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Register a new userSharp conditions on global existence and blow-up in a degenerate two-species and cross-attraction system
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We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $dgeq 3$ and find two critical curves intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.
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We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term. We show that for small enough initial data, if $rho(x)sim frac{1}{left(log|x|right)^{alpha}|x|^{2}}$ as $|x|to infty$, then solutions globally exist for any $p>1$. On the other hand, when $rho(x)simfrac{left(log|x|right)^{alpha}}{|x|^{2}}$ as $|x|to infty$, if the initial datum is small enough then one has global existence of the solution for any $p>m$, while if the initial datum is large enough then the blow-up of the solutions occurs for any $p>m$. Such results generalize those established in [27] and [28], where it is supposed that $rho(x)sim |x|^{-q}$ for $q>0$ as $|x|to infty$.
Perhaps the most classical diffusion model for chemotaxis is the Patlak-Keller-Segel system begin{equation} label{ks0} left{ begin{aligned} u_t =& Delta u - abla cdot(u abla v) quad inn R^2times(0,infty), v =& (-Delta_{R^2})^{-1} u := frac 1{2pi} int_{R^2} , log frac 1{|x-z|} u(z,t), dz, & qquad u(cdot ,0) = u_0 ge 0quadhbox{in } R^2. end{aligned} right. end{equation} We consider the {em critical mass case} $int_{R^2} u_0(x), dx = 8pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^*$ with mass $8pi$ such that for any initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$ of equ{ks0} is globally defined and blows-up in infinite time. As $tto+infty $ it has the approximate profile $$ u(x,t) approx frac 1{la^2} U_0left (frac {x-xi(t)}{la(t)} right ), quad U_0(y)= frac{8}{(1+|y|^2)^2}, $$ where $la(t) approx frac c{sqrt{log t}}, xi(t)to q $ for some $c>0$ and $qin R^2$
We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term $rho(x) u^p$ with $p>1$; this is a mathematical model of a thermal evolution of a heated plasma (see [25]). The density decays slowly at infinity, in the sense that $rho(x)lesssim |x|^{-q}$ as $|x|to +infty$ with $qin [0, 2).$ We show that for large enough initial data, solutions blow-up in finite time for any $p>1$. On the other hand, if the initial datum is small enough and $p>bar p$, for a suitable $bar p$ depending on $rho, m, N$, then global solutions exist. In addition, if $p<underline p$, for a suitable $underline pleq bar p$ depending on $rho, m, N$, then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypotehsis that $qin [0, epsilon)$ for $epsilon>0$ small enough, when $mleq p<underline p$. Observe that $underline p=bar p$, if $rho(x)$ is a multiple of $|x|^{-q}$ for $|x|$ large enough. Such results are in agreement with those established in [41], where $rho(x)equiv 1$. The case of fast decaying density at infinity, i.e. $qgeq 2$, is examined in [31].
We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {it fast} at infinity, in the sense that $rho(x)sim |x|^{-q}$ as $|x|to +infty$ with $qge 2.$ In the case when $q=2$, if $p$ is bigger than $m$, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when $q>2$, we show that existence of global in time solutions always prevails. The case of {it slowly} decaying density at infinity, i.e. $qin [0,2)$, is examined in [41].
This work considers a chemotaxis system for multi-species that includes birth or death rate terms, which implies no mass preservation of the populations. We aim to show the convergence to a $L^{infty} - $weak solutions, that is local in time, of the JKO - scheme arising from the Optimal Transport Theory, in the spirit of [35,14]. Currently, $L^{infty}$ solutions have shown to be important in order to get uniqueness. Since death rate case does not ensure global solutions, for arbitrary initial data, in this framework, it could be interest to analyze the Blowing-up phenomenon of this system. Therefore, in the last section, we get sufficient conditions that implies blowing-up phenomenon in finite time and we draw several stages where this occurs. This last part can be seen as a partial generalization of the blowing-up results in [16].
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