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

  • Received: 01 August 2019 Revised: 01 September 2019
  • Primary: 35A01; Secondary: 47D06

  • 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.

    Citation: Jon Johnsen. Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators[J]. Electronic Research Archive, 2019, 27: 20-36. doi: 10.3934/era.2019008

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  • 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.



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    [1] Almog Y., Helffer B. (2015) On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent. Comm. PDE 40: 1441-1466.
    [2] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, Abstract Linear Theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995.

    10.1007/978-3-0348-9221-6

    MR1345385

    [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.

    10.1007/978-3-0348-0087-7

    MR2798103

    [4] Christensen A.-E., Johnsen J. (2018) On parabolic final value problems and well-posedness. C. R. Acad. Sci. Paris, Ser. I 356: 301-305.
    [5] Christensen A.-E., Johnsen J. (2018) Final value problems for parabolic differential equations and their well-posedness. Axioms 7: 31.
    [6] R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953.

    MR0065391

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

    MR591851

    [8] Eldén L. (1987) Approximations for a Cauchy problem for the heat equation. Inverse Problems 3: 263-273.
    [9] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, second ed., American Mathematical Society, Providence, RI, 2010.

    10.1090/gsm/019

    MR2597943

    [10] Grebenkov D.S., Helffer B., Henry R. (2017) The complex Airy operator on the line with a semipermeable barrier. SIAM J. Math. Anal. 49: 1844-1894.
    [11] Grebenkov D. S., Helffer B. (2018) On the spectral properties of the Bloch–Torrey operator in two dimensions. SIAM J. Math. Anal. 50: 622-676.
    [12] G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009.

    MR2453959

    [13] Grubb G., Solonnikov V. A. (1990) Solution of parabolic pseudo-differential initial-boundary value problems. J. Differential Equations 87: 256-304.
    [14] B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, vol. 139, Cambridge University Press, Cambridge, 2013.

    MR3027462

    [15] Herbst I. W. (1979) Dilation analyticity in constant electric field. I. The two body problem. Comm. Math. Phys. 64: 279-298.
    [16] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin, 1983.

    10.1007/978-3-642-96750-4

    MR717035

    [17] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications, vol. 26, Springer Verlag, Berlin, 1997.

    MR1466700

    [18] Janas J. (1994) On unbounded hyponormal operators Ⅲ. Studia Mathematica 112: 75-82.
    [19] John F. (1955) Numerical solution of the equation of heat conduction for preceding times. Ann. Mat. Pura Appl. (4) 40: 129-142.
    [20] Johnsen J. (2018) Characterization of log-convex decay in non-selfadjoint dynamics. Elec. Res. Ann. Math. 25: 72-86.
    [21] J. Johnsen, A class of well-posed parabolic final value problems, Appl. Num. Harm. Ana., Birkhäuser (to appear). arXiv: 1904.05190.
    [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.

    MR0241822

    [23] Lions J.-L., Malgrange B. (1960) Sur l'unicité rétrograde dans les problèmes mixtes parabolic. Math. Scand. 8: 227-286.
    [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.

    MR0350177

    [25] Miranker W. L. (1961) A well posed problem for the backward heat equation. Proc. Amer. Math. Soc. 12: 243-247.
    [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.

    10.1007/978-1-4612-5561-1

    MR710486

    [27] J. Rauch, Partial Differential Equations, Springer, 1991.

    10.1007/978-1-4612-0953-9

    MR1223093

    [28] L. Schwartz, Théorie Des Distributions, revised and enlarged ed., Hermann, Paris, 1966.

    MR0209834

    [29] Showalter R. E. (1974) The final value problem for evolution equations. J. Math. Anal. Appl. 47: 563-572.
    [30] H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman, Boston, Mass., 1979.

    MR533824

    [31] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, Elsevier Science Publishers B.V., Amsterdam, 1984.

    MR769654

    [32] K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin-New York, 1980.

    MR617913

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