| ^{\gamma }) $, Guedda and Kirane \cite{G-K-2} also studied blow up. Other related results (when $\alpha _1=\alpha _2=2$) can be found, for example in \cite{andre,fila,koby, moch} and references therein. It is worth while to mention that Guedda and Kirane \cite{G-K-2} observed that to reduce the study of blow up of \eqref{wcs} to a system of ordinary differential equations we must have a comparison result between $p_i(t,x) $ and $p_{j}(t,x) $. Therefore, the goal of this paper is to use the comparison result given in \cite[Lemma 2.4]{M-V} to follow the usual approach, see among others \cite{Sug} or \cite{G-K-1}. When $\alpha _1=\alpha _2=2$, $\rho _1=\rho _2=1$ and $\sigma _1=\sigma _2=0$ the Uda condition \eqref{dUda}, the P\'{e}rez condition \eqref{dAro} and the condition \eqref{conexpledi} become \begin{gather} d \leq \frac{2(\max \{\beta _1,\beta _2\}+1)}{\beta _1\beta _2-1} =C_{U}, \label{dUda} \\ d <\frac{2}{\max \{\beta _1,\beta _2\}-1}=C_{A}, \label{dAro} \\ d \leq \frac{\beta _1+\beta _2+2}{\beta _1\beta _2-1}=C_{V}, \label{dVilla} \end{gather} respectively. Since $C_{A}\leq C_{V}\leq C_{U}$ we see that the Uda condition \eqref{dUda} is the best. Also, from this we see that $C_{V}$, given in \eqref{dVilla}, is not the optimal bound (critical dimension), but we believe that it is the best we can get by constructing a convenient subsolution of the solution of \eqref{ecintegrl}. In fact, the condition \eqref{conexpledi} coincides with the condition for blow up given by P\'erez and Villa \cite{P-V}. This article is organized as follows. In Section 1 we prove the existence of local solutions for the equation \eqref{ecintegrl}. In Section 2 we give some preliminary results and discuses a sufficient condition for blow up of a system of ordinary differential equations, finally in Section 3 we prove the main result and its corollary. \section{Existence of local solution} The existence of local solutions for the weakly coupled system \eqref{ecintegrl} follows form the fix-point theorem of Banach. We begin introducing some normed linear spaces. By $L^{\infty }(\mathbb{R}^{d}) $ we denote the space of all real-valued functions essentially bounded defined on $\mathbb{R}^{d}$. Let $\tau >0$ be a real number that we will fix later. Define \begin{equation*} E_{\tau }=\{ (u_1,u_2) :[0,\tau ] \to L^{\infty }(\mathbb{R}^{d}) \times L^{\infty }( \mathbb{R}^{d}) ,\,|||(u_1,u_2) |||<\infty \} , \end{equation*} where \begin{equation*} |||(u_1,u_2) |||=\sup_{0\leq t\leq \tau }\{ \| u_1(t) \| _{\infty }+\| u_2(t)\| _{\infty }\} . \end{equation*} Then $E_{\tau }$ is a Banach space and the sets, $R>0$, \begin{gather*} P_{\tau } = \{ (u_1,u_2) \in E_{\tau }\text{, } u_1\geq 0,u_2\geq 0\} , \\ B_{\tau } = \{ (u_1,u_2) \in E_{\tau }\text{, } |||(u_1,u_2) |||\leq R\} \text{,} \end{gather*} are closed subspaces of $E_{\tau }$. \begin{theorem}\label{exisloc} There exists a $\tau =\tau (\varphi _1,\varphi _2) >0$ such that the integral system \eqref{ecintegrl} has a local solution in $B_{\tau }\cap P_{\tau }$. \end{theorem} \begin{proof} Define the operator $\Psi :B_{\tau }\cap P_{\tau }\to B_{\tau }\cap P_{\tau }$, by \begin{align*} &\Psi (u_1,u_2) (t,x) \\ & =\Big(\int_{\mathbb{R}^{d}}p_1(G_1(t) ,y-x) \varphi _1(y) dy,\int_{\mathbb{R} ^{d}}p_2(G_2(t) ,y-x) \varphi _2( y) dy\Big) \\ &\quad +\Big(\int_{0}^{t}\int_{\mathbb{R}^{d}}p_1( G_1(s,t) ,y-x) h_1(s) u_2^{\beta _1}(s,y) \,dy\,ds, \\ & \quad \int_{0}^{t}\int_{\mathbb{R}^{d}}p_2( G_2(s,t) ,y-x) h_2(s) u_1^{\beta _2}(s,y) \,dy\,ds\Big) . \end{align*} We choose $R$ sufficiently large such that $\Psi$ is onto $B_{\tau }\cap P_{\tau }$. We are going to show that $\Psi $ is a contraction, therefore $\Psi $ has a fix point. Let $(u_1,u_2) ,(\tilde{u}_1,\tilde{u} _2) \in B_{\tau }\cap P_{\tau }$ with $u_i(0)=\widetilde{u}_i(0)$, \begin{align*} &|||\Psi (u_1,u_2) -\Psi (\tilde{u}_1,\tilde{u} _2) ||| \\ &= |||\Big(\int_{0}^{t}\int_{\mathbb{R}^{d}}p_1( G_1(s,t) ,y-x) h_1(s) [u_2^{\beta _1}(s,y) -\tilde{u}_2^{\beta _1}(s,y) ] \,dy\,ds, \\ &\quad \int_{0}^{t}\int_{\mathbb{R}^{d}}p_2( G_2(s,t) ,y-x) h_2(s) [u_1^{\beta _2}(s,y) -\tilde{u}_1^{\beta _2}(s,y) ] \,dy\,ds\Big) ||| \\ &\leq \sum_{i=1}^{2}\sup_{t\in [0,\tau ] }\int _{0}^{t}\int_{\mathbb{R}^{d}}p_i(G_i( s,t) ,y-x) h_i(s) \| u_{j}^{\beta _i}(s) -\tilde{u}_{j}^{\beta _i}(s) \| _{\infty }\,dy\,ds. \end{align*} Let $w,z>0$ and $p\geq 1$, then \begin{equation*} | w^{p}-z^{p}| \leq p(w\vee z)^{p-1}| w-z| . \end{equation*} Using the previous elementary inequality we obtain \begin{align*} | u_{j}^{\beta _i}(s,x) -\tilde{u}_{j}^{\beta_i}(s,x) | &\leq \beta _i(u_{j}(s,x) \vee \tilde{u}_{j}(s,x) ) ^{\beta _i-1}| u_{j}(s,x) -\tilde{u}_{j}(s,x)| \\ &\leq \beta _iR^{\beta _i-1}\| u_{j}-\tilde{u}_{j}\| _{\infty }\text{,} \end{align*} from this we deduce \begin{align*} |||\Psi (u_1,u_2) -\Psi (\tilde{u}_1,\tilde{u} _2) ||| &\leq \sum_{i=1}^{2}\sup_{t\in [0,\tau ] }\int_{0}^{t}h_i(s) \beta _iR^{\beta _i-1}\| u_i(s)- \tilde{u}_i(s) \| _{\infty }ds \\ &\leq (\sum_{i=1}^{2}\beta _iR^{\beta _i-1}\int_{0}^{\tau }h_i( s) ds)|||(u_1,u_2) -(\tilde{u}_1,\tilde{u}_2) |||. \end{align*} Since $\lim_{t\to 0}\int_{0}^{t}h_i(s) ds=0$ , we can choose $\tau >0$ small enough such that $\Psi $ is a contraction. \end{proof} \section{Preliminary results} \begin{lemma} \label{pden} For any $s,t>0$ and any $x,y\in \mathbb{R}^{d}$, we have \begin{itemize} \item[(i)] $p_i(ts,x) =t^{-d/\alpha_i }p_i(s,t^{-1/\alpha_i}x)$. \item[(ii)] $p_i(t,x) \geq (\frac{s}{t}) ^{d/\alpha_i}p_i(s,x) $, for $t\geq s$. \item[(iii)] $p_i(t,\frac{1}{\tau }(x-y) ) \geq p_i(t,x) p_i(t,y) $, if $p_i(t,0) \leq 1$ and $\tau \geq 2$. \item[(iv)] There exist constants $c_i\in (0,1] $ such that \begin{equation} p_i(t,x) \geq c_ip_b(t^{\alpha _b/\alpha _i},x), \label{reldensi} \end{equation} where $b$ is as in \eqref{defa}. \end{itemize} \end{lemma} For the proof of (i)-(iii) see \cite[Section 2]{Sug}, and for (iv) see \cite[Lemma 2.4]{M-V}. \begin{lemma} \label{cotini} Let $u_i$ be a positive solution of \eqref{ecintegrl}, then \begin{equation} u_i(t_{0},x) \geq c_i(t_{0})p_b\big(2^{-\alpha_b}G_i(t_{0}) ^{\alpha _b/\alpha _i},x\big) ,\quad \forall x\in \mathbb{R}^{d}, \label{estuini} \end{equation} where \begin{equation*} c_i(t_{0})=c_i2^{-d}\int_{\mathbb{R}^{d}}p_b\big( G_i(t_{0}) ^{\alpha _b/\alpha _i},2y\big) \varphi_i(y)dy \end{equation*} and $t_{0}>1$ is large enough such that \begin{equation} p_b(G_i(t_{0}) ^{\alpha _b/\alpha _i},0)\leq 1. \label{tre} \end{equation} \end{lemma} \begin{proof} By (i) of Lemma \ref{pden} and \eqref{condG} there exist $t_{0}$ large enough such that \begin{equation} p_b\big(G_i(t_{0}) ^{\alpha _b/\alpha _i},0\big) =G_i(t_{0}) ^{-d/\alpha _i}p_b(1,0) \leq 1. \label{cpsm1} \end{equation} Using (iii) and (i) of Lemma \ref{pden}, we obtain \begin{align*} p_b\big(G_i(t_{0}) ^{\alpha _b/\alpha _i},y-x\big) &\geq p_b\big(G_i(t_{0}) ^{\alpha _b/\alpha _i},2x\big) p_b(G_i(t_{0}) ^{\alpha _b/\alpha_i},2y) \\ &= 2^{-d}p_b\big(2^{-\alpha _b}G_i(t_{0}) ^{\alpha _b/\alpha _i},x\big) p_b(G_i(t_{0}) ^{\alpha _b/\alpha _i},2y) . \end{align*} From \eqref{ecintegrl}, (iv) of Lemma \ref{pden} and the previous inequality we conclude \begin{equation*} u_i(t_{0},x)\geq (c_i2^{-d}\int_{\mathbb{R} ^{d}}p_b(G_i(t_{0}) ^{\alpha _b/\alpha _i},2y) \varphi _i(y)dy) p_b(2^{-\alpha _b}G_i(t_{0}) ^{\alpha _b/\alpha _i},x) . \end{equation*} Getting the desired result.\hfill \end{proof} Observe that the semigroup property implies \begin{equation} \begin{aligned} &u_i(t+t_{0},x)\\ &=\int_{\mathbb{R}^{d}}p_i( G_i(t_{0},t+t_{0}) ,y-x) u_i(t_{0},y)dy \\ &\quad +\int_{0}^{t}\int_{\mathbb{R}^{d}}p_i( G_i(s+t_{0},t+t_{0}) ,y-x) h_i(s+t_{0}) u_{j}^{\beta _i}(s+t_{0},y) \,dy\,ds. \end{aligned} \label{umtei} \end{equation} Let \begin{equation} \bar{u}_i(t) =\int_{\mathbb{R}^{d}}p_b( G_b(t),x) u_i(t,x) dx,\quad t\geq 0. \label{dubar} \end{equation} \begin{lemma} \label{caresp} If $\overline{u}_i$ blow up in finite time, then $u_i$ also does. \end{lemma} \begin{proof} Let $t_{0}$ be given in Lemma \ref{pden}. Take $t_{0}t_{j}$ large enough such that \begin{equation*} G_i(t_{j}+t_{0},t+t_{0}) >2^{\alpha _i}G_b(t_{j}+t_{0}) ^{\alpha _i/\alpha _b}. \end{equation*} Thus, for each $0\leq s\leq t_{j}$, \begin{align*} \int_{s+t_{0}}^{t+t_{0}}g_i(r) dr &\geq \int_{t_{j}+t_{0}}^{t+t_{0}}g_i(r) dr \\ &>2^{\alpha _i}\Big(\int_{0}^{t_{j}+t_{0}}g_b(r) dr\Big) ^{\alpha _i/\alpha _b}\\ &\geq 2^{\alpha _i}\Big( \int_{0}^{s+t_{0}}g_b(r) dr\Big) ^{\alpha _i/\alpha _b}, \end{align*} hence \begin{equation*} \tau _i=\frac{G_i(s+t_{0},t+t_{0}) ^{1/\alpha _i}}{ G_b(s+t_{0}) ^{1/\alpha _b}}\geq 2. \end{equation*} On the other hand, \eqref{cpsm1} implies \begin{equation*} p_b(G_b(s+t_{0}) ,0) \leq p_b( G_b(t_{0}) ,0) =G_b(t_{0}) ^{-d/\alpha_b}p_b(1,0) \leq 1. \end{equation*} Using (i) and (iii) of Lemma \ref{pden} we obtain \begin{align*} p_b(G_i(s+t_{0},t+t_{0}) ^{\alpha _b/\alpha_i},y-x) &= \tau _i^{-d}p_b(G_b(s+t_{0}) , \frac{1}{\tau _i}(y-x)) \\ &\geq \tau _i^{-d}p_b(G_b(s+t_{0}) ,x) p_b(G_b(s+t_{0}) ,y) . \end{align*} From \eqref{umtei}, (iv) of Lemma \ref{pden} and Jensen's inequality we deduce that \begin{align*} &u_i(t+t_{0},x) \\ &\geq c_i\int_{0}^{t_{j}}h_i(s+t_{0}) \int_{\mathbb{R}^{d}}p_b\big( G_i(s+t_{0},t+t_{0})^{\alpha _b/\alpha _i},y-x\big) u_{j}(s+t_{0},y) ^{\beta _i}\,dy\,ds \\ &\geq c_i\int_{0}^{t_{j}}\tau _i^{-d}h_i( s+t_{0}) p_b(G_b(s+t_{0}) ,x) \overline{u} _{j}(s+t_{0}) ^{\beta _i}ds. \end{align*} Then $u_i(t+t_{0},x) =\infty $. The definition \eqref{dubar} of $\overline{u}_i$\ implies that $\overline{u}_i$ blows up in finite time, and working as before we conclude that $u_{j}$ also blows up in finite time.\hfill \end{proof} In what follows by $c$ we mean a positive constant that may change from place to place. The following result is interesting in itself. \begin{proposition}\label{explocombi} Let $v_i,f_i:[t_{0},\infty )\to \mathbb{R}$ be continuous functions such that \begin{equation*} v_i(t)\geq k+k\int_{t_{0}}^{t}f_i(s)v_{j}(s) ^{\beta _i}ds,\quad t\geq t_{0}, \end{equation*} where $k>0$ is a constant. Then $v_i$ blow up in finite time if \begin{equation*} \int_{t_{0}}^{\infty }\Big(f_i(s)^{1/(\beta _i+1)}f_{j}(s)^{1/(\beta _{j}+1)}\Big) ^{(\beta _i+1) (\beta _{j}+1)/(\beta _i+\beta _{j}+2)}ds=\infty . \end{equation*} \end{proposition} \begin{proof} Consider the system \begin{equation} z_i(t)=\frac{k}{2}+k\int_{t_{0}}^{t}f_i(s)z_{j}(s) ^{\beta _i}ds,\quad t\geq t_{0}. \label{ecauxcomexplo} \end{equation} Let $N_i=\{t>t_{0}:z_i(s)0$, \begin{equation*} y^{\beta _i+1}+x^{\beta _{j}+1} \geq c(xy)^{(\beta _i+1) (\beta _{j}+1)/(\beta _i+\beta _{j}+2)}. \end{equation*} Using this and \eqref{dprosol} we obtain \begin{align*} Z'(t) &\geq c\Big(f_i(t)^{1/(\beta _i+1)}f_{j}(t)^{1/(\beta _{j}+1)}\Big) ^{(\beta _i+1) (\beta _{j}+1)/(\beta_i+\beta _{j}+2)}\\ &\quad\times \big(z_i(t)z_{j}(t)\big) ^{(\beta _i\beta _i-1) /(\beta _i+\beta _{j}+2)} \\ &= cF(t)\exp (\frac{\beta _i\beta _i-1}{\beta _i+\beta _{j}+2} Z(t)) , \end{align*} where $F$ is like \eqref{dfexpl}. Consider the equation \begin{equation*} H'(t)=cF(t)\exp (cH(t)) ,\quad t>t_{0},\; H(t_{0})=2\log \frac{k}{2}. \end{equation*} whose solution is \begin{equation*} H(t)=\log \Big(e^{-cH(t_{0})}-c^{2}\int_{t_{0}}^{t}F(s)ds\Big) ^{-1/c}. \end{equation*} Since $H\leq Z$ then the result follows from \eqref{cenfexplo}.\hfill \end{proof} \section{Blow up results} \begin{proof}[Proof of Theorem \protect\ref{TeoPr}] From \eqref{umtei} and \eqref{reldensi}, \begin{align*} & u_i(t+t_{0},x)\\ &\geq \int_{\mathbb{R}^{d}}c_ip_b (G_i(t_{0},t+t_{0}) ^{\alpha _b/\alpha _i},y-x) u_i(t_{0},y)dy \\ &\quad +\int_{0}^{t}h_i(s+t_{0}) \int_{\mathbb{R} ^{d}}c_ip_b(G_i(s+t_{0},t+t_{0}) ^{\alpha _b/\alpha _i},y-x) u_{j}^{\beta _i}(s+t_{0},y) \,dy\,ds. \end{align*} Multiplying by $p_b(G_b(t+t_{0}) ,x) $ and integrating with respect to $x$ we obtain \begin{align*} \bar{u}_i(t+t_{0}) &\geq c_i\int_{\mathbb{R }^{d}}p_b(G_i(t_{0},t+t_{0}) ^{\alpha _b/\alpha _i}+G_b(t+t_{0}) ,y) u_i(t_{0},y)dy \\ &\quad +c_i\int_{0}^{t}h_i(s+t_{0}) \int_{ \mathbb{R}^{d}}p_b(G_i(s+t_{0},t+t_{0}) ^{\alpha _b/\alpha _i}+G_b(t+t_{0}) ,y) \\ &\quad \times u_{j}^{\beta _i}(s+t_{0},y) \,dy\,ds. \end{align*} The property (ii) of Lemma \ref{pden} and Jensen's inequality, rendering \begin{align*} \bar{u}_i(t+t_{0}) &\geq c_i\int_{\mathbb{R}^{d}}p_b\Big(G_i(t_{0},t+t_{0})^{\alpha _b/\alpha _i} +G_b( t+t_{0}) ,y\Big) u_i(t_{0},y)dy \\ &\quad +c_i\int_{0}^{t}(\frac{G_b(s+t_{0}) }{ G_i(s+t_{0},t+t_{0}) ^{\alpha _b/\alpha _i} +G_b(t+t_{0}) }) ^{d/\alpha _b} \\ &\quad \times h_i(s+t_{0}) (\bar{u}_{j}(s+t_{0})) ^{\beta _i}ds. \end{align*} Moreover, \eqref{estuini} and that $G_i(s,\cdot ) $ is increasing implies \begin{align*} \bar{u}_i(t+t_{0}) &\geq c_ic_i(t_{0})p_b( 1,0) (2G_i(t+t_{0})^{\alpha _b/\alpha _i}+2G_b( t+t_{0}) ) ^{-d/\alpha _b} \\ &\quad +c_i\int_{0}^{t}h_i(s+t_{0}) (\frac{ G_b(s+t_{0}) }{2G_i(t+t_{0}) ^{\alpha _b/\alpha _i}+2G_b(t+t_{0}) }) ^{d/\alpha _b}(\bar{u}_{j}(s+t_{0}) ) ^{\beta _i}ds. \end{align*} Let \begin{equation*} v_i(t+t_{0})=\bar{u}_i(t+t_{0}) (G_i(t+t_{0}) ^{\alpha _b/\alpha _i}+G_b(t+t_{0}) )^{d/\alpha _b}, \end{equation*} then \begin{equation*} v_i(t+t_{0})\geq c+c\int_{0}^{t}f_i(s+t_{0})v_{j}( s+t_{0}) ^{\beta _i}ds, \end{equation*} where $f_i$ is defined in \eqref{dfi}. The result follows from Proposition \ref{explocombi} and Lemma \ref{caresp}. \end{proof} \begin{proof}[Proof of Corollary \protect\ref{SegRe}] Let \begin{equation*} f_i(t)=\frac{t^{\sigma _i+d\rho _b/\alpha _b}}{(t^{\rho _{j}\alpha _b/\alpha _{j}}+t^{\rho _b})^{d\beta _i/\alpha _b}}, \end{equation*} then \begin{equation*} F(t)=\frac{t^{\theta _1}}{(t^{\theta _2}+t^{\theta _3})^{\theta _{4}}(t^{\theta _5}+t^{\theta _3})^{\theta _6}} \end{equation*} where \begin{gather*} \theta _1 = \frac{d\rho _b}{\alpha _b}+\frac{\sigma _i(1+\beta _{j})+\sigma _{j}(1+\beta _i)}{2+\beta _i+\beta _{j}}, \\ \theta _2 = \frac{\rho _{j}\alpha _b}{\alpha _{j}},\quad \theta _3=\rho _b,\ \ \theta _{4}=\frac{d\beta _i(\beta _{j}+1)}{\alpha _b(2+\beta _i+\beta _{j})}, \\ \theta _5 = \frac{\rho _i\alpha _b}{\alpha _i},\quad \theta _6=\frac{d\beta _{j}(\beta _i+1)}{\alpha _b(2+\beta _i+\beta _{j})}. \end{gather*} Using the elementary inequality \begin{equation*} (t^{\theta _2}+t^{\theta _3})^{\theta _{4}}(t^{\theta _5}+t^{\theta _3})^{\theta _6}\leq (2t^{\max \{\theta _2,\theta _3\}})^{\theta _{4}}(2t^{\max \{\theta _5,\theta _3\}})^{\theta _6},\quad t>1, \end{equation*} the result follows. \end{proof} \begin{thebibliography}{99} \bibitem{andre} Andreucci, D.; Herrero, M. 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