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Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities

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Abstract

This paper is concerned with a class of semilinear hyperbolic equations with singular potentials on the manifolds with conical singularities, which was introduced to describe a field propagating on the spacetime of a true string. We prove the local existence and uniqueness of the solution by using the contraction mapping principle. In the spirit of variational principle and mountain pass theorem, a class of initial data are precisely divided into three different energy levels. The main ingredient of this paper is to conduct a comprehensive and systematic study on the dynamic behavior of the solution with three different energy levels. We introduce a family of potential wells to derive a threshold of the existence of global solutions and blow up in finite time of solution in both cases with sub-critical and critical initial energy. Moreover, two sets of sufficient conditions for initial data leading to blow up result are established at arbitrarily positive initial energy level.

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References

  1. Hawking, S.W., Israel, W.: Three Hundred Years of Gravitation. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  2. Aryal, M., Ford, L.H., Vilenkin, A.: Cosmic strings and black holes. Phys. Rev. D 34(8), 2263–2266 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Filgueiras, C., Moraes, F.: On the quantum dynamics of a point particle in conical space. Ann. Phys. 323(12), 3150–3157 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Filgueiras, C., Silva, E.O., Andrade, F.M.: Nonrelativistic quantum dynamics on a cone with and without a constraining potential. J. Math. Phys. 53(12), 122106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Nazaikinskii, V., Savin, AYu., Schulze, B.W., Sternin, Y.B.: Elliptic Theory on Singular Manifolds, Differential and Integral Equations and Their Applications, vol. 7. Chapman Hall/CRC, Boca Raton (2006)

    MATH  Google Scholar 

  6. Melrose, R.B., Mendoza, G.A.: Elliptic Operators of Totally Characteristic Types, pp. 47–83. Mathematical Sciences Research Institute, MRSI, Berkeley (1983)

    Google Scholar 

  7. Cheeger, J.: On the spectral geometry of spaces with cone-like singularities. Proc. Natl. Acad. Sci. U. S. A. 76(5), 2103–2106 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18(4), 575–657 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kay, B.S., Studer, U.M.: Boundary conditions for quantum mechanics on cones and fields around cosmic strings. Commun. Math. Phys. 139(1), 103–139 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pitelli, J.M., Letelier, P.S.: Quantum singularities in static spacetimes. Int. J. Modern Phys. D 20(5), 729–743 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krtouš, P.: Electromagnetic field near a cosmic string. Phys. Rev. D 74, 065006 (2006)

    Article  Google Scholar 

  12. Schrohe, E.: Noncommutative residues and manifolds with conical singularities. J. Funct. Anal. 150(1), 146–174 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Harutyunyan, G., Schulze, B.W.: Elliptic Mixed, Transmission and Singular Crack Problems, EMS Tracts in Mathematics, vol. 4. European Mathematical Society (EMS), Zürich (2008)

    MATH  Google Scholar 

  14. Schulze, B.W.: Elliptic complexes on manifolds with conical singularities. Teubner Texte Zur Math. 106, 170–223 (1988)

    MathSciNet  MATH  Google Scholar 

  15. Schulze, B.W.: Pseudo-differential Operators on Manifolds with Singularities. Studies in Mathematics and Its Applications, vol. 24. North-Holland Publishing Co., Amsterdam (1991)

    Google Scholar 

  16. Schulze, B.W., Sternin, B.Y., Shatalow, V.: Differential Equations on Singular Manifolds, Semiclassical Theory and Operator Algebras. Wiley-VCH, Berlin (1998)

    Google Scholar 

  17. Schulze, B.W.: Boundary Value Problems and Singular Pseudo-differential Operators. Wiley, Chichester (1998)

    MATH  Google Scholar 

  18. Guillarmou, C., Hassell, A., Sikora, A.: Resolvent at low energy III: the spectral measure. Trans. Am. Math. Soc. 365(11), 6103–6148 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guillarmou, C., Hassell, A., Sikora, A.: Restriction and spectral multiplier theorems on asymptotically conic manifolds. Anal. PDE 6(4), 893–950 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hassell, A., Tao, T., Wunsch, J.: Sharp Strichartz estimates on non-trapping asymptotically conic manifolds. Am. J. Math. 128(4), 963–1024 (2006)

    Article  MATH  Google Scholar 

  21. Hassell, A., Wunsch, J.: The Schrödinger propagator for scattering metrics. Ann. Math. 162(1), 487–523 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 209–292 (1967)

    MathSciNet  Google Scholar 

  23. Amann, H.: Function spaces on singular manifolds. Math. Nachr. 286(5–6), 436–475 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Egnell, H.: Positive solutions of semilinear equations in cones. Trans. Am. Math. Soc. 330(1), 191–201 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Melrose, R., Wunsch, J.: Propagation of singularities for the wave equation on conic manifolds. Invent. Math. 156(2), 235–299 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, H.Q.: \(L^{p}\)-estimates for the wave equation on manifolds with conical singularities. Math. Z. 272(1–2), 551–575 (2012)

    Article  MathSciNet  Google Scholar 

  27. Chen, H., Liu, X.C., Wei, Y.W.: Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equalitions on manifold with conical singularities. Calc. Var. Partial Differ. Equ. 43(3–4), 463–484 (2012)

    Article  MATH  Google Scholar 

  28. Roidos, N., Schrohe, E.: The Cahn–Hilliard equation and the Allen–Cahn equation on manifolds with conical singularities. Commun. Partial Differ. Equ. 38(5), 925–943 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Roidos, N., Schrohe, E.: Bounded imaginary powers of cone differential operators on higher order Mellin–Sobolev spaces and applications to the Cahn–Hilliard equation. J. Differ. Equ. 257(3), 611–637 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shao, Y.Z.: Singular parabolic equations of second order on manifolds with singularities. J. Differ. Equ. 260(2), 1747–1800 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hassell, A., Zhang, J.Y.: Global-in-time Strichartz estimates on nontrapping, asymptotically conic manifolds. Anal. PDE 9(1), 151–192 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Roidos, N., Schrohe, E.: Existence and maximal \(L_{p}\)-regularity of solutions for the porous medium equation on manifolds with conical singularities. Commun. Partial Differ. Equ. 41(9), 1441–1471 (2016)

    Article  MATH  Google Scholar 

  33. Zhang, J.Y.: Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifold. Adv. Math. 271, 91–111 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, J.Y., Zheng, J.Q.: Strichariz estimates and wave equation in a conic singular space. Math. Ann. 376(1–2), 525–581 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Alimohammady, M., Kalleji, M.K.: Existence result for a class of semi-linear totally characteristic hypoelliptic equations with conical degeneration. J. Funct. Anal. 265(10), 2331–2356 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Chen, H., Liu, X.C., Wei, Y.W.: Existence theory for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents. Ann. Global Anal. Geom. 39(1), 27–43 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Chen, H., Wei, Y.W., Zhou, B.: Existence of solutions for degenerate elliptic equations with singular potential on singular manifolds. Math. Nachr. 285(11–12), 1370–1384 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Chen, H., Liu, G.W.: Global existence and nonexistence for semilinear parabolic equalitions with conical degeneration. J. Pseudo-Differ. Oper. Appl. 3(3), 329–349 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, Inc., Haup-pauge (2003)

    MATH  Google Scholar 

  40. Xu, R.Z.: Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Q. Appl. Math. 68(3), 459–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu, Y.C., Xu, R.Z.: A class of fourth order wave equations with dissipative and nonlinear strain terms. J. Differ. Equ. 244(1), 200–228 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 30, 148–172 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  43. Bressan, A.: Lecture Notes on Functional Analysis with Applications to Linear Partial Differential Equations. American Mathematical Society, Providence (2013)

    MATH  Google Scholar 

  44. Bühler, T., Salamon, D.A.: Function Analysis. American Mathematical Society, Providence (2008)

    Google Scholar 

  45. Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11871017, 12271122), the China Postdoctoral Science Foundation (2013M540270), the Fundamental Research Funds for the Central Universities, the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072022GIP2403). In the process of this work, Runzhang Xu visited Mathematical Institute, University of Oxford and The Institute of Mathematical Sciences, The Chinese University of Hong Kong.

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Authors and Affiliations

  1. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, People’s Republic of China

    Yongbing Luo, Runzhang Xu & Chao Yang

  2. Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059, Kraków, Poland

    Chao Yang

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  1. Yongbing Luo
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  2. Runzhang Xu
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  3. Chao Yang
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Correspondence to Runzhang Xu.

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Communicated by A. Mondino.

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Luo, Y., Xu, R. & Yang, C. Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities. Calc. Var. 61, 210 (2022). https://doi.org/10.1007/s00526-022-02316-2

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