Geodesic
From weak discontinuity to gradient catastrophe
Sbornik. Mathematics, Tome 192 (2001) no. 10, pp. 1417-1433 
S. V. Zakharov; A. M. Il'in. From weak discontinuity to gradient catastrophe. Sbornik. Mathematics, Tome 192 (2001) no. 10, pp. 1417-1433. http://geodesic.mathdoc.fr/item/SM_2001_192_10_a0/
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     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_10_a0/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Cauchy problem for a quasilinear parabolic equation with small parameter at the highest derivative is considered in the case when the solution of the degenerate equation has a weak discontinuity subsequently turning into a strong discontinuity. The singularities that the coefficients of the asymptotic formula representing the solution in the boundary layer of the weak discontinuity develop on approaching the point of the gradient catastrophe are analysed.

[1] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[2] Sushko V. G., “Ob asimptoticheskikh razlozheniyakh reshenii odnogo parabolicheskogo uravneniya s malym parametrom”, Differents. uravneniya, 21:10 (1985), 1794–1798 | MR | Zbl

[3] Hopf E., “The partial differential equation $u_t+uu_x=\mu u_{xx}$”, Comm. Pure Appl. Math., 3:3 (1950), 201–230 | DOI | MR | Zbl

[4] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[5] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl