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Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production. (English. French summary) Zbl 1460.35197

Summary: This paper deals with the chemotaxis system with nonlinear signal secretion \[ \begin{cases} u_t=\nabla\cdot(D(u)\nabla u-S(u)\nabla v), & x\in\Omega,\quad t>0,\\ v_t=\Delta v-v+g(u), & x\in \Omega,\quad t>0, \end{cases} \] under homogeneous Neumann boundary conditions in a bounded domain \(\Omega\subset\mathbb{R}^n\) \((n\ge 2)\). The diffusion function \(D(s)\in C^2([0,\infty))\) and the chemotactic sensitivity function \(S(s)\in C^2([0,\infty))\) are given by \(D(s)\geq C_d(1+s)^{-\alpha}\) and \(0<S(s)\leq C_ss(1+s)^{\beta-1}\) for all \(s\geq 0\) with \(C_d,C_s>0\) and \(\alpha,\beta\in\mathbb{R}\). The nonlinear signal secretion function \(g(s)\in C^1([0,\infty))\) is supposed to satisfy \(g(s)\leq C_g s^{\gamma}\) for all \(s\geq 0\) with \(C_g,\gamma>0\). Global boundedness of solution is established under the specific conditions: \[ 0<\gamma\leq 1\text{ and }\alpha+\beta<\min\left\lbrace 1+\frac{1}{n},1+\frac{2}{n}-\gamma \right\rbrace. \] The purpose of this work is to remove the upper bound of the diffusion condition assumed in [X. Tao et al., J. Math. Anal. Appl. 474, No. 1, 733–747 (2019; Zbl 1483.35111)], and we also give the necessary constraint \(\alpha+\beta<1+\frac{1}{n}\), which is ignored in [loc. cit., Theorem 1.1].

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
35B44 Blow-up in context of PDEs
35B35 Stability in context of PDEs
92C17 Cell movement (chemotaxis, etc.)

Citations:

Zbl 1483.35111
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References:

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