Sharp condition for blow-up and global existence in a two species chemotactic Keller–Segel system in 2

2012 ◽  
Vol 24 (2) ◽  
pp. 297-313 ◽  
Author(s):  
ELIO ESPEJO ◽  
KARINA VILCHES ◽  
CARLOS CONCA
Keyword(s):  
Global Existence ◽  
Global Solution ◽  
Blow Up ◽  
Critical Mass ◽  
Single Species ◽  
Radial Symmetry ◽  
Initial Mass ◽  
Threshold Curve ◽  
Sharp Condition ◽  
Additional Question

For the parabolic–elliptic Keller–Segel system in 2 it has been proved that if the initial mass is less than 8π/χ, a global solution exists, and in case the initial mass is larger than 8π/χ, blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the two species case that allows us to determine if the system is a blow-up or a global in time solution. No radial symmetry is assumed.

Remarks on the blowup and global existence for a two species chemotactic Keller–Segel system in2

2011 ◽  
Vol 22 (6) ◽  
pp. 553-580 ◽  
Author(s):  
CARLOS CONCA ◽  
ELIO ESPEJO ◽  
KARINA VILCHES
Keyword(s):  
New York ◽  
Global Existence ◽  
Critical Mass ◽  
Single Species ◽  
Two Dimensions ◽  
Threshold Condition ◽  
Two Dimensional ◽  
Additional Question ◽  
General Data ◽  
Dimensional Domain

For the Keller–Segel model, it was conjectured by Childress and Percus (1984, Chemotactic collapse in two dimensions. InLecture Notes in Biomath. Vol. 55, Springer, Berlin-Heidelberg-New York, 1984, pp. 61–66) that in a two-dimensional domain there exists a critical numberCsuch that if the initial mass is strictly less thanC, then the solution exists globally in time and if it is strictly larger thanCblowup happens. For different versions of the Keller–Segel model, the conjecture has essentially been proved. The case of several chemotactic species introduces an additional question: What is the analogue for the critical mass obtained for the single species system? In this paper, we investigate for a two-species model for chemotaxis in2the conditions on the initial data, which determine blowup or global existence in time. Specifically, we find a curve in the plane of masses such that outside of it there is blowup and inside of it global existence in time is proved when the initial masses satisfy a threshold condition. Optimality of this condition is discussed through an analysis in the radial case. Finally, we show in the case of blowup for general data how it is possible to obtain a balance between entropies and prove what species should aggregates first.

scholarly journals Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yongbin Wang ◽  
Binhua Feng
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Choquard Equation ◽  
Power Type ◽  
Scale Invariant ◽  
Nonlinear Schrödinger ◽  
Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

scholarly journals Global Existence and Blow-Up for the Pseudo-parabolic p(x)-Laplacian Equation with Logarithmic Nonlinearity

2021 ◽  
Author(s):  
Fugeng Zeng ◽  
Qigang Deng ◽  
Dongxiu Wang
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Global Solution ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Potential Well ◽  
Laplacian Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem of the pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity. The existence of the global solution is obtained by using the potential well method and the logarithmic inequality. In addition, the sufficient conditions of the blow-up are obtained by concavity method.

scholarly journals Wave-breaking phenomena and global existence for the weakly dissipative generalized Novikov equation

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Shuguan Ji ◽  
Yonghui Zhou
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Global Solution ◽  
Wave Breaking ◽  
Blow Up ◽  
Momentum Density ◽  
Well Posedness ◽  
Novikov Equation ◽  
Blow Up Rate

In this paper, we mainly study several problems on the weakly dissipative generalized Novikov equation. We first establish the local well-posedness of solutions. We then give the precise blow-up scenarios for the generalized Novikov equation provided the momentum density associated with their initial data changes sign, and obtain the blow-up rate of blow-up solutions. Finally, we prove that the equation has a global solution provided the momentum density associated with their initial data do not change sign.

scholarly journals Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Logarithmic Sobolev Inequality ◽  
Decay Rates ◽  
Infinite Time ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
T Tau

AbstractIn this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

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