Skip to main content
Log in

Multi-dimensional initial-boundary value problems with strong nonlinearities

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

References

  1. H. Amann, Periodic Solutions of Semilinear Parabolic Equations, in Nonlinear Analysis, ed. by Cesari et al., Academic Press, New York, 1978, pp. 1–29.

    Google Scholar 

  2. D. G. Aronson, Linear Parabolic Differential Equations Containing a Small Parameter, J. Rational Mech. Anal. 5 (1956), 1003–1014.

    Google Scholar 

  3. N. S. Bakhvalov, On the Asymptotics with Small ε of the Solution of the Equation ut+(φ(u))x=εuxx, Corresponding to a Rarefaction Wave, U.S.S.R. Comp. Math. Physics 6 (1966), 152–160.

    Google Scholar 

  4. C. Bardos, A. Y. Le Roux & J. C. Nedelec, First Order Quasilinear Equations with Boundary Conditions, Comm. Partial Diff. Eqns. 4 (1979), 1017–1034.

    Google Scholar 

  5. L. E. Bobisud, Second-Order Linear Parabolic Equations with a Small Parameter, Arch. Rational Mech. Anal. 27 (1968), 385–397.

    Google Scholar 

  6. L. E. Bobisud, Parabolic Equations with a Small Parameter and Discontinuous Data, J. Math. Anal. Appl. 26 (1969), 208–220.

    Google Scholar 

  7. J. D. Cole, On a Quasilinear Parabolic Equation Occurring in Aerodynamics, Quart. Appl. Math. 9 (1951), 225–236.

    Google Scholar 

  8. M. G. Crandall, Viscosity Solutions of Hamilton-Jacobi Equations, in Nonlinear Problems: Present and Future, ed. by Bishop et al., North-Holland, Amsterdam, 1982, pp. 117–125.

    Google Scholar 

  9. C. M. Dafermos, Asymptotic Behavior of Solutions of Hyperbolic Balance Laws, in Bifurcation Phenomena in Mathematical Physics and Related Topics, ed. by C. Bardos & D. Bessis, D. Reidel, Holland, 1980, pp. 521–533.

    Google Scholar 

  10. P. C. Fife, Propagating Fronts in Reactive Media, in Nonlinear Problems: Present and Future, ed. by Bishop et al., North-Holland, Amsterdam, 1982, pp. 267–285.

    Google Scholar 

  11. W. H. Fleming, The Cauchy Problem for a Nonlinear First Order Partial Differential Equation, J. Diff. Eqns. 5 (1969), 515–530.

    Google Scholar 

  12. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.

    Google Scholar 

  13. A. van Harten, Feed-Back Control of Singularly Perturbed Heating Problems, in Lecture Notes in Math., Vol. 711, Springer-Verlag, New York, 1979, pp. 33–62.

    Google Scholar 

  14. C. J. Holland, Singular Perturbations in the First Boundary Value Problem for Parabolic Equations, SIAM J. Math. Anal. 8 (1977), 368–374.

    Google Scholar 

  15. E. Hopf, The Partial Differential Equation ut+uux=μuxx, Comm. Pure Appl. Math. 3 (1950), 201–230.

    Google Scholar 

  16. F. A. Howes, Perturbed Boundary Value Problems Whose Reduced Solutions are Nonsmooth, Indiana U. Math. J. 30 (1981), 267–280.

    Google Scholar 

  17. S. L. Kamenomostskaya, On Equations of Elliptic and Parabolic Type with a Small Parameter in the Highest Derivatives (in Russian), Mat. Sbornik 31 (1952), 703–708.

    Google Scholar 

  18. J. Kevorkian & J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.

    Google Scholar 

  19. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS Series in Appl. Math., Vol. 11, SIAM, Philadelphia, 1973.

    Google Scholar 

  20. N. Levinson, The First Boundary Value Problem for εΔu + A(x,y) u x + B(x,y) u y + C(x,y) u = D(x,y) for Small ε, Ann. Math. 51 (1950), 428–445.

    Google Scholar 

  21. J. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Math., vol. 323, Springer-Verlag, New York, 1973.

    Google Scholar 

  22. P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, 1982.

    Google Scholar 

  23. J. D. Murray, Singular Perturbations of a Class of Nonlinear Hyperbolic and Parabolic Equations, J. Math. and Physics 47 (1968), 111–133.

    Google Scholar 

  24. O. A. Oleinik, Discontinuous Solutions of Non-Linear Differential Equations, Uspehi Mat. Nauk 12 (1975), 3–73; Amer. Math. Soc. Translations (Ser. 2) 26 (1962) 95–172.

    Google Scholar 

  25. A. B. Vasil'eva & V. F. Butuzov, Singularly Perturbed Equations of Parabolic Type, in Lecture Notes in Math., Vol. 985, Springer-Verlag, New York, 1983, pp. 38–75.

    Google Scholar 

  26. M. I. Vishik & L. A. Liusternik, Regular Degeneration and Boundary Layer for Linear Differential Equations with a Small Parameter, Uspehi Mat. Nauk 12 (1957), 3–122; Amer. Math. Soc. Translations (Ser. 2) 20 (1962), 239–364.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by C. Dafermos

Rights and permissions

Reprints and permissions

About this article

Cite this article

Howes, F.A. Multi-dimensional initial-boundary value problems with strong nonlinearities. Arch. Rational Mech. Anal. 91, 153–168 (1986). https://doi.org/10.1007/BF00276861

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00276861

Keywords

Navigation