A simultaneous blow-up problem arising in tumor modeling

Espejo, Elio; Vilches, Karina; Conca, Carlos

doi:10.1007/s00285-019-01397-6

Cited by 22 publications

(16 citation statements)

References 30 publications

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“…Let Ω ⊂ R n (n ≥ 1) be a smooth and bounded domain, 4). Assume that χ ij , h i (i = 1, 2, j = 1, 2, 3, 4) satisfy (H1)-(H4).…”

Section: Lemma 32mentioning

confidence: 99%

“…Without respect to the kinetic terms, Espejo et al derived the simultaneous blow-up phenomenon in [4] for the parabolic-elliptic case of (1) in the whole space R 2 . Considering the Lotka-Volterra-type competition, whether the parabolic-elliptic case or the fully parabolic case of (1), the global dynamics of solutions were detected, it was found that the solution of ( 1) is globally bounded without any requirement on the size of the parameters for the fully parabolic case in the lower dimensions n ≤ 2 [14], while the largeness of parameters µ 1 , µ 2 is needed to guarantee the global solvability of (1) for n = 3 [15], and the global solution of this system exponentially approaches to a steady state for all n ≥ 1 [14], specifically, the system was shown to exhibit the large population densities phenomenon in [16], that is, the solution exhibits unbounded peculiarity for the proper choice of initial data.…”

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confidence: 99%

“…Let Ω ⊂ R 2 be a smooth and bounded domain, 2,3,4). Assume that the conditions in Lemma 3.6 are satisfied, and…”

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confidence: 99%

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Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

Huang

2021

era

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In this work, the fully parabolic chemotaxis-competition system with loop<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula>is considered under the homogeneous Neumann boundary condition, where <inline-formula><tex-math id="M1">\begin{document}$ x\in\Omega, t>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^{n} (n\leq 3) $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters <inline-formula><tex-math id="M3">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula> are sufficiently large, then the system possesses a unique and global classical solution for <inline-formula><tex-math id="M4">\begin{document}$ n\leq 3 $\end{document}</tex-math></inline-formula>. Specifically, when <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, the global boundedness can be attained without any constraints on <inline-formula><tex-math id="M6">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula>.

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“…Let Ω ⊂ R n (n ≥ 1) be a smooth and bounded domain, 4). Assume that χ ij , h i (i = 1, 2, j = 1, 2, 3, 4) satisfy (H1)-(H4).…”

Section: Lemma 32mentioning

confidence: 99%

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confidence: 99%

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Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

Huang

2021

era

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“…Inspired by [7](see also [9,12]), for modeling the interaction and motion of two cell populations in breast cancer cell invasion models in R d , d ≥ 3 we propose the following chemotaxis kind of system with two chemicals and a nonlinear diffusion term:…”

Section: Introductionmentioning

confidence: 99%

On a two-species cross attraction system in higher dimensions

Cárlen¹,

Ulusoy²

2021

Preprint

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We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension d ≥ 3. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of free-energy solutions, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a set of critical values: there is a critical value of a parameter pair in the system of equations for which there is a global-in-time energy solution and there exist blowing-up free-energy solutions under a criticality condition is violated for the parameter pair. 2d d+2 0 ∈ L 2 (R d ), w 0 ≥ 0.

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“…x ∈ Ω, t > 0, (1.3) in case f 1 (u 1 ) = f 2 (u 2 ) = 0, Espejo et al [6] obtained the solution of system blow-up simultaneously in finite time under some specific parameter conditions. Moreover, in case f 1 (u 1 ) = µ 1 u 1 (1 − u 1 − a 1 u 2 ) and f 2 (u 2 ) = µ 2 u 2 (1 − u 2 − a 2 u 1 ), α 11 = α 22 = χ 12 = χ 21 = 0, the global boundedness solution has been studied by Black [3] for all µ i > 0 (i = 1, 2) in n = 2, and when µ1 were large enough, the large time behavior of the bounded solution was discussed.…”

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confidence: 99%