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
Hawking, S.W., Israel, W.: Three Hundred Years of Gravitation. Cambridge University Press, Cambridge (1987)
Aryal, M., Ford, L.H., Vilenkin, A.: Cosmic strings and black holes. Phys. Rev. D 34(8), 2263–2266 (1986)
Filgueiras, C., Moraes, F.: On the quantum dynamics of a point particle in conical space. Ann. Phys. 323(12), 3150–3157 (2008)
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)
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)
Melrose, R.B., Mendoza, G.A.: Elliptic Operators of Totally Characteristic Types, pp. 47–83. Mathematical Sciences Research Institute, MRSI, Berkeley (1983)
Cheeger, J.: On the spectral geometry of spaces with cone-like singularities. Proc. Natl. Acad. Sci. U. S. A. 76(5), 2103–2106 (1979)
Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18(4), 575–657 (1983)
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)
Pitelli, J.M., Letelier, P.S.: Quantum singularities in static spacetimes. Int. J. Modern Phys. D 20(5), 729–743 (2011)
Krtouš, P.: Electromagnetic field near a cosmic string. Phys. Rev. D 74, 065006 (2006)
Schrohe, E.: Noncommutative residues and manifolds with conical singularities. J. Funct. Anal. 150(1), 146–174 (1997)
Harutyunyan, G., Schulze, B.W.: Elliptic Mixed, Transmission and Singular Crack Problems, EMS Tracts in Mathematics, vol. 4. European Mathematical Society (EMS), Zürich (2008)
Schulze, B.W.: Elliptic complexes on manifolds with conical singularities. Teubner Texte Zur Math. 106, 170–223 (1988)
Schulze, B.W.: Pseudo-differential Operators on Manifolds with Singularities. Studies in Mathematics and Its Applications, vol. 24. North-Holland Publishing Co., Amsterdam (1991)
Schulze, B.W., Sternin, B.Y., Shatalow, V.: Differential Equations on Singular Manifolds, Semiclassical Theory and Operator Algebras. Wiley-VCH, Berlin (1998)
Schulze, B.W.: Boundary Value Problems and Singular Pseudo-differential Operators. Wiley, Chichester (1998)
Guillarmou, C., Hassell, A., Sikora, A.: Resolvent at low energy III: the spectral measure. Trans. Am. Math. Soc. 365(11), 6103–6148 (2013)
Guillarmou, C., Hassell, A., Sikora, A.: Restriction and spectral multiplier theorems on asymptotically conic manifolds. Anal. PDE 6(4), 893–950 (2013)
Hassell, A., Tao, T., Wunsch, J.: Sharp Strichartz estimates on non-trapping asymptotically conic manifolds. Am. J. Math. 128(4), 963–1024 (2006)
Hassell, A., Wunsch, J.: The Schrödinger propagator for scattering metrics. Ann. Math. 162(1), 487–523 (2005)
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)
Amann, H.: Function spaces on singular manifolds. Math. Nachr. 286(5–6), 436–475 (2013)
Egnell, H.: Positive solutions of semilinear equations in cones. Trans. Am. Math. Soc. 330(1), 191–201 (1992)
Melrose, R., Wunsch, J.: Propagation of singularities for the wave equation on conic manifolds. Invent. Math. 156(2), 235–299 (2004)
Li, H.Q.: \(L^{p}\)-estimates for the wave equation on manifolds with conical singularities. Math. Z. 272(1–2), 551–575 (2012)
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)
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)
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)
Shao, Y.Z.: Singular parabolic equations of second order on manifolds with singularities. J. Differ. Equ. 260(2), 1747–1800 (2016)
Hassell, A., Zhang, J.Y.: Global-in-time Strichartz estimates on nontrapping, asymptotically conic manifolds. Anal. PDE 9(1), 151–192 (2016)
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)
Zhang, J.Y.: Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifold. Adv. Math. 271, 91–111 (2015)
Zhang, J.Y., Zheng, J.Q.: Strichariz estimates and wave equation in a conic singular space. Math. Ann. 376(1–2), 525–581 (2020)
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)
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)
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)
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)
Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, Inc., Haup-pauge (2003)
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)
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)
Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 30, 148–172 (1968)
Bressan, A.: Lecture Notes on Functional Analysis with Applications to Linear Partial Differential Equations. American Mathematical Society, Providence (2013)
Bühler, T., Salamon, D.A.: Function Analysis. American Mathematical Society, Providence (2008)
Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)
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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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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DOI: https://doi.org/10.1007/s00526-022-02316-2