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

<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>

Valuation of the American put option as a free boundary problem through a high-order difference scheme

Valuation of the American options encountered commonly in finance is quite difficult due to the possibility of early exercise alternatives. Since an exact solution for the American options does not exist, effective numerical methods are needed to understand the behavior of option pricing models. Therefore, in this paper, a new approach based on a high-order difference scheme is proposed to discuss the valuation of an American put option as a free boundary problem. Using a front-fixing approach that transforms the unknown free boundary (optimal stopping) into a fixed one, a sixth-order finite difference scheme (FD6) in space and a third-order strong-stability preserving Runge–Kutta (SSPRK3) in time are applied to the model converted to a nonlinear partial differential equation. The computed results revealed that the combined method is seen to attempt to pull up the capacity of the algorithm to achieve higher accuracy. It is seen that the quantitative and qualitative results produced by the method proposed with minimal computational effort are sufficiently accurate and meaningful. Therefore, this article provides some new insights about the physical characteristics of financial problems and such realistic phenomena.

Two-phase problem with a free boundary for systems of parabolic equations with a nonlinear term of convection

Эта статья посвящена задаче со свободной границей для полулинейных параболических уравнений, в которой описывается феномен сегрегации местообитаний в популяционной экологии. Основная цель — показать глобальное существование, единственность решений проблемы. Предлагается двухфазная математическая модель со свободными границами для параболических уравнений типа реакция-диффузия. Установлены априорные оценки щаудеровского типа, на основе которых доказана однозначная разрешимость задачи. Неустойчивость каждого решения полностью определяется с помощью теоремы сравнения. This article is concerned with a free boundary problem for semilinear parabolic equations, wbich describes the habitat segregation phenomenon in population ecology. The main goal is to show global existence, the uniqueness of solutions to the problem. A two-phase mathematical model with free boundaries for parabolic equations of the reaction-diffusion type is proposed. A priori estimates of Schauder type are established, on the basis of which the unique solvability of the problem is proved. The instability of each solution is fully determined using the comparison theorem.