Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/well-posedness-for-constrained-hamilton-jacobi-equations-Ig0dBTpas0
References (21)
-
S. Geritz, J. Metz, É. Kisdi, G. Meszéna (1997)
Dynamics of Adaptation and Evolutionary BranchingPhysical Review Letters, 78
-
S. Geritz, É. Kisdi, G. Meszéna, J. Metz (2004)
Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary treeEvolutionary Ecology, 12
-
M. Crandall, H. Ishii, P. Lions (1992)
User’s guide to viscosity solutions of second order partial differential equationsBulletin of the American Mathematical Society, 27
-
(1995)
On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions -
Nicolas Champagnat, P. Jabin, S. M'el'eard (2013)
Adaptation in a stochastic multi-resources chemostat modelJournal de Mathématiques Pures et Appliquées, 101
-
G. Barles (1993)
Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visitNonlinear Analysis-theory Methods & Applications, 20
-
O. Diekmann (2002)
A beginners guide to adaptive dynamics -
G. Barles (2007)
10.1090/conm/439/08463Contemp. Math., 439
-
Yeoneung Kim (2018)
On the Uniqueness for One-Dimensional Constrained Hamilton-Jacobi Equations.arXiv: Analysis of PDEs
-
M. James (1994)
Controlled markov processes and viscosity solutionsStochastics and Stochastics Reports, 49
-
G. Meszéna, M. Gyllenberg, F. Jacobs, J. Metz (2005)
Link between population dynamics and dynamics of Darwinian evolution.Physical review letters, 95 7
-
Samuel Nordmann, B. Perthame, C'ecile Taing (2018)
Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical TraitActa Applicandae Mathematicae, 155
-
G. Barles, S. Mirrahimi, Benoit Lmpt, Ljll (2009)
Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence ResultMethods and applications of analysis, 16
-
N. Le, Hiroyoshi Mitake, H. Tran (2017)
Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations -
M. Crandall, L. Evans, P. Lions (1984)
Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations.Transactions of the American Mathematical Society, 282
-
(2007)
Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics -
Nicolas Champagnat, P. Jabin (2010)
The evolutionary limit for models of populations interacting competitively via several resourcesJournal of Differential Equations, 251
-
O. Diekmann, P. Jabin, S. Mischler, B. Perthame (2005)
The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach.Theoretical population biology, 67 4
-
S. Mirrahimi, J. Roquejoffre (2015)
A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approacharXiv: Analysis of PDEs
-
S. Armstrong, H. Tran (2013)
Viscosity solutions of general viscous Hamilton–Jacobi equationsMathematische Annalen, 361
-
B. Perthame, G. Barles (2007)
Dirac concentrations in Lotka-Volterra parabolic PDEsarXiv: Analysis of PDEs
- Publisher
- Springer Journals
- Copyright
- Copyright © Springer Nature B.V. 2019
- Subject
- Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-019-00267-z
- Publisher site
- See Article on Publisher Site
Abstract
The goal of this paper is to study a Hamilton-Jacobi equation{ut=H(Du)+R(x,I(t))in Rn×(0,∞),supRnu(⋅,t)=0on [0,∞),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \textstyle\begin{cases} u_{t}=H(Du)+R(x,I(t)) &\text{in }\mathbb{R}^{n} \times (0,\infty ), \\ \sup_{\mathbb{R}^{n}} u(\cdot ,t)=0 &\text{on }[0,\infty ), \end{cases} $$\end{document} with initial conditions I(0)=I0>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$I(0)=I_{0}>0$\end{document}, u(x,0)=u0(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u(x,0)=u_{0}(x)$\end{document} on Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathbb{R}^{n}$\end{document}. Here (u,I)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(u,I)$\end{document} is a pair of unknowns and the Hamiltonian H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$H$\end{document} and the reaction term R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$R$\end{document} are given. Moreover, I(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$I(t)$\end{document} is an unknown constraint (Lagrange multiplier) that constrains the supremum of u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u$\end{document} to be always zero. We construct a solution in the viscosity setting using a fixed point argument when the reaction term R(x,I)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$R(x,I)$\end{document} is strictly decreasing in I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$I$\end{document}. We also discuss both uniqueness and nonuniqueness. For uniqueness, a certain structural assumption on R(x,I)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$R(x,I)$\end{document} is needed. We also provide an example with infinitely many solutions when the reaction term is not strictly decreasing in I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$I$\end{document}.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Jun 24, 2020