Table of Contents
International Journal of Partial Differential Equations
Volume 2014, Article ID 461046, 13 pages
http://dx.doi.org/10.1155/2014/461046
Research Article

Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

1Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università degli Studi di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy
2Dipartimento di Matematica, Università degli Studi di Roma “La Sapienza”, Piazzle Aldo Moro 2, 00185 Roma, Italy

Received 30 April 2013; Accepted 17 October 2013; Published 22 January 2014

Academic Editor: William E. Fitzgibbon

Copyright © 2014 Maria Rosaria Lancia and Paola Vernole. 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. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, vol. 83 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at MathSciNet
  2. 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
  3. M. R. Lancia, “A transmission problem with a fractal interface,” Zeitschrift für Analysis und ihre Anwendungen, vol. 21, no. 1, pp. 113–133, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. R. Lancia, “Second order transmission problems across a fractal surface,” Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni, vol. 27, no. 5, pp. 191–213, 2003. View at Google Scholar · View at MathSciNet
  5. M. R. Lancia and M. A. Vivaldi, “On the regularity of the solutions for transmission problems,” Advances in Mathematical Sciences and Applications, vol. 12, no. 1, pp. 455–466, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. R. Lancia and M. A. Vivaldi, “Asymptotic convergence of transmission energy forms,” Advances in Mathematical Sciences and Applications, vol. 13, no. 1, pp. 315–341, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. R. Lancia and P. Vernole, “Irregular heat flow problems,” SIAM Journal on Mathematical Analysis, vol. 42, no. 4, pp. 1539–1567, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. D. Ventcel', “On boundary conditions for multidimensional diffusion processes,” Teoriya Veroyatnostei i ee Primeneniya, vol. 4, pp. 172–185, 1959, English translation, Theory of Probability and Its Applications, vol. 4, pp. 164–177, 1959. View at Google Scholar
  9. H. Fujita, “On the blowing up of solutions of the Cauchy problem for ut=Δu+u1+α,” Journal of the Faculty of Science. University of Tokyo, vol. 13, p. 109–124 (1966), 1966. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. A. Levine, “The role of critical exponents in blowup theorems,” SIAM Review, vol. 32, no. 2, pp. 262–288, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. F. B. Weissler, “Local existence and nonexistence for semilinear parabolic equations in Lp,” Indiana University Mathematics Journal, vol. 29, no. 1, pp. 79–102, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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
  13. F. B. Weissler, “Semilinear evolution equations in Banach spaces,” Journal of Functional Analysis, vol. 32, no. 3, pp. 277–296, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. F. Gazzola and T. Weth, “Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level,” Differential and Integral Equations, vol. 18, no. 9, pp. 961–990, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. K. Falconer and J. Hu, “Nonlinear diffusion equations on unbounded fractal domains,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 606–624, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Matzeu, “Mountain pass and linking type solutions for semilinear Dirichlet forms,” in Recent Trends in Nonlinear Analysis, vol. 40 of Progress in Nonlinear Differential Equations and Their Applications, pp. 217–231, Birkhäuser, Basel, Switzerland, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. Grigor'yan, J. Hu, and K.-S. Lau, “Heat kernels on metric measure spaces and an application to semilinear elliptic equations,” Transactions of the American Mathematical Society, vol. 355, no. 5, pp. 2065–2095, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. R. Lancia and P. Vernole, “Semilinear evolution transmission problems across fractal layers,” Nonlinear Analysis, vol. 75, no. 11, pp. 4222–4240, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. R. Lancia and P. Vernole, “Semilinear fractal problems: approximation and regularity results,” Nonlinear Analysis, vol. 80, pp. 216–232, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Cefalo, G. Dell'Acqua, and M. R. Lancia, “Numerical approximation of transmission problems across Koch-type highly conductive layers,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5453–5473, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. R. Lancia and P. Vernole, “Convergence results for parabolic transmission problems across highly conductive layers with small capacity,” Advances in Mathematical Sciences and Applications, vol. 16, no. 2, pp. 411–445, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, UK, 2nd edition, 1990. View at MathSciNet
  23. U. Freiberg and M. R. Lancia, “Energy form on a closed fractal curve,” Zeitschrift für Analysis und ihre Anwendungen, vol. 23, no. 1, pp. 115–137, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. U. Mosco, “Lagrangian metrics on fractals,” in Proceedings of Symposia in Applied Mathematics, R. Spigler and S. Venakides, Eds., vol. 54, pp. 301–323, American Mathematical Society, 1998.
  25. M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, vol. 19 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  26. U. Mosco and M. A. Vivaldi, “Variational problems with fractal layers,” Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni, vol. 27, no. 5, pp. 237–251, 2003. View at Google Scholar · View at MathSciNet
  27. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 2nd edition, 1977. View at MathSciNet
  28. R. Rammal and G. Tolouse, “Walks on fractal structures and percolation clusters,” Journal de Physique Lettres, vol. 44, pp. L-13–L-22, 1983. View at Google Scholar
  29. M. Fukushima and T. Shima, “On a spectral analysis for the Sierpiński gasket,” Potential Analysis, vol. 1, no. 1, pp. 1–35, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, UK, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  31. T. Kumagai, “Brownian motion penetrating fractals,” Journal of Functional Analysis, vol. 170, no. 1, pp. 69–92, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. A. Grigor'yan, “Heat kernel upper bounds on a complete non-compact manifold,” Revista Matemática Iberoamericana, vol. 10, no. 2, pp. 395–452, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  33. U. Mosco, “Convergence of convex sets and of solutions of variational inequalities,” Advances in Mathematics, vol. 3, pp. 510–585, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. U. Mosco, “Composite media and asymptotic Dirichlet forms,” Journal of Functional Analysis, vol. 123, no. 2, pp. 368–421, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free—Boundary Value Problems, John Wiley & Sons, New York, NY, USA, 1984. View at MathSciNet
  36. D. Jerison and C. E. Kenig, “The inhomogeneous Dirichlet problem in Lipschitz domains,” Journal of Functional Analysis, vol. 130, no. 1, pp. 161–219, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. A. Buffa and P. Ciarlet,, “On traces for functional spaces related to Maxwell's Equations—part I: an integration by parts formula in Lipschitz Polyhedra,” Mathematical Methods in the Applied Sciences, vol. 21, no. 1, pp. 9–30, 2001. View at Google Scholar
  38. J. Necas, Les mèthodes directes en thèorie des èquationes elliptiques, Masson, Paris, France, 1967.
  39. F. Brezzi and G. Gilardi, “Foundamentals of P.D.E. for numerical analysis,” in Finite Element Handbook, H. Kardestuncer and D. H. Norrie, Eds., McGraw-Hill, New York, NY, USA, 1987. View at Google Scholar
  40. P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24, Pitman, Boston, Mass, USA, 1985. View at MathSciNet
  41. P. Grisvard, “Théorèmes de traces relatifs à un polyèdre,” Comptes Rendus de l'Académie des Sciences A, vol. 278, pp. 1581–1583, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin, Germany, 1966. View at Publisher · View at Google Scholar · View at MathSciNet
  43. H. Triebel, Fractals and Spectra related to Fourier Analysis and Function Spaces, vol. 91 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  44. A. Jonsson and H. Wallin, Function Spaces on Subset of ℝn, vol. 2, part 1 of Mathematical Reports, Harwood Academic Publishers, London, UK, 1984.