Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 428909, 17 pages
http://dx.doi.org/10.1155/2014/428909
Research Article

Space-Time Estimates on Damped Fractional Wave Equation

1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
3School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China

Received 26 July 2013; Accepted 27 January 2014; Published 4 May 2014

Academic Editor: Nasser-eddine Tatar

Copyright © 2014 Jiecheng Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Bellout and A. Friedman, “Blow-up estimates for a nonlinear hyperbolic heat equation,” SIAM Journal on Mathematical Analysis, vol. 20, no. 2, pp. 354–366, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. D. Blair, H. F. Smith, and C. D. Sogge, “On Strichartz estimates for Schrödinger operators in compact manifolds with boundary,” Proceedings of the American Mathematical Society, vol. 136, no. 1, pp. 247–256, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, NY, USA, 2003. View at MathSciNet
  4. T. Cazenave and F. B. Weissler, “Rapidly decaying solutions of the nonlinear Schrödinger equation,” Communications in Mathematical Physics, vol. 147, no. 1, pp. 75–100, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Chen, D. Fan, and L. Sun, “Hardy space estimates for the wave equation on compact Lie groups,” Journal of Functional Analysis, vol. 259, no. 12, pp. 3230–3264, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Giga, “Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system,” Journal of Differential Equations, vol. 62, no. 2, pp. 186–212, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Hieber and J. Prüss, “Heat kernels and maximal Lp-Lq estimates for parabolic evolution equations,” Communications in Partial Differential Equations, vol. 22, no. 9-10, pp. 1647–1669, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. Keel and T. Tao, “Endpoint Strichartz estimates,” American Journal of Mathematics, vol. 120, no. 5, pp. 955–980, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. Lindblad and C. D. Sogge, “On existence and scattering with minimal regularity for semilinear wave equations,” Journal of Functional Analysis, vol. 130, no. 2, pp. 357–426, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. Marshall, W. Strauss, and S. Wainger, “Lp-Lq estimates for the Klein-Gordon equation,” Journal de Mathématiques Pures et Appliquées, vol. 59, no. 4, pp. 417–440, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C. Miao, B. Yuan, and B. Zhang, “Strong solutions to the nonlinear heat equation in homogeneous Besov spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1329–1343, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. A. Miyachi, “On some Fourier multipliers for Hp(n),” Journal of the Faculty of Science. University of Tokyo IA, vol. 27, no. 1, pp. 157–179, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. Mockenhaupt, A. Seeger, and C. D. Sogge, “Local smoothing of Fourier integral operators and Carleson-Sjölin estimates,” Journal of the American Mathematical Society, vol. 6, no. 1, pp. 65–130, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. Müller and E. M. Stein, “Lp-estimates for the wave equation on the Heisenberg group,” Revista Matemática Iberoamericana, vol. 15, no. 2, pp. 297–334, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Nakao and K. Ono, “Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,” Mathematische Zeitschrift, vol. 214, no. 2, pp. 325–342, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. C. Peral, “Lp estimates for the wave equation,” Journal of Functional Analysis, vol. 36, no. 1, pp. 114–145, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. Seeger, C. D. Sogge, and E. M. Stein, “Regularity properties of Fourier integral operators,” Annals of Mathematics, vol. 134, no. 2, pp. 231–251, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. Staffilani and D. Tataru, “Strichartz estimates for a Schrödinger operator with nonsmooth coefficients,” Communications in Partial Differential Equations, vol. 27, no. 7-8, pp. 1337–1372, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Stefanov, “Strichartz estimates for the Schrödinger equation with radial data,” Proceedings of the American Mathematical Society, vol. 129, no. 5, pp. 1395–1401, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. E. Terraneo, “Non-uniqueness for a critical non-linear heat equation,” Communications in Partial Differential Equations, vol. 27, no. 1-2, pp. 185–218, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  21. G. Todorova and B. Yordanov, “Critical exponent for a nonlinear wave equation with damping,” Journal of Differential Equations, vol. 174, no. 2, pp. 464–489, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. C. Vilela, “Inhomogeneous Strichartz estimates for the Schrödinger equation,” Transactions of the American Mathematical Society, vol. 359, no. 5, pp. 2123–2136, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. F. B. Weissler, “Existence and nonexistence of global solutions for a semilinear heat equation,” Israel Journal of Mathematics, vol. 38, no. 1-2, pp. 29–40, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  24. N. Hayashi, E. I. Kaikina, and P. I. Naumkin, “Damped wave equation with a critical nonlinearity,” Transactions of the American Mathematical Society, vol. 358, no. 3, pp. 1165–1185, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. R. Ikehata and K. Tanizawa, “Global existence of solutions for semilinear damped wave equations in N with noncompactly supported initial data,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 7, pp. 1189–1208, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. Ikehata, Y. Miyaoka, and T. Nakatake, “Decay estimates of solutions for dissipative wave equations in N with lower power nonlinearities,” Journal of the Mathematical Society of Japan, vol. 56, no. 2, pp. 365–373, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. P. Marcati and K. Nishihara, “The Lp-Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media,” Journal of Differential Equations, vol. 191, no. 2, pp. 445–469, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  28. A. Matsumura, “On the asymptotic behavior of solutions of semi-linear wave equations,” Publications of the Research Institute for Mathematical Sciences, vol. 12, no. 1, pp. 169–189, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  29. C. Miao, B. Yuan, and B. Zhang, “Well-posedness of the Cauchy problem for the fractional power dissipative equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 3, pp. 461–484, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. C. Miao and B. Zhang, “The Cauchy problem for semilinear parabolic equations in Besov spaces,” Houston Journal of Mathematics, vol. 30, no. 3, pp. 829–878, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. T. Narazaki, “Lp-Lq estimates for damped wave equations and their applications to semi-linear problem,” Journal of the Mathematical Society of Japan, vol. 56, no. 2, pp. 585–626, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. T. Narazaki, “Lp-Lq estimates for damped wave equations with odd initial data,” Electronic Journal of Differential Equations, vol. 2005, pp. 1–17, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. K. Nishihara, “Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application,” Mathematische Zeitschrift, vol. 244, no. 3, pp. 631–649, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. H. Volkmer, “Asymptotic expansion of L2-norms of solutions to the heat and dissipative wave equations,” Asymptotic Analysis, vol. 67, no. 1-2, pp. 85–100, 2010. View at Google Scholar · View at MathSciNet
  35. Q. S. Zhang, “A blow-up result for a nonlinear wave equation with damping: the critical case,” Comptes Rendus de l'Académie des Sciences I, vol. 333, no. 2, pp. 109–114, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. J. Chen, Q. Deng, Y. Ding, and D. Fan, “Estimates on fractional power dissipative equations in function spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 5, pp. 2959–2974, 2012. View at Google Scholar
  37. Z. Zhai, “Strichartz type estimates for fractional heat equations,” Journal of Mathematical Analysis and Applications, vol. 356, no. 2, pp. 642–658, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. A. Miyachi, “On some estimates for the wave equation in Lp and Hp,” Journal of the Faculty of Science. University of Tokyo IA, vol. 27, no. 2, pp. 331–354, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. A. Miyachi, “On some singular Fourier multipliers,” Journal of the Faculty of Science. University of Tokyo IA, vol. 28, no. 2, pp. 267–315, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. K. M. Rogers, “A local smoothing estimate for the Schrödinger equation,” Advances in Mathematics, vol. 219, no. 6, pp. 2105–2122, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. L. Carleson, “Some analytic problems related to statistical mechanics,” in Euclidean Harmonic Analysis, vol. 779 of Lecture Notes in Mathematics, pp. 5–45, Springer, Berlin, Germany, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. S. Lee, “On pointwise convergence of the solutions to Schrödinger equations in 2,” International Mathematics Research Notices, vol. 2006, Article ID 32597, 21 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. P. Sjölin, “Regularity of solutions to the Schrödinger equation,” Duke Mathematical Journal, vol. 55, no. 3, pp. 699–715, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. L. Vega, “Schrödinger equations: pointwise convergence to the initial data,” Proceedings of the American Mathematical Society, vol. 102, no. 4, pp. 874–878, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, no. 223, Springer, Berlin, Germany, 1976. View at MathSciNet
  46. H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  47. A.-P. Calderón and A. Torchinsky, “Parabolic maximal functions associated with a distribution. II,” Advances in Mathematics, vol. 24, no. 2, pp. 101–171, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993. View at MathSciNet
  49. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, no. 32, Princeton University Press, Princeton, NJ, USA, 1971. View at MathSciNet
  50. S. L. Wang, “On the weighted estimate of the solution associated with the Schrödinger equation,” Proceedings of the American Mathematical Society, vol. 113, no. 1, pp. 87–92, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet