Abstract
We study the Cauchy–Dirichlet problem for monotone semilinear uniformly elliptic second-order parabolic systems in divergence form with measure data. We show that under mild integrability conditions on the data, there exists a unique probabilistic solution of the system. We also show that if the operator and the data do not depend on time, then the solution of the parabolic system converges as t → ∞ to the solution of the Dirichlet problem for an associated elliptic system. In fact, we prove some results on the rate of the convergence.
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References
Aronson D.G.: Non-Negative Solutions of Linear Parabolic Equations. Ann. Sc. Norm. Super. Pisa 22, 607–693 (1968)
Bénilan P., Boccardo L., Gallouët T., Gariepy R., Pierre M., Vazquez J.-L.: L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 241–273 (1995)
Briand Ph., Delyon B., Hu Y., Pardoux E., Stoica L.: L p solutions of Backward Stochastic Differential Equations. Stochastic Process. Appl. 108, 109–129 (2003)
Briand Ph., Carmona R.: BSDEs with polynomial growth generators. J. Appl. Math. Stochastic Anal. 13, 207–238 (2000)
Chung K.L., Zhao Z.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin Heidelberg (1995)
J. Droniou, A. Porretta, A. Prignet, Parabolic Capacity and Soft Measures for Nonlinear Equations. Potential Anal. 19 (2003) 99–161.
N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.C. Quenez, Reflected solutions of backward SDEs, and related obstacle problems for PDE’s. Ann. Probab. 25 (1997) 702–737.
A. Friedman, Partial differential equations of parabolic type. Prentice–Hall, Englewood Cliffs, N.J., 1964.
M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994.
Hamadène S., Hassani M.: BSDEs with two reflecting barriers: the general result. Probab. Theory Relat. Fields 132, 237–264 (2005)
L.V. Kantorowich, B.Z. Vulih, A.G. Pinsker, Functional Analysis in Partially Ordered Spaces (in Russian). Gostekhizdat, Moscow, 1950.
Kisyński J.: Convergence du type L. Colloq. Math. 7, 205–211 (1960)
Klimsiak T.: Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form. Stochastic Process. Appl. 122, 134–169 (2012)
Klimsiak T.: Cauchy problem for semilinear parabolic equation with time-dependent obstacles: a BSDEs approach. Potential Anal. 39, 99–140 (2013)
T. Klimsiak, Semilinear elliptic systems with measure data. Ann. Mat. Pura Appl. (4) (2013) doi:10.1007/s10231-013-0364-4.
Klimsiak T.: On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators. J. Theoret. Probab. 26, 437–473 (2013)
T. Klimsiak, Semi-Dirichlet forms, Feynman-Kac functionals and the Cauchy problem for semilinear parabolic equations (2014) arXiv:1401.3643v1
T. Klimsiak, A. Rozkosz, Dirichlet forms and semilinear elliptic equations with measure data. J. Funct. Anal. 265 (2013) 890–925.
T. Klimsiak, A. Rozkosz, Obstacle problem for semilinear parabolic equations with measure data (2012). arXiv:1301.5795
Lejay A.: A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE. Stochastic Process. Appl. 110, 145–176 (2004)
Leonori T., Petitta F.: Asymptotic behavior for solutions of parabolic equations with natural growth terms and irregular data. Asymptot. Anal. 48, 219–233 (2006)
Meyer P.-A.: Fonctionnelles multiplicatives et additives de Markov. Ann. Inst. Fourier 12, 125–230 (1962)
L. Orsina, A.C. Ponce, Semilinear elliptic equations and systems with diffuse measures. J. Evol. Equ. 8 (2008) 781–812.
Y. Oshima, On construction of Markov processes associated with time dependent Dirichlet spaces. Forum Math. 4 (1992) 395–415.
Oshima Y.: Time-dependent Dirichlet forms and related stochastic calculus. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 281–316 (2004)
E. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998). Kluwer Academic Publishers, Dordrecht, 1999, pp. 503–549.
F. Petitta, Asymptotic behavior of solutions for linear parabolic equations with general measure data. C. R. Math. Acad. Sci. Paris 344 (2007) 571-576.
Petitta F.: Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data. Adv. Differential Equations 12, 867–891 (2007)
F. Petitta, Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data. Electron. J. Differential Equations (2008) No. 132, 10 pp.
Revuz D.: Mesures associees aux fonctionnelles additives de Markov I. Trans. Amer. Math. Soc. 148, 501–531 (1970)
Rozkosz A.: Weak convergence of diffusions corresponding to divergence form operators. Stochastics Stochastics Rep. 57, 129–157 (1996)
Rozkosz A.: Backward SDEs and Cauchy problem for semilinear equations in divergence form. Probab. Theory Related Fields 125, 393–407 (2003)
L. Słomiński, Stability of stochastic differential equations driven by general semimartingales. Dissertationes Math. 349 (1996) 113 pp.
G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus. Séminaire de Mathémtiques Supérieures 16 (1966) 326 pp.
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Klimsiak, T. Existence and large-time asymptotics for solutions of semilinear parabolic systems with measure data. J. Evol. Equ. 14, 913–947 (2014). https://doi.org/10.1007/s00028-014-0244-4
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DOI: https://doi.org/10.1007/s00028-014-0244-4