scholarly journals Optimal Control of a Free Boundary Problem: Analysis with Second-Order Sufficient Conditions

2014 ◽  
Vol 52 (5) ◽  
pp. 2771-2799 ◽  
Author(s):  
Harbir Antil ◽  
Ricardo H. Nochetto ◽  
Patrick Sodré
Keyword(s):  
Optimal Control ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Sufficient Conditions ◽  
Second Order ◽  
Problem Analysis ◽  
A Free Boundary Problem ◽  
Second Order Sufficient Conditions

scholarly journals A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuki Kaneko ◽  
Hiroshi Matsuzawa ◽  
Yoshio Yamada
Keyword(s):  
Free Boundary ◽  
Diffusion Equation ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Reaction Diffusion Equation ◽  
Bistable Nonlinearity ◽  
High Space ◽  
A Free Boundary Problem

<p style='text-indent:20px;'>We study a free boundary problem of a reaction-diffusion equation <inline-formula><tex-math id="M1">\begin{document}$ u_t = \Delta u+f(u) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ t&gt;0,\ |x|&lt;h(t) $\end{document}</tex-math></inline-formula> under a radially symmetric environment in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The reaction term <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> has positive bistable nonlinearity, which satisfies <inline-formula><tex-math id="M5">\begin{document}$ f(0) = 0 $\end{document}</tex-math></inline-formula> and allows two positive stable equilibrium states and a positive unstable equilibrium state. The problem models the spread of a biological species, where the free boundary represents the spreading front and is governed by a one-phase Stefan condition. We show multiple spreading phenomena in high space dimensions. More precisely the asymptotic behaviors of solutions are classified into four cases: big spreading, small spreading, transition and vanishing, and sufficient conditions for each dynamical behavior are also given. We determine the spreading speed of the spherical surface <inline-formula><tex-math id="M6">\begin{document}$ \{x\in \mathbb{R}^N:\ |x| = h(t)\} $\end{document}</tex-math></inline-formula>, which expands to infinity as <inline-formula><tex-math id="M7">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>, even when the corresponding semi-wave problem does not admit solutions.</p>

Qualitative analysis of a free boundary problem for tumor growth under the action of periodic external inhibitors

2018 ◽  
Vol 11 (01) ◽  
pp. 1850008
Author(s):  
Shihe Xu ◽  
Meng Bai ◽  
Zhong Wang ◽  
Fangwei Zhang
Keyword(s):  
Free Boundary ◽  
Tumor Growth ◽  
Periodic Function ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Time Delays ◽  
Sufficient Conditions ◽  
External Concentration ◽  
A Free Boundary Problem ◽  
Avascular Tumor

In this paper, a free boundary problem for a solid avascular tumor growth under the action of periodic external inhibitors with time delays in proliferation is studied. Sufficient conditions for the global stability of tumor-free equilibrium are given. Moreover, if external concentration of nutrients is large, we also prove that the tumor will not disappear and determine the conditions under which there exist periodic solutions to the model. The results show that the periodicity of the inhibitor may imply periodicity of the size of the tumor. More precisely, if [Formula: see text] (the concentration of external nutrients) is greater than [Formula: see text], where [Formula: see text] are two constants; [Formula: see text]; [Formula: see text] is a periodic function which can be interpreted as a treatment and [Formula: see text] is the period of [Formula: see text]. Results are illustrated by computer simulations.

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