# American Institute of Mathematical Sciences

February  2020, 19(2): 1017-1036. doi: 10.3934/cpaa.2020047

## Elliptic approximation of forward-backward parabolic equations

 Dipartimento di Matematica Tullio Levi Civita, Università degli Studi di Padova, Via Trieste 63 - Padova, Italy

Received  March 2018 Revised  June 2019 Published  October 2019

In this note we give existence and uniqueness result for some elliptic problems depending on a small parameter and show that their solutions converge, when this parameter goes to zero, to the solution of a mixed type equation, elliptic-parabolic, parabolic both forward and backward. The aim is to give an approximation result via elliptic equations of a changing type equation.

Citation: Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1017-1036. doi: 10.3934/cpaa.2020047
##### References:
 [1] R. Beals, On an equation of mixed type from electron scattering theory, J. Math. Anal. Appl., 58 (1977), 32-45. doi: 10.1016/0022-247X(77)90225-6. [2] R. Beals, An abstract treatment of some forward-backward problems of transport and scattering, J. Funct. Anal., 34 (1979), 1-20. doi: 10.1016/0022-1236(79)90021-1. [3] I. M. Karabash, Abstract kinetic equations with positive collision operators, in Spectral Theory in Inner Product Spaces and Applications, vol. 188 of Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, 2009,175-195. doi: 10.1007/978-3-7643-8911-6_9. [4] J. L. Lions, Équations linéaires du 1er ordre, in Equazioni Differenziali Astratte, vol. 29, C.I.M.E. Seminar, 1963, 15-28. [5] V. Moretti, Spectral theory and quantum mechanics, vol. 64 of Unitext, Springer, Milan, 2013., With an introduction to the algebraic formulation., doi: 10.1007/978-88-470-2835-7. [6] C. D. Pagani - G. Talenti, On a forward-backward parabolic equation, Ann. Mat. Pura Appl., 90 (1971), 1-57. doi: 10.1007/BF02415041. [7] F. Paronetto, Further existence results for evolution equations of mixed type and for a generalized Tricomi equation, to appear. [8] F. Paronetto, Existence results for a class of evolution equations of mixed type, J. Funct. Anal., 212 (2004), 324-356. doi: 10.1016/j.jfa.2004.03.014. [9] F. Paronetto, Homogenization of degenerate elliptic-parabolic equations, Asymptotic Anal., 37 (2004), 21-56. [10] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997. [11] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II A and II B, Springer Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

##### References:
 [1] R. Beals, On an equation of mixed type from electron scattering theory, J. Math. Anal. Appl., 58 (1977), 32-45. doi: 10.1016/0022-247X(77)90225-6. [2] R. Beals, An abstract treatment of some forward-backward problems of transport and scattering, J. Funct. Anal., 34 (1979), 1-20. doi: 10.1016/0022-1236(79)90021-1. [3] I. M. Karabash, Abstract kinetic equations with positive collision operators, in Spectral Theory in Inner Product Spaces and Applications, vol. 188 of Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, 2009,175-195. doi: 10.1007/978-3-7643-8911-6_9. [4] J. L. Lions, Équations linéaires du 1er ordre, in Equazioni Differenziali Astratte, vol. 29, C.I.M.E. Seminar, 1963, 15-28. [5] V. Moretti, Spectral theory and quantum mechanics, vol. 64 of Unitext, Springer, Milan, 2013., With an introduction to the algebraic formulation., doi: 10.1007/978-88-470-2835-7. [6] C. D. Pagani - G. Talenti, On a forward-backward parabolic equation, Ann. Mat. Pura Appl., 90 (1971), 1-57. doi: 10.1007/BF02415041. [7] F. Paronetto, Further existence results for evolution equations of mixed type and for a generalized Tricomi equation, to appear. [8] F. Paronetto, Existence results for a class of evolution equations of mixed type, J. Funct. Anal., 212 (2004), 324-356. doi: 10.1016/j.jfa.2004.03.014. [9] F. Paronetto, Homogenization of degenerate elliptic-parabolic equations, Asymptotic Anal., 37 (2004), 21-56. [10] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997. [11] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II A and II B, Springer Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
 [1] Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043 [2] Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks and Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941 [3] Jiongmin Yong. Forward-backward evolution equations and applications. Mathematical Control and Related Fields, 2016, 6 (4) : 653-704. doi: 10.3934/mcrf.2016019 [4] Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761 [5] Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920 [6] G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783 [7] Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115 [8] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control and Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613 [9] Jiongmin Yong. Forward-backward stochastic differential equations: Initiation, development and beyond. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022011 [10] Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507 [11] Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010 [12] Gary Lieberman. Oblique derivative problems for elliptic and parabolic equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2409-2444. doi: 10.3934/cpaa.2013.12.2409 [13] Tomasz Dlotko, Tongtong Liang, Yejuan Wang. Critical and super-critical abstract parabolic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1517-1541. doi: 10.3934/dcdsb.2019238 [14] Wolf-Jürgen Beyn, Sergey Piskarev. Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 19-42. doi: 10.3934/dcdsb.2008.10.19 [15] Flavia Smarrazzo, Andrea Terracina. Sobolev approximation for two-phase solutions of forward-backward parabolic problems. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1657-1697. doi: 10.3934/dcds.2013.33.1657 [16] Li Ma, Chong Li, Lin Zhao. Monotone solutions to a class of elliptic and diffusion equations. Communications on Pure and Applied Analysis, 2007, 6 (1) : 237-246. doi: 10.3934/cpaa.2007.6.237 [17] Giuseppe Da Prato, Alessandra Lunardi. On a class of elliptic and parabolic equations in convex domains without boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 933-953. doi: 10.3934/dcds.2008.22.933 [18] J.I. Díaz, D. Gómez-Castro. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Conference Publications, 2015, 2015 (special) : 379-386. doi: 10.3934/proc.2015.0379 [19] Ciprian G. Gal, Mahamadi Warma. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations and Control Theory, 2016, 5 (1) : 61-103. doi: 10.3934/eect.2016.5.61 [20] Tommaso Leonori, Ireneo Peral, Ana Primo, Fernando Soria. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6031-6068. doi: 10.3934/dcds.2015.35.6031

2020 Impact Factor: 1.916