Skip to main content
SpringerLink
Log in

Fundamental Solution for Cauchy Initial Value Problem for Parabolic PDEs with Discontinuous Unbounded First-Order Coefficient at the Origin. Extension of the Classical Parametrix Method

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method of E.E. Levi.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aronson, D.G.: Gaussian estimates: a brief history (14 Jul 2017). arXiv:1707.04620v1 [math.AP]

  2. Bally, V., Kohatsu-Higa, A.: A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25(6), 3095–3138 (2015)

    Article  MathSciNet  Google Scholar 

  3. Chaudru de Raynal, P-E., Menozzi, S., Priola, E.: Schauder estimates for drifted fractional operators in the supercritical case. J. Funct. Anal. 278(8), 108425 (2020). https://doi.org/10.1016/j.jfa.2019.108425

    Article  MathSciNet  MATH  Google Scholar 

  4. Corielli, F., Foschi, P., Pascucci, A.: Parametrix approximation of diffusion transition densities. SIAM J. Financ. Math. 1(1), 833–867 (2010)

    Article  MathSciNet  Google Scholar 

  5. Deck, T., Kruse, S.: Parabolic differential equations with unbounded coefficients – a generalization of the parametrix method. Acta Appl. Math. 74(1), 71–91 (2002)

    Article  MathSciNet  Google Scholar 

  6. Dressel, F.: The fundamental solution of the parabolic equation. Duke Math. J. 7(1), 186–203 (1940)

    Article  MathSciNet  Google Scholar 

  7. Dressel, F.: The fundamental solution of the parabolic equation. II. Duke Math. J. 13(1), 61–70 (1946)

    Article  MathSciNet  Google Scholar 

  8. Èĭdel’man, S.D.: On fundamental solutions of parabolic systems. II. Mat. Sb. 53(95), 73–136 (1961)

    MathSciNet  MATH  Google Scholar 

  9. Èĭdel’man, S.D., Porper, F.O.: Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them. Usp. Mat. Nauk 39(3), 107–156 (1984)

    MathSciNet  MATH  Google Scholar 

  10. Evgrafov, M.A.: Asymptotic Estimates and Entire Functions. Gordon and Breach, Inc., New York (1961)

    MATH  Google Scholar 

  11. Freĭdlin, M.I.: Quasilinear parabolic equations, and measures on a function space. Funkc. Anal. Prilozh. 1(3), 74–82 (1967). (In Russian)

    MathSciNet  Google Scholar 

  12. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs (1964)

    MATH  Google Scholar 

  13. Gagliardo, E.: Teoremi di esistenza e di unicità per problemi al contorno relativi ad equazioni paraboliche lineari e quasi lineari in \(n\) variabili. Ric. Mat. 5, 239–257 (1956)

    MATH  Google Scholar 

  14. Glagoleva, R.J.: A priori estimate of the Hölder norm and the Harnack inequality for the solution of a second order linear parabolic equation with discontinuous coefficients. Mat. Sb. (N.S.) 76(118), 167–185 (1968)

    MathSciNet  Google Scholar 

  15. Gruber, M.: Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants. Math. Z. 185(1), 23–43 (1984)

    Article  MathSciNet  Google Scholar 

  16. Ilyin, A.M.: Degenerate elliptic and parabolic equations. Mat. Sb. (N.S.) 50(92), 443–498 (1960)

    MathSciNet  Google Scholar 

  17. Ilyin, A.M., Kalashnikov, A.S., Oleynik, O.A.: Linear second-order partial differential equations of the parabolic type. J. Math. Sci. 108(4), 435–542 (2002)

    Article  MathSciNet  Google Scholar 

  18. Inoue, M.: A stochastic method for solving quasilinear parabolic equations and its application to an ecological model. Hiroshima Math. J. 13(2), 379–391 (1983)

    Article  MathSciNet  Google Scholar 

  19. Kamynin, L.I., Maslennikov, V.N.: On the maximum principle for a parabolic equation with discontinuous coefficients. Sib. Math. J. 2(3), 384–399 (1961)

    Google Scholar 

  20. Korzeniowski, A.: A probabilistic approach to numerical solution of the nonlinear diffusion equation. Numer. Methods Partial Differ. Equ. 6(4), 335–342 (1990)

    Article  MathSciNet  Google Scholar 

  21. Krasnoselsky, M.A., Krein, S.G., Sobolevsky, P.E.: On differential equations with nonbounded operators in Banach spaces. Dokl. Akad. Nauk SSSR 111(1), 19–22 (1956)

    MathSciNet  Google Scholar 

  22. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Mathematics and Its Applications. Springer, Berlin (1987)

    Book  Google Scholar 

  23. Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  24. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow (1967)

    MATH  Google Scholar 

  25. Lax, P., Milgram, A.: Parabolic equations. Ann. Math. Stud. 33, 167–190 (1954)

    MATH  Google Scholar 

  26. Levi, E.E.: Sulle Equazioni Lineari Alle Derivate Parziali Totalmente Ellittiche. Rend. R. Acc. Lincei, Classe Sci. (V), vol. 16 (1907)

    MATH  Google Scholar 

  27. Levi, E.E.: Sulle equazioni lineari totalmente ellittiche alle derivate parziali. Rend. Circ. Mat. Palermo 24, 275–317 (1907)

    Article  Google Scholar 

  28. Lieberman, G.M.: Second Order Parabolic Partial Differential Equations. World Scientific Publishing House, New-York (1996)

    Book  Google Scholar 

  29. Mittag-Leffler, G.: Sur la nouvelle function Ea. C. R. Math. Acad. Sci. Paris 137, 554–558 (1903)

    MATH  Google Scholar 

  30. Molchanov, S.A., Ostrovskii, E.: Symmetric stable processes as traces of degenerate diffusion processes. Teor. Veroâtn. Primen. 14(1), 127–130 (1969)

    MathSciNet  MATH  Google Scholar 

  31. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  Google Scholar 

  32. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Fundamental Principles of Mathematical Sciences. Springer, Berlin (1999)

    Book  Google Scholar 

  33. Sachkov, V.N.: Combinatorial Methods in Discrete Mathematics. Encyclopedia of Mathematics and Its Applications, vol. 55. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  34. Samarskiĭ, A.A.: Equations of the parabolic type with discontinuous coefficients. Dokl. Akad. Nauk SSSR 121(2), 225–228 (1958)

    MathSciNet  Google Scholar 

  35. Shiga, T., Watanabe, S.: Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 27, 37–46 (1973)

    Article  MathSciNet  Google Scholar 

  36. Shukla, A.K., Prajapati, J.C.: On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336(2), 797–811 (1997)

    Article  MathSciNet  Google Scholar 

  37. Slobodetskiĭ, L.N.: The fundamental solution and the Cauchy problem for a parabolic system. Mat. Sb. 46(88) 2, 229–258 (1958)

    MathSciNet  MATH  Google Scholar 

  38. Sobolevskiĭ, P.E.: Equations of parabolic type in a Banach space. Tr. Mosk. Mat. Obš. 10, 297–350 (1961)

    MathSciNet  Google Scholar 

  39. van Diejen, J.F., Vinet, L.: Calogero-Sutherland-Moser Models. CRM Series in Mathematical Physics. Springer, Berlin (2000)

    Book  Google Scholar 

  40. Voit, M., Woerner, J.H.C.: Functional Central Limit Theorem in the Freezing Regime. arXiv:1901.08390v1 [math.PR] (24 Jan 2019)

Download references

Acknowledgement

The first author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Università degli Studi di Napoli Parthenope through the project “sostegno alla Ricerca individuale”.

Author information

Authors and Affiliations

  1. Parthenope University of Naples, via Generale Parisi 13, Palazzo Pacanowsky, 80132, Naples, Italy

    Maria Rosaria Formica

  2. Department of Mathematics and Statistics, Bar-Ilan University, 59200, Ramat Gan, Israel

    Eugeny Ostrovsky & Leonid Sirota

Authors
  1. Maria Rosaria Formica
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eugeny Ostrovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leonid Sirota
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Rosaria Formica.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Formica, M.R., Ostrovsky, E. & Sirota, L. Fundamental Solution for Cauchy Initial Value Problem for Parabolic PDEs with Discontinuous Unbounded First-Order Coefficient at the Origin. Extension of the Classical Parametrix Method. Acta Appl Math 170, 399–413 (2020). https://doi.org/10.1007/s10440-020-00339-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-020-00339-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation