Mathematical Research for Models Which is Related to Chemotaxis System
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Cited by 3 publications
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We study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - \frac{1}{|\Omega |} \int \limits _\Omega u + u \end{array}\right. } \end{aligned}$$ u t = ∇ · ( ( u + 1 ) m - 1 ∇ u - u ∇ v ) + λ u - μ u 1 + κ , 0 = Δ v - 1 | Ω | ∫ Ω u + u and $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - v + u, \end{array}\right. } \end{aligned}$$ u t = ∇ · ( ( u + 1 ) m - 1 ∇ u - u ∇ v ) + λ u - μ u 1 + κ , 0 = Δ v - v + u , where $$\lambda $$ λ and $$\mu $$ μ are given spatially radial nonnegative functions and $$m, \kappa > 0$$ m , κ > 0 are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $$m,\kappa $$ m , κ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $$\lambda , \mu > 0$$ λ , μ > 0 , we find that there are initial data which lead to blow-up in (JL) if $$\begin{aligned} 0 \le \kappa&< \min \left\{ \frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&\quad \text {if } m\in \left[ \frac{2}{n},\frac{2n-2}{n}\right) \\ \text { or }\quad 0 \le \kappa&<\min \left\{ \frac{1}{2},\frac{n-1}{n}-\frac{m}{2}\right\}&\quad \text {if } m\in \left( 0,\frac{2}{n}\right) , \end{aligned}$$ 0 ≤ κ < min 1 2 , n - 2 n - ( m - 1 ) + if m ∈ 2 n , 2 n - 2 n or 0 ≤ κ < min 1 2 , n - 1 n - m 2 if m ∈ 0 , 2 n , and in (PE) if $$m \in [1, \frac{2n-2}{n})$$ m ∈ [ 1 , 2 n - 2 n ) and $$\begin{aligned} 0 \le \kappa < \min \left\{ \frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\} . \end{aligned}$$ 0 ≤ κ < min ( m - 1 ) n + 1 2 ( n - 1 ) , n - 2 - ( m - 1 ) n n ( n - 1 ) .
We study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - \frac{1}{|\Omega |} \int \limits _\Omega u + u \end{array}\right. } \end{aligned}$$ u t = ∇ · ( ( u + 1 ) m - 1 ∇ u - u ∇ v ) + λ u - μ u 1 + κ , 0 = Δ v - 1 | Ω | ∫ Ω u + u and $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - v + u, \end{array}\right. } \end{aligned}$$ u t = ∇ · ( ( u + 1 ) m - 1 ∇ u - u ∇ v ) + λ u - μ u 1 + κ , 0 = Δ v - v + u , where $$\lambda $$ λ and $$\mu $$ μ are given spatially radial nonnegative functions and $$m, \kappa > 0$$ m , κ > 0 are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $$m,\kappa $$ m , κ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $$\lambda , \mu > 0$$ λ , μ > 0 , we find that there are initial data which lead to blow-up in (JL) if $$\begin{aligned} 0 \le \kappa&< \min \left\{ \frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&\quad \text {if } m\in \left[ \frac{2}{n},\frac{2n-2}{n}\right) \\ \text { or }\quad 0 \le \kappa&<\min \left\{ \frac{1}{2},\frac{n-1}{n}-\frac{m}{2}\right\}&\quad \text {if } m\in \left( 0,\frac{2}{n}\right) , \end{aligned}$$ 0 ≤ κ < min 1 2 , n - 2 n - ( m - 1 ) + if m ∈ 2 n , 2 n - 2 n or 0 ≤ κ < min 1 2 , n - 1 n - m 2 if m ∈ 0 , 2 n , and in (PE) if $$m \in [1, \frac{2n-2}{n})$$ m ∈ [ 1 , 2 n - 2 n ) and $$\begin{aligned} 0 \le \kappa < \min \left\{ \frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\} . \end{aligned}$$ 0 ≤ κ < min ( m - 1 ) n + 1 2 ( n - 1 ) , n - 2 - ( m - 1 ) n n ( n - 1 ) .
Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems
We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider
<p style='text-indent:20px;'>This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t>0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t>0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} (*)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^N(N\geq1) $\end{document}</tex-math></inline-formula> under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> are positive. Based on an <b>new</b> energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mu>\left\{ \begin{array}{ll} {0, \; \; \; {\rm{if}}\; \; N\leq4},\\ {\frac{(N-4)_{+}}{N-2}\max\{1,\lambda_{0}\},\; \; \; {\rm{if}}\; \; N\geq5}\\ \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the initial data <inline-formula><tex-math id="M5">\begin{document}$ (u_0,v_0,w_0) $\end{document}</tex-math></inline-formula> are sufficiently regular. Here <inline-formula><tex-math id="M6">\begin{document}$ \lambda_0 $\end{document}</tex-math></inline-formula> is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.</p>
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