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  2019, 27: 20-36. doi: 10.3934/era.2019008

Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators

Jon Johnsen

Department of Mathematics, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmark

Received  August 2019 Revised  September 2019 Published  October 2019

This paper treats parabolic final value problems generated by coercive Lax–Milgram operators, and well-posedness is proved for this large class. The result is obtained by means of an isomorphism between Hilbert spaces containing the data and solutions. Like for elliptic generators, the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, and the resulting compatibility condition extends to the coercive context. Lax–Milgram operators in vector distribution spaces is the main framework, but the crucial tool that analytic semigroups always are invertible in the class of closed operators is extended to unbounded semigroups, and this is shown to yield a Duhamel formula for the Cauchy problems in the set-up. The final value heat conduction problem with the homogeneous Neumann boundary condition on a smooth open set is also proved to be well posed in the sense of Hadamard.

Keywords: Duhamel formula, coercive operator, final value data, Neumann heat problem, well-posed.
Mathematics Subject Classification: Primary: 35A01; Secondary: 47D06.
Citation: Jon Johnsen. Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators. Electronic Research Archive, 2019, 27: 20-36. doi: 10.3934/era.2019008
References:
[1]

Y. Almog and B. Helffer, On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent, Comm. PDE, 40 (2015), 1441-1466.  doi: 10.1080/03605302.2015.1025978.  Google Scholar

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A.-E. Christensen and J. Johnsen, On parabolic final value problems and well-posedness, C. R. Acad. Sci. Paris, Ser. I, 356 (2018), 301-305.  doi: 10.1016/j.crma.2018.01.019.  Google Scholar

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A.-E. Christensen and J. Johnsen, Final value problems for parabolic differential equations and their well-posedness, Axioms, 7 (2018), 31.  doi: 10.3390/axioms7020031.  Google Scholar

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R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953.  Google Scholar

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E. B. Davies, One-parameter Semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, Inc., London-New York, 1980.  Google Scholar

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L. Eldén, Approximations for a Cauchy problem for the heat equation, Inverse Problems, 3 (1987), 263-273.  doi: 10.1088/0266-5611/3/2/009.  Google Scholar

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L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, second ed., American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

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D.S. Grebenkov, B. Helffer and R. Henry, The complex Airy operator on the line with a semipermeable barrier, SIAM J. Math. Anal., 49 (2017), 1844-1894.  doi: 10.1137/16M1067408.  Google Scholar

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G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009.  Google Scholar

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G. Grubb and V. A. Solonnikov, Solution of parabolic pseudo-differential initial-boundary value problems, J. Differential Equations, 87 (1990), 256-304.  doi: 10.1016/0022-0396(90)90003-8.  Google Scholar

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B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, vol. 139, Cambridge University Press, Cambridge, 2013.  Google Scholar

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I. W. Herbst, Dilation analyticity in constant electric field. I. The two body problem, Comm. Math. Phys., 64 (1979), 279-298.  doi: 10.1007/BF01221735.  Google Scholar

[16]

L. Hörmander, The Analysis of Linear Partial Differential Operators I, Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

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L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications, vol. 26, Springer Verlag, Berlin, 1997.  Google Scholar

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F. John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. (4), 40 (1955), 129-142.  doi: 10.1007/BF02416528.  Google Scholar

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J. Johnsen, Characterization of log-convex decay in non-selfadjoint dynamics, Elec. Res. Ann. Math., 25 (2018), 72-86.  doi: 10.3934/era.2018.25.008.  Google Scholar

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J. Johnsen, A class of well-posed parabolic final value problems, Appl. Num. Harm. Ana., Birkhäuser (to appear). arXiv: 1904.05190. Google Scholar

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O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of mathematical monographs, vol. 23, Amer. Math. Soc., 1968.  Google Scholar

[23]

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[25]

W. L. Miranker, A well posed problem for the backward heat equation, Proc. Amer. Math. Soc., 12 (1961), 243-247.  doi: 10.1090/S0002-9939-1961-0120462-2.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

J. Rauch, Partial Differential Equations, Springer, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

[28]

L. Schwartz, Théorie Des Distributions, revised and enlarged ed., Hermann, Paris, 1966.  Google Scholar

[29]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  doi: 10.1016/0022-247X(74)90008-0.  Google Scholar

[30]

H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman, Boston, Mass., 1979.  Google Scholar

[31]

R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, Elsevier Science Publishers B.V., Amsterdam, 1984.  Google Scholar

[32]

K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin-New York, 1980.  Google Scholar

show all references

References:
[1]

Y. Almog and B. Helffer, On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent, Comm. PDE, 40 (2015), 1441-1466.  doi: 10.1080/03605302.2015.1025978.  Google Scholar

[2]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, Abstract Linear Theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.  Google Scholar

[4]

A.-E. Christensen and J. Johnsen, On parabolic final value problems and well-posedness, C. R. Acad. Sci. Paris, Ser. I, 356 (2018), 301-305.  doi: 10.1016/j.crma.2018.01.019.  Google Scholar

[5]

A.-E. Christensen and J. Johnsen, Final value problems for parabolic differential equations and their well-posedness, Axioms, 7 (2018), 31.  doi: 10.3390/axioms7020031.  Google Scholar

[6]

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953.  Google Scholar

[7]

E. B. Davies, One-parameter Semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, Inc., London-New York, 1980.  Google Scholar

[8]

L. Eldén, Approximations for a Cauchy problem for the heat equation, Inverse Problems, 3 (1987), 263-273.  doi: 10.1088/0266-5611/3/2/009.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, second ed., American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

D.S. Grebenkov, B. Helffer and R. Henry, The complex Airy operator on the line with a semipermeable barrier, SIAM J. Math. Anal., 49 (2017), 1844-1894.  doi: 10.1137/16M1067408.  Google Scholar

[11]

D. S. Grebenkov and B. Helffer, On the spectral properties of the Bloch–Torrey operator in two dimensions, SIAM J. Math. Anal., 50 (2018), 622-676.  doi: 10.1137/16M1088387.  Google Scholar

[12]

G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009.  Google Scholar

[13]

G. Grubb and V. A. Solonnikov, Solution of parabolic pseudo-differential initial-boundary value problems, J. Differential Equations, 87 (1990), 256-304.  doi: 10.1016/0022-0396(90)90003-8.  Google Scholar

[14]

B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, vol. 139, Cambridge University Press, Cambridge, 2013.  Google Scholar

[15]

I. W. Herbst, Dilation analyticity in constant electric field. I. The two body problem, Comm. Math. Phys., 64 (1979), 279-298.  doi: 10.1007/BF01221735.  Google Scholar

[16]

L. Hörmander, The Analysis of Linear Partial Differential Operators I, Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[17]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications, vol. 26, Springer Verlag, Berlin, 1997.  Google Scholar

[18]

J. Janas, On unbounded hyponormal operators Ⅲ, Studia Mathematica, 112 (1994), 75-82.  doi: 10.4064/sm-112-1-75-82.  Google Scholar

[19]

F. John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. (4), 40 (1955), 129-142.  doi: 10.1007/BF02416528.  Google Scholar

[20]

J. Johnsen, Characterization of log-convex decay in non-selfadjoint dynamics, Elec. Res. Ann. Math., 25 (2018), 72-86.  doi: 10.3934/era.2018.25.008.  Google Scholar

[21]

J. Johnsen, A class of well-posed parabolic final value problems, Appl. Num. Harm. Ana., Birkhäuser (to appear). arXiv: 1904.05190. Google Scholar

[22]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of mathematical monographs, vol. 23, Amer. Math. Soc., 1968.  Google Scholar

[23]

J.-L. Lions and B. Malgrange, Sur l'unicité rétrograde dans les problèmes mixtes parabolic, Math. Scand., 8 (1960), 227-286.  doi: 10.7146/math.scand.a-10611.  Google Scholar

[24]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.  Google Scholar

[25]

W. L. Miranker, A well posed problem for the backward heat equation, Proc. Amer. Math. Soc., 12 (1961), 243-247.  doi: 10.1090/S0002-9939-1961-0120462-2.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

J. Rauch, Partial Differential Equations, Springer, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

[28]

L. Schwartz, Théorie Des Distributions, revised and enlarged ed., Hermann, Paris, 1966.  Google Scholar

[29]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  doi: 10.1016/0022-247X(74)90008-0.  Google Scholar

[30]

H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman, Boston, Mass., 1979.  Google Scholar

[31]

R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, Elsevier Science Publishers B.V., Amsterdam, 1984.  Google Scholar

[32]

K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin-New York, 1980.  Google Scholar

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