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

Elliptic approximation of forward-backward parabolic equations

Fabio Paronetto

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.

Keywords: Abstract equations, elliptic equations, elliptic-parabolic equations, forward-backward parabolic equations, monotone operators, approximation by singular perturbations.
Mathematics Subject Classification: 35J60, 35M10, 35K90, 34D15.
Citation: Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. L. Lions, Équations linéaires du 1er ordre, in Equazioni Differenziali Astratte, vol. 29, C.I.M.E. Seminar, 1963, 15-28. Google Scholar

[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.  Google Scholar

[6]

C. D. Pagani - G. Talenti, On a forward-backward parabolic equation, Ann. Mat. Pura Appl., 90 (1971), 1-57. doi: 10.1007/BF02415041.  Google Scholar

[7]

F. Paronetto, Further existence results for evolution equations of mixed type and for a generalized Tricomi equation, to appear.  Google Scholar

[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.  Google Scholar

[9]

F. Paronetto, Homogenization of degenerate elliptic-parabolic equations, Asymptotic Anal., 37 (2004), 21-56.  Google Scholar

[10]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. L. Lions, Équations linéaires du 1er ordre, in Equazioni Differenziali Astratte, vol. 29, C.I.M.E. Seminar, 1963, 15-28. Google Scholar

[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.  Google Scholar

[6]

C. D. Pagani - G. Talenti, On a forward-backward parabolic equation, Ann. Mat. Pura Appl., 90 (1971), 1-57. doi: 10.1007/BF02415041.  Google Scholar

[7]

F. Paronetto, Further existence results for evolution equations of mixed type and for a generalized Tricomi equation, to appear.  Google Scholar

[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.  Google Scholar

[9]

F. Paronetto, Homogenization of degenerate elliptic-parabolic equations, Asymptotic Anal., 37 (2004), 21-56.  Google Scholar

[10]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997.  Google Scholar

[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.  Google Scholar

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