We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed point-wise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behavior of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.
Accepté le :
DOI : 10.1051/cocv/2016077
Mots clés : Nonlocal operators, evolution equations, sub- supersolutions
@article{COCV_2018__24_1_105_0,
author = {Punzo, Fabio and Valdinoci, Enrico},
title = {Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {105--127},
publisher = {EDP-Sciences},
volume = {24},
number = {1},
year = {2018},
doi = {10.1051/cocv/2016077},
mrnumber = {3764136},
zbl = {1395.35197},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016077/}
}
TY - JOUR AU - Punzo, Fabio AU - Valdinoci, Enrico TI - Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 105 EP - 127 VL - 24 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016077/ DO - 10.1051/cocv/2016077 LA - en ID - COCV_2018__24_1_105_0 ER -
%0 Journal Article %A Punzo, Fabio %A Valdinoci, Enrico %T Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 105-127 %V 24 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016077/ %R 10.1051/cocv/2016077 %G en %F COCV_2018__24_1_105_0
Punzo, Fabio; Valdinoci, Enrico. Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 105-127. doi : 10.1051/cocv/2016077. http://www.numdam.org/articles/10.1051/cocv/2016077/
and , The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka J. Math. 46 (2009) 661–683. | MR | Zbl
and , Uniqueness of solutions to the Cauchy problem for parabolic equations. J. Math. Anal. Appl. 13 (1966) 516–526. | DOI | MR | Zbl
, and , On the Dirichlet Problem for Second-Order Elliptic Integro-Differential Equations. Indiana Univ. Math. J. 57 (2008) 213–146. | DOI | MR | Zbl
, and , Holder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. 13 (2010) 1–26. | MR | Zbl
R.F. Bass, Stochastic Processes. Cambridge Series in Statistical and Probabilistic Mathematics (2011) | MR | Zbl
and , Some theorems on stable processes. Trans. Amer. Math. Soc. 95 (1960) 263–273. | DOI | MR | Zbl
, and , Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. | DOI | MR | Zbl
, and , Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity. Asympt. Anal. 22 (2000) 349–358. | MR | Zbl
, First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 (1961) 75–90. | DOI | MR | Zbl
, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. 36 (1999) 135–249. | DOI | MR | Zbl
, and , Conditions at infinity for the inhomogeneous filtration equation. Ann. I. Henri Poincaré-AN 31 (2014) 413–428 | DOI | Numdam | MR | Zbl
, and , Linear equations of the second order of parabolic type. Russian Math. Surveys 17 (1962) 1–144. | DOI
, and , Admissible conditions for parabolic equations degenerating at infinity. St. Petersburg Math. J. 19 (2008) 239–251. | DOI | MR | Zbl
and , Dirichlet conditions at infinity for parabolic and elliptic equations. Nonl. Anal. TMA 138 (2016) 156–175. | DOI | MR | Zbl
and , Prescribed conditions at infinity for parabolic equations. Commun. Cont. Math. 17 (2015). | DOI | MR | Zbl
and , On the Cauchy problem for integro-differential operators in Holder classes and the uniqueness of the martingale problem. Potential Anal. 40 (2014) 539–563. | DOI | MR | Zbl
and , On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem. J. Diff. Eq. 256 (2014) 1581–1626. | DOI | MR | Zbl
, On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients. Math. Z. 223 (1996) 569–586. | DOI | MR | Zbl
and , Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients. Ann. Inst. Henri Poincaré (C) Non Linear Analysis 26 (2009) 2001–2024. | DOI | Numdam | MR | Zbl
and , Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations. J. Diff. Equ. 258 (2015) 555–587. | DOI | MR | Zbl
, Monotone Methods in Nonlinear Elliptic and Parabolic Boundary Value Problems. Indiana Univ. Math. J. 21 (1972) 979–1000. | DOI | MR | Zbl
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator (Ph.D. Thesis). The University of Texas at Austin (2005). | MR
L. Schwartz, Théorie des distributions, Hermann, Paris (1966). | MR | Zbl
, Théorèmes d’unicité pour l’équation de la chaleur. Mat. Sb. 42 (1935) 199–215. | Zbl
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