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Year 2022, Volume: 10 Issue: 1, 11 - 29, 15.04.2022

Abstract

References

  • [1] E.J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1991.
  • [2] C.M. Bender, and S.A. Orszag, Advanced mathematical methods for scientists and engineers: Asymptotic methods and perturbation theory, Vol. 1. New York: Springer Verlag, 1999.
  • [3] R.B. Dingle, Asymptotic expansions: Their derivation and interpretation. London: Academic Press; 1973.
  • [4] F. Say, On the asymptotic behavior of a second-order general differential equation, Numer. Methods Partial Differ. Equ., 38(2)(2021), 262-271. DOI: https://doi.org/10.1002/num.22774
  • [5] F. Say, Late-order terms of second order ODEs in terms of pre-factors, Hacettepe J. Math. Stat., 50(2)(2021), 342 - 350.
  • [6] H. Poincar´e, Sur les int´egrales irr´eguli`eres. Acta math., 8(1)(1886), 295-344.
  • [7] J.P. Boyd, The devil’s invention: Asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Math., 56(1)(1999),1-98.
  • [8] F.W. Olver, D. W. Lozier, R.F. Boisvert, and C.W. Clark, NIST handbook of mathematical functions, Cambridge university press, New York 2010.
  • [9] J. Serrin, Pathological Solutions of Elliptic Differential Equations. Ann Scuola Norm-Sci, 18(3)(1964), 385-389.
  • [10] G. Stampacchia, E´quations Elliptiques du Second Ordre a` Coefficients Discontinus. Seminaire Jean Leray, 3(1963-1964), 1-77.
  • [11] F. Petitta, Asymptotic Behaviour of Solutions for Linear Parabolic Equations with General Measure Data. C. R. Math. Acad. Sci. Paris, 344(9)(2007), 571-576.
  • [12] L. Boccardo, and T. Gallou¨et, Nonlinear Elliptic and Parabolic Equations Involving Measure Data. J Funct Anal, 87(1)(1989), 149-169.
  • [13] L. Boccardo, A. Dall’Aglio, T. Gallou¨et, and L. Orsina, Nonlinear Parabolic Equation with Measure Data, J Funct Anal, 147(1)(1997), 237-258.
  • [14] D. Blanchard, and F. Murat, Renormalised Solutions of Nonlinear Parabolic Equation with L1 Data: Existence and Uniqueness, P. Roy. Soc. Edinb. A., 127(6)(1997), 1137-1152.
  • [15] R. DiPerna, and P. Lions, On the Cauchy Problem for Boltzmann Equations: Global Existence and Weak Stability. Ann Math, 130(1989), 321-366.
  • [16] D. Blanchard, and H. Redwane, Renormalized Solutions for a Class of Evolution Problem. J. Math. Anal. Appl., 77(2)(1998), 117-151.
  • [17] D. Blanchard, F. Murat, and H. Redwane, Existence and Uniqueness of Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems. J. Differ. Equ., 177(2)(2001), 331-374.
  • [18] J. Droniou, A. Porretta, and Prignet, A., Parabolic Capacity and Soft Measures for Nonlinear Equations, potential Anal., 19(2)(2003), 99-161.
  • [19] F. Petitta, Renormalized Solutions of Nonlinear Parabolic Equations with General Measure Data. Ann. Mat. Pura Appl., 187(4)(2008), 563-604.
  • [20] F. A. P. Petitta, and A. Porretta, Diffuse Measures and Nonlinear Parabolic Equation. J. Evol. Equ., 11(4)(2011), 861-905.
  • [21] D. Blanchard, F. Petitta, and H. Redwane, Renormalised Solutions of Nonlinear Parabolic Equations with General Measure Data. Manuscr Math, 141 (2013).
  • [22] T. Leonori, I. Peral, A. Primo, and F. Soria, Basic Estimates for Solutions of a Class of Nonlocal Elliptic and Parabolic Equations, Discrete Contin Dyn Syst Ser A, 35(2)(2015), 6031-6068.
  • [23] F. Petitta, and A. Porretta, On the Notion of Renormalized Solution to Nonlinear Parabolic Equations with General Measure Data, (2017).
  • [24] P. B´enilan, L. Boccardo, T. Gallou¨et, R. Gariepy, M. Pierre, and J. L. Vazquez, An L1- Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations. Ann Scuola Norm-Sci, 22(2)(1995), 241-273.
  • [25] A. Prignet, Existence and Uniqueness of ”Entropy’ Solutions of Parabolic Problems with L1 Data. Nonlinear Anal. Theory Methods Appl., 28(12)(1997), 1943-1954.
  • [26] J. Droniou, and A. Prignet, 2007, Equivalence Between Entropy and Renormalized Solution for Parabolic Equations with Smooth Measure Data. Nonlinear Differ. Equ. Appl., 14(2007), 181- 205.
  • [27] T. Klimsiak, and A. Rozkosz, Dirichlet Forms and Semilinear Elliptic Equations with Measure Data. J Funct Anal, 265(6)(2013), 890-925.
  • [28] T. Klimsiak, and A. Rozkosz, Semilinear Elliptic Equation with Measure Data and Quasi-Regular Dirichlet Forms. Colloq. Math, 145(1)(2013), 35-67.
  • [29] T. Klimsiak, Existence and Large-time Asymptotic for Solutions of Semilinear Parabolic Systems with Measure Data. J. Evol. Equ., 14(2014), 913–947.
  • [30] T. Klimsiak, and A. Rozkosz, Obstacle Problem for Semilinear Parabolic Equation with Measure Data. J. Evol. Equ., 15(2015), 457-491.
  • [31] T. Klimsiak, and A. Rozkosz, Renormalised Solutions of Semilinear Equations Involving Measure Data and Operator Corresponding to Dirichlet Form. Nonlinear Differ. Equ. Appl., 22(2015), 1911-1934.
  • [32] T. Klimsiak, Semi-Dirichlet Forms, Feynman-Kac Functionals and the Cauchy Problem for Semilinear Parabolic Equations. J Funct Anal, 268(5)(2015), 1205-1240.
  • [33] T. Klimsiak, Semilinear Elliptic Systems with Measure Data. Ann. Mat. Pura Appl., 194(1)(2015), 55-76.
  • [34] T. Klimsiak, Cauchy Problem for Semilinear Parabolic Equation with Time-Dependent Obstacle: A BSDEs Approach. Potential Anal, 39(2013), 99-140.
  • [35] P. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Stoica, Lp Solutions of Backward Stochastic Differential Equations. Stoch Process Their Appl, 108(2003), 109-129.
  • [36] E. Pardoux, and S. Peng, Adapted Solution of a Backward Stochastic Differential Equation. Syst Control Lett, 14(1)(1990), 55–61.
  • [37] A. Rozkosz, Backward SDEs and Cauchy Problem for Semilinear Equations in Divergence Form. Probab Theory Relat Fields, 125(3)(2003), 393-407.
  • [38] A. Lejay, A Probabilistic Representation of the Solution of some Quasi-Linear PDE with a Divergence Form Operator: Application to Existence of Weak Solution of FBSDE. Stoch Process Their Appl, 110(1)(2004), 145-176.
  • [39] D. G. Aronson, Non-negative Solutions of Linear Parabolic Equations. Ann Scuola Norm-Sci, 22(4)(1968), 607-694.
  • [40] S. Hamad`ene, and M. Hassani, BSDEs with Two Reflecting Barriers: The General Result. Probab Theory Relat Fields, 132(2)(2005), 237-264.
  • [41] E. Pardoux, and A. R˘as¸canu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic Modelling and Applied Probability, 69, 680, Springer International Publishing, Switzerland.

Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System with Measure Data Using BSDEs Approach

Year 2022, Volume: 10 Issue: 1, 11 - 29, 15.04.2022

Abstract

This paper considers the existence and uniqueness of an asymptotic solution of a monotone semilinear parabolic system in divergence form with measure data. The proof of the main result is probabilistic, which are those of stochastic analysis, Markov process and primarily Backward Stochastic Differential Equations (BSDEs). The probabilistic solution to the system is considered as some generalization of the notion of renormalized (or entropy) solution. It is shown for a Cauchy-Dirichlet problem of a monotone semilinear parabolic system in divergence form with measure data, there exists a unique probabilistic solution of the system under a mild integrability condition on the data.

References

  • [1] E.J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1991.
  • [2] C.M. Bender, and S.A. Orszag, Advanced mathematical methods for scientists and engineers: Asymptotic methods and perturbation theory, Vol. 1. New York: Springer Verlag, 1999.
  • [3] R.B. Dingle, Asymptotic expansions: Their derivation and interpretation. London: Academic Press; 1973.
  • [4] F. Say, On the asymptotic behavior of a second-order general differential equation, Numer. Methods Partial Differ. Equ., 38(2)(2021), 262-271. DOI: https://doi.org/10.1002/num.22774
  • [5] F. Say, Late-order terms of second order ODEs in terms of pre-factors, Hacettepe J. Math. Stat., 50(2)(2021), 342 - 350.
  • [6] H. Poincar´e, Sur les int´egrales irr´eguli`eres. Acta math., 8(1)(1886), 295-344.
  • [7] J.P. Boyd, The devil’s invention: Asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Math., 56(1)(1999),1-98.
  • [8] F.W. Olver, D. W. Lozier, R.F. Boisvert, and C.W. Clark, NIST handbook of mathematical functions, Cambridge university press, New York 2010.
  • [9] J. Serrin, Pathological Solutions of Elliptic Differential Equations. Ann Scuola Norm-Sci, 18(3)(1964), 385-389.
  • [10] G. Stampacchia, E´quations Elliptiques du Second Ordre a` Coefficients Discontinus. Seminaire Jean Leray, 3(1963-1964), 1-77.
  • [11] F. Petitta, Asymptotic Behaviour of Solutions for Linear Parabolic Equations with General Measure Data. C. R. Math. Acad. Sci. Paris, 344(9)(2007), 571-576.
  • [12] L. Boccardo, and T. Gallou¨et, Nonlinear Elliptic and Parabolic Equations Involving Measure Data. J Funct Anal, 87(1)(1989), 149-169.
  • [13] L. Boccardo, A. Dall’Aglio, T. Gallou¨et, and L. Orsina, Nonlinear Parabolic Equation with Measure Data, J Funct Anal, 147(1)(1997), 237-258.
  • [14] D. Blanchard, and F. Murat, Renormalised Solutions of Nonlinear Parabolic Equation with L1 Data: Existence and Uniqueness, P. Roy. Soc. Edinb. A., 127(6)(1997), 1137-1152.
  • [15] R. DiPerna, and P. Lions, On the Cauchy Problem for Boltzmann Equations: Global Existence and Weak Stability. Ann Math, 130(1989), 321-366.
  • [16] D. Blanchard, and H. Redwane, Renormalized Solutions for a Class of Evolution Problem. J. Math. Anal. Appl., 77(2)(1998), 117-151.
  • [17] D. Blanchard, F. Murat, and H. Redwane, Existence and Uniqueness of Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems. J. Differ. Equ., 177(2)(2001), 331-374.
  • [18] J. Droniou, A. Porretta, and Prignet, A., Parabolic Capacity and Soft Measures for Nonlinear Equations, potential Anal., 19(2)(2003), 99-161.
  • [19] F. Petitta, Renormalized Solutions of Nonlinear Parabolic Equations with General Measure Data. Ann. Mat. Pura Appl., 187(4)(2008), 563-604.
  • [20] F. A. P. Petitta, and A. Porretta, Diffuse Measures and Nonlinear Parabolic Equation. J. Evol. Equ., 11(4)(2011), 861-905.
  • [21] D. Blanchard, F. Petitta, and H. Redwane, Renormalised Solutions of Nonlinear Parabolic Equations with General Measure Data. Manuscr Math, 141 (2013).
  • [22] T. Leonori, I. Peral, A. Primo, and F. Soria, Basic Estimates for Solutions of a Class of Nonlocal Elliptic and Parabolic Equations, Discrete Contin Dyn Syst Ser A, 35(2)(2015), 6031-6068.
  • [23] F. Petitta, and A. Porretta, On the Notion of Renormalized Solution to Nonlinear Parabolic Equations with General Measure Data, (2017).
  • [24] P. B´enilan, L. Boccardo, T. Gallou¨et, R. Gariepy, M. Pierre, and J. L. Vazquez, An L1- Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations. Ann Scuola Norm-Sci, 22(2)(1995), 241-273.
  • [25] A. Prignet, Existence and Uniqueness of ”Entropy’ Solutions of Parabolic Problems with L1 Data. Nonlinear Anal. Theory Methods Appl., 28(12)(1997), 1943-1954.
  • [26] J. Droniou, and A. Prignet, 2007, Equivalence Between Entropy and Renormalized Solution for Parabolic Equations with Smooth Measure Data. Nonlinear Differ. Equ. Appl., 14(2007), 181- 205.
  • [27] T. Klimsiak, and A. Rozkosz, Dirichlet Forms and Semilinear Elliptic Equations with Measure Data. J Funct Anal, 265(6)(2013), 890-925.
  • [28] T. Klimsiak, and A. Rozkosz, Semilinear Elliptic Equation with Measure Data and Quasi-Regular Dirichlet Forms. Colloq. Math, 145(1)(2013), 35-67.
  • [29] T. Klimsiak, Existence and Large-time Asymptotic for Solutions of Semilinear Parabolic Systems with Measure Data. J. Evol. Equ., 14(2014), 913–947.
  • [30] T. Klimsiak, and A. Rozkosz, Obstacle Problem for Semilinear Parabolic Equation with Measure Data. J. Evol. Equ., 15(2015), 457-491.
  • [31] T. Klimsiak, and A. Rozkosz, Renormalised Solutions of Semilinear Equations Involving Measure Data and Operator Corresponding to Dirichlet Form. Nonlinear Differ. Equ. Appl., 22(2015), 1911-1934.
  • [32] T. Klimsiak, Semi-Dirichlet Forms, Feynman-Kac Functionals and the Cauchy Problem for Semilinear Parabolic Equations. J Funct Anal, 268(5)(2015), 1205-1240.
  • [33] T. Klimsiak, Semilinear Elliptic Systems with Measure Data. Ann. Mat. Pura Appl., 194(1)(2015), 55-76.
  • [34] T. Klimsiak, Cauchy Problem for Semilinear Parabolic Equation with Time-Dependent Obstacle: A BSDEs Approach. Potential Anal, 39(2013), 99-140.
  • [35] P. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Stoica, Lp Solutions of Backward Stochastic Differential Equations. Stoch Process Their Appl, 108(2003), 109-129.
  • [36] E. Pardoux, and S. Peng, Adapted Solution of a Backward Stochastic Differential Equation. Syst Control Lett, 14(1)(1990), 55–61.
  • [37] A. Rozkosz, Backward SDEs and Cauchy Problem for Semilinear Equations in Divergence Form. Probab Theory Relat Fields, 125(3)(2003), 393-407.
  • [38] A. Lejay, A Probabilistic Representation of the Solution of some Quasi-Linear PDE with a Divergence Form Operator: Application to Existence of Weak Solution of FBSDE. Stoch Process Their Appl, 110(1)(2004), 145-176.
  • [39] D. G. Aronson, Non-negative Solutions of Linear Parabolic Equations. Ann Scuola Norm-Sci, 22(4)(1968), 607-694.
  • [40] S. Hamad`ene, and M. Hassani, BSDEs with Two Reflecting Barriers: The General Result. Probab Theory Relat Fields, 132(2)(2005), 237-264.
  • [41] E. Pardoux, and A. R˘as¸canu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic Modelling and Applied Probability, 69, 680, Springer International Publishing, Switzerland.
There are 41 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Deborah Danıel 0000-0003-4025-1448

Publication Date April 15, 2022
Submission Date October 8, 2020
Acceptance Date April 21, 2022
Published in Issue Year 2022 Volume: 10 Issue: 1

Cite

APA Danıel, D. (2022). Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System with Measure Data Using BSDEs Approach. Konuralp Journal of Mathematics, 10(1), 11-29.
AMA Danıel D. Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System with Measure Data Using BSDEs Approach. Konuralp J. Math. April 2022;10(1):11-29.
Chicago Danıel, Deborah. “Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System With Measure Data Using BSDEs Approach”. Konuralp Journal of Mathematics 10, no. 1 (April 2022): 11-29.
EndNote Danıel D (April 1, 2022) Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System with Measure Data Using BSDEs Approach. Konuralp Journal of Mathematics 10 1 11–29.
IEEE D. Danıel, “Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System with Measure Data Using BSDEs Approach”, Konuralp J. Math., vol. 10, no. 1, pp. 11–29, 2022.
ISNAD Danıel, Deborah. “Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System With Measure Data Using BSDEs Approach”. Konuralp Journal of Mathematics 10/1 (April 2022), 11-29.
JAMA Danıel D. Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System with Measure Data Using BSDEs Approach. Konuralp J. Math. 2022;10:11–29.
MLA Danıel, Deborah. “Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System With Measure Data Using BSDEs Approach”. Konuralp Journal of Mathematics, vol. 10, no. 1, 2022, pp. 11-29.
Vancouver Danıel D. Existence and Uniqueness of Asymptotic Solution of a Semilinear Parabolic System with Measure Data Using BSDEs Approach. Konuralp J. Math. 2022;10(1):11-29.
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