scholarly journals Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary

2005 ◽  
Vol 4 (3) ◽  
pp. 523-536 ◽  
Author(s):  
C. Brändle ◽  
◽  
F. Quirós ◽  
Julio D. Rossi ◽  
Keyword(s):  
Parabolic System ◽  
Blow Up ◽  
Quasilinear Parabolic System ◽  
Quasilinear Parabolic

scholarly journals Blow-Up and Global Existence for a Quasilinear Parabolic System

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Chunchen Wu
Keyword(s):  
Global Existence ◽  
Parabolic System ◽  
Blow Up ◽  
Quasilinear Parabolic System ◽  
Quasilinear Parabolic

The problem of solutions to a class of quasilinear coupling parabolic system was studied. By constructing weak upper-solutions and weak lower-solutions, we obtain the global existence and blow-up of solutions under appropriate conditions.

scholarly journals Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Juntang Ding ◽  
Chenyu Dong
Keyword(s):  
Positive Solution ◽  
Parabolic System ◽  
Blow Up ◽  
Sufficient Condition ◽  
Quasilinear Parabolic System ◽  
Spatial Region ◽  
Weakly Coupled ◽  
Blow Up Rate ◽  
Upper Estimate ◽  
Quasilinear Parabolic

<p style='text-indent:20px;'>The main purpose of the present paper is to study the blow-up problem of a weakly coupled quasilinear parabolic system as follows:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{ll} u_{t} = \nabla\cdot\left(r(u)\nabla u\right)+f(u,v,x,t), &amp; \\ v_{t} = \nabla\cdot\left(s(v)\nabla v\right)+g(u,v,x,t) &amp;{\rm in} \ \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), \ \frac{\partial v}{\partial\nu} = k(v) &amp;{\rm on} \ \partial\Omega\times(0,t^{*}), \\ u(x,0) = u_{0}(x), \ v(x,0) = v_{0}(x) &amp;{\rm in} \ \overline{\Omega}. \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a spatial bounded region in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{n} \ (n\geq2) $\end{document}</tex-math></inline-formula> and the boundary <inline-formula><tex-math id="M3">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> of the spatial region <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is smooth. We give a sufficient condition to guarantee that the positive solution <inline-formula><tex-math id="M5">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> of the above problem must be a blow-up solution with a finite blow-up time <inline-formula><tex-math id="M6">\begin{document}$ t^* $\end{document}</tex-math></inline-formula>. In addition, an upper bound on <inline-formula><tex-math id="M7">\begin{document}$ t^* $\end{document}</tex-math></inline-formula> and an upper estimate of the blow-up rate on <inline-formula><tex-math id="M8">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> are obtained.</p>

scholarly journals A PLANKTON ALLELOPATHIC MODEL DESCRIBED BY A DELAYED QUASILINEAR PARABOLIC SYSTEM

2012 ◽  
Vol 17 (4) ◽  
pp. 485-497 ◽  
Author(s):  
Canrong Tian ◽  
Peng Zhu
Keyword(s):  
Parabolic System ◽  
Upper And Lower Solutions ◽  
Asymptotically Stable ◽  
Stability Of Solutions ◽  
Quasilinear Parabolic System ◽  
Monotone Iterations ◽  
Engineering Problems ◽  
Existence And Stability ◽  
Uniform Equilibrium ◽  
Quasilinear Parabolic

The quasilinear parabolic system has been applied to a variety of physical and engineering problems. However, most works lack effective techniques to deal with the asymptotic stability. This paper is concerned with the existence and stability of solutions for a plankton allelopathic model described by a quasilinear parabolic system, in which the diffusions are density-dependent. By the coupled upper and lower solutions and its associated monotone iterations, it is shown that under some parameter conditions the positive uniform equilibrium is asymptotically stable. Some biological interpretations for our results are given.

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