NDLI: Analysis of the Minimal Traveling Wave Speed via the Methods of Young Measures

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ito, Ryo
Copyright Year	2018
Abstract	In this paper, we consider the equation $u_t = u_{xx} + h(b(x)) (1-u)u,\ x \in \mathbb{R}$, where $\gamma \mapsto h(\gamma)$ is a nonnegative continuous function and $b(x)$ is a periodic function. In some sense, $b(x)$ represents a controllable parameter and the function $\gamma \mapsto h(\gamma)$ describes how the intrinsic growth rate of $u$ depends on the parameter $b$. It is known that there exists a nonnegative number $c^{}(h(b))$ such that the traveling wave with average speed $c$ exists if and only if $c \geq c^{}(h(b))$. We study minimizing problems and maximizing problems of the minimal speed $c^{*}(h(b))$ by varying $b(x)$ under the constraint $\frac{1}{L} \int_{0}^{L} b(x) dx = \alpha$, where $\alpha$ is a given positive constant. We prove the existence of minimizers and maximizers of the minimal speed, but it turns out that the minimizers for certain choices of $h$ do not exist in the class of functions but do exist in the space of Young measures. This means that the minimizing sequence exhibits indefinitely rapid oscillations. Similar oscillation appears in the maximizing problems as we let the spatial period $L$ of $b$ tend to $0$. We give sufficient condition on the occurrence and nonoccurrence of such oscillation. As far as we know, this work gives the first application of Young measures to the theory of traveling waves.
Sponsorship	Ministry of Education, Culture, Sports, Science and Technology
Starting Page	3478
Ending Page	3534
Page Count	57
File Format	PDF
ISSN	00361410
DOI	10.1137/17M1111218
e-ISSN	10957154
Journal	SIAM Journal on Mathematical Analysis (SJMAAH)
Issue Number	4
Volume Number	50
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-07-03
Access Restriction	Subscribed
Subject Keyword	traveling wave Young measure Variational methods extended variational problem Ecology Reaction-diffusion equations minimal speed Traveling wave solutions KPP equation Nonlinear parabolic equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Mathematics

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