On the solution of the equation

Authors:
K. T. Joseph and P. L. Sachdev

Journal:
Quart. Appl. Math. **52** (1994), 519-527

MSC:
Primary 35L65; Secondary 35Q53

DOI:
https://doi.org/10.1090/qam/1292202

MathSciNet review:
MR1292202

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the equation and derive a transformation relating it to . Special cases of the equation appearing in applications are discussed. Initial value problems and asymptotic behaviour of the solution are studied.

**[1]**C. Bardos, A. Y. Leroux, and J. C. Nedelec,*First-order quasilinear equation with boundary condition*, Comm. Partial Differential Equations**4**, 1017-1037 (1979)**[2]**D. G. Crighton,*Model equations of nonlinear acoustics*, Ann. Rev. Fluid Mech.**11**, 11-33 (1979)**[3]**D. G. Crighton and J. F. Scott,*Asymptotic solutions of model equations in nonlinear acoustics*, Philos. Trans. Roy. Soc. London Ser. A**292**, 101-134 (1979)**[4]**C. M. Dafermos,*Regularity and large time behaviour of solutions of conservation law without convexity condition*, Proc. Roy. Soc. Edinburgh Sect. A**99**, 201-239 (1985)**[5]**F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin,*Self-similar spin-up spin-down in a cylinder of small ratio of height to diameter*, J. Fluid Mech.**234**, 473-486 (1992)**[6]**E. Hopf,*The partial differential equation*, Comm. Pure Appl. Math.**3**, 201-230 (1950)**[7]**P. D. Lax,*Hyperbolic systems of conservation laws*. II, Comm. Pure Appl. Math.**10**, 537-566 (1957)**[8]**P. Lefloch,*Explicit formula for scalar non-linear conservation laws with boundary condition*, Math. Methods Appl. Sci.**10**, 265-287 (1988)**[9]**J. D. Murray,*Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations*, Siam J. Appl. Math.**19**, 135-160 (1970)**[10]**J. D. Murray,*Lectures on nonlinear differential equation models in biology*, Oxford Univ. Press, New York, 1977**[11]**J. J. C. Nimmo and D. G. Crighton,*Geometrical and diffusive effects in non-linear acoustic propagation over long ranges*, Philos. Trans. Roy. Soc. London Ser. A**320**, 1-35 (1986)**[12]**P. L. Sachdev,*Nonlinear diffusive waves*, Cambridge Univ. Press, Cambridge, 1987**[13]**P. L. Sachdev, V. G. Tikekar, and K. R. C. Nair,*Evolution and decay of spherical and cylindrical N waves*, J. Fluid Mech.**172**, 347-371 (1986)**[14]**P. L. Sachdev, K. T. Joseph, and K. R. C. Nair,*Exact N-wave solutions for the non-planar Burgers equation*, Proc. Roy. Soc. London (A), to appear (1994)**[15]**C. C. Shih,*Attenuation characteristics of nonlinear pressure waves propagating in pipes*, Finite Amplitude Wave Effects in Fluids (Bjorno, ed.), IPC Sci. Technol. Press, Guildford, 1974, pp. 81-87**[16]**E. H. Wedemeyer,*The unsteady flow within a spinning cylinder*, J. Fluid Mech.**20**, 383-399 (1964)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35L65,
35Q53

Retrieve articles in all journals with MSC: 35L65, 35Q53

Additional Information

DOI:
https://doi.org/10.1090/qam/1292202

Article copyright:
© Copyright 1994
American Mathematical Society