Abstract
We study large time behavior of renormalized solutions of the Cauchy problem for equations of the form ∂tu − Lu + λu = f(x, u) + g(x, u) ⋅ μ, where L is the operator associated with a regular lower bounded semi-Dirichlet form and μ is a nonnegative bounded smooth measure with respect to the capacity determined by . We show that under the monotonicity and some integrability assumptions on f, g as well as some assumptions on the form , u(t, x) → v(x) as t →∞ for quasi-every x, where v is a solution of some elliptic equation associated with our parabolic equation. We also provide the rate convergence. Some examples illustrating the utility of our general results are given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aronson, D.G.: Non–Negative Solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa 22, 607–693 (1968)
Bass, R.F., Hsu, P.: Some potential theory for reflecting Brownian motion in hölder and Lipschitz domains. Ann. Probab. 19, 486–508 (1991)
Boccardo, L., Gallouët, T.: Strongly nonlinear elliptic equations having natural growth terms and l 1 data. Nonlinear Anal. 19, 573–579 (1992)
Briand, P., Delyon, B., Hu, Y., Pardoux, E., Stoica, L.: l p solutions of backward stochastic differential equations. Stoch. Process Appl. 108, 109–129 (2003)
Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré, Probab. Statist. 23(2, suppl.), 245–287 (1987)
Chen, Z.-Q., Fukushima, M.: Symmetric Markov processes, time change, and boundary theory. Princeton University Press, Princeton (2012)
Chung, K.L.: Greenian bounds for Markov processes. Potential Anal. 1, 83–92 (1992)
Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin Heidelberg (1995)
Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28, 741–808 (1999)
Dellacherie, C., Meyer, P. -A.: Probabilités at Potentiel Chaptires V à VIII. Théorie des martingales. Revised edition. Hermann, Paris (1980)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Second Revised and Extended Edition. Walter de Gruyter, Berlin (2011)
Klimsiak, T.: Existence and large-time asymptotics for solutions of semilinear parabolic systems with measure data. J. Evol. Equ. 14, 913–947 (2014)
Klimsiak, T.: Semilinear elliptic systems with measure data. Ann. Mat. Pura Appl. (4) 194, 55–76 (2015)
Klimsiak, T.: Semi-dirichlet forms, Feynman-Kac functionals and the Cauchy problem for semilinear parabolic equations. J. Funct. Anal. 268, 1205–1240 (2015)
Klimsiak, T., Rozkosz, A.: Dirichlet forms and semilinear elliptic equations with measure data. J. Funct. Anal. 265, 890–925 (2013)
Klimsiak, T., Rozkosz, A.: Obstacle problem for semilinear parabolic equations with measure data. J. Evol. Equ. 15, 457–491 (2015)
Klimsiak, T., Rozkosz, A.: Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form. NoDEA Nonlinear Diff. Equ. Appl. 22, 457–491 (2015)
Klimsiak, T., Rozkosz, A.: Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms. Colloq. Math. 145, 35–67 (2016)
Kulczycki, T.: Properties of Green function of symmetric stable processes. Probab. Math. Statist. 17, 339–364 (1997)
Leonori, T., Petitta, F.: Asymptotic behavior for solutions of parabolic equations with natural growth terms and irregular data. Asymptot. Anal. 48, 219–233 (2006)
Liptser, R., Shiryayev, A.N.: Theory of Martingales. Nauka Moscow, 1986; English transl. Kluwer, Dordrecht (1989)
Ma, Z. -M., Röckner, M.: Introduction to the Theory of (Non–Symmetric) Dirichlet Forms. Springer, Berlin (1992)
Marcus, M., Véron, L.: Nonlinear Second Order Elliptic Equations Involving Measures. De Gruyter, Berlin (2014)
Murat, F., Porretta, A.: Stability properties, existence, and nonexistence of renormalized solutions for elliptic equations with measure data. Comm. Partial Diff. Equ. 27, 2267–2310 (2002)
Oshima, Y.: Some properties of Markov processes associated with time dependent Dirichlet forms. Osaka J. Math. 29, 103–127 (1992)
Oshima, Y.: Semi-Dirichlet Forms and Markov Processes. Walter de Gruyter, Berlin (2013)
Pardoux, E., Zhang, S.: Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields 110, 535–558 (1998)
Petitta, F.: Asymptotic behavior of solutions for linear parabolic equations with general measure data. C. R. Math. Acad. Sci. Paris 344, 535–558 (2007)
Petitta, F.: Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data. Adv. Diff. Equ. 12, 867–891 (2007)
Petitta, F.: Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data. Electron. J. Differential Equations No. 132, pp. 10 (2008)
Petitta, F., Ponce, A.C., Porretta, A.: Diffuse measures and nonlinear parabolic equations. J. Evol. Equ. 11, 861–905 (2011)
Pierre, M.: Representant Precis d’Un Potentiel Parabolique, Seminaire de Theorie du Potentiel. Lect. Notes Math. 814, 186–228 (1980)
Porretta, A.: Existence results for nonlinear parabolic equations via strong convergence truncations. Ann. Mat. Pura Appl. (IV) 177, 143–172 (1999)
Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004)
Stannat, W.: The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics, vol. 142 (1999)
Acknowledgments
Research supported by Polish National Science Center (grant no. 2012/07/B/ST1/03508).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Klimsiak, T., Rozkosz, A. Large Time Behavior of Solutions to Parabolic Equations with Dirichlet Operators and Nonlinear Dependence on Measure Data. Potential Anal 51, 255–289 (2019). https://doi.org/10.1007/s11118-018-9711-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-018-9711-9
Keywords
- Semilinear equation
- Dirichlet operator
- Mesure data
- Large time behavior of solutions
- Rate of convergence
- Backward stochastic differential equation