On the propagation of perturbations for essentially nonautonomous quasilinear first-order equations
- Yu. G. Rykov
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
This paper deals with generalized solutions of the Cauchy problem for the equationu t + [A(t, x, u)] x +B(t, x, u) = 0 (t, x) ∈ ℝ+ × ℝ. Here A, B may depend essentially on t, x; for example, they may tend to zero or to infinity as t becomes infinite. Sufficient conditions are obtained for the presence and the absence of finite time extinction and space localization. These phenomena have been studied earlier mainly for degenerate parabolic equations. In the case of first-order equations the situation is more complicated due to the discontinuity of solutions. The essential dependence of the coefficients on t, x gives rise to a threshold phenomenon: the presence of the finite time extinction depends on the maximum of the modulus of the initial function. Bibliography: 29 titles.
- A.I. Vol'pert,Spaces BV and quasilinear equations, Mat. Sb.73 (1967), no. 2, 255–302.
- I.M. Gel'fand,Some problems of the theory of quasilinear equations, Usp. Mat. Nauk14 (1959), no. 2, 87–158.
- V.M. Il'in and O.A. Oleinik,Asymptotic behavior of solutions of the Cauchy problem for some quasilinear equations for large values of time, Mat. Sb.51 (1960), no. 2, 191–216.
- S.G. Krein, and M.I. Khazan,Differential equations in Banach space, Itogi Nauki i Tekhniki. Mat. Analiz21 (1983), 130–264.
- S.N. Kruzhkov, and N.S. Petrosjan,Asymptotic behavior of solutions of the Cauchy problem for first-order nonlinear equations, Usp. Mat. Nauk42 (1987), no. 5, 3–40.
- S.N. Kruzhkov, and F. Hidebrand,The Cauchy problem for first-order nonlinear equations with unbounded domains of dependence on initial data, Vestn. Mosk. Univ., Ser. 1. Mat., Mekh. (1974), no. 1, 93–100.
- S.N. Kruzhkov,First-order quasilinear equations with several independent variables, Mat. Sb.81 (1970), no. 2, 228–255.
- S.N. Kruzhkov and P.A. Andreyanov,On nonlocal theory of the Cauchy problem for first-order quasilinear equations in the class of locally summable functions, Dokl. Akad. Nauk SSSR220 (1975), no. 1, 23–26.
- S.N. Kruzhkov,Nonlinear Partial Differential Equations, Part 2, Moscow, 1970.
- O.A. Ladyzhenskaya,On construction of discontinuous solutions of quasilinear hyperbolic equation as limits of corresponding parabolic equations with vanishing ‘viscosity coefficient’, Dokl. Akad. Nauk SSSR111 (1956), no. 2, 291–294.
- O.A. Oleinik,On the Cauchy problem for nonlinear equations in a class of discontinuous functions, Dokl. Akad. Nauk SSSR95 (1954, no. 3, 451–455.
- O.A. Oleinik,The Cauchy problem for first-order nonlinear differential equations with discontinuous initial data, Trudy Mosk. Mat. Obshch.5 (1956), 433–454.
- O.A. Oleinik,Discontinuous solutions of nonlinear differential equations, Usp. Mat. Mauk12 (1957), no. 3, 3–73.
- B.L. Rozhdestvenskii, and N.N. Yanenko,Systems of Quasilinear Equations and Their Applications in Gas Dynamics, Nauka, Moscow, 1978. (in Russian).
- Yu.G. Rykov,On the behavior of the supports of generalized solutions to first-order quasilinear equations for small and large values of time, Usp. Mat. Nauk41 (1986), no. 4, 172.
- Yu.G. Rykov,On Finite Time Stabilization of Generalized Solutions to the Cauchy Problem for Essentially Non-autonomous First-Order Quasilinear Equations, VINITI Akad. Nauk SSSR, no. 485-B88 Dep.
- Yu.G. Rykov,On the nature of the perturbation's propagation in the Cauchy problem for first-order quasilinear equations, Mat. Zametki42 (1987), no. 5, 712–722.
- A.N. Tikhonov, and A.A. Samarskii,On discontinuous solutions of first-order quasilinear equations, Dokl. Akad. Nauk SSSR99 (1954), no. 1, 27–30.
- M.I. Khazan,Nonlinear and quasilinear evolution equations: existence, uniqueness, and comparis on of solutions; rate of convergence of the difference method, Zap. Nauch. Semin. LOMI127 (1983), 181–200.
- P. Bauman, and D. Phillips,Large-time behavior of solutions to a scalar conservation law in several space dimensions, Trans. Amer. Math. Soc.298 (1986), no. 1, 401–419.
- Ph. Benilan,Equations d'evolution dans un espace de Banach quelconque et applications, These, Orsay, 1972.
- E.D. Conway E.D, and J. Smoller,Global solutions of the Cauchy problem for quasi-linear first-order equations in several space variables, Comm. Pure Appl Math.19 (1966), no. 1, 95–105.
- M.G. Crandall, and T.M. Ligget,Generation of semigroups of nonlinear transformations of general Banach spaces, Amer. J. Math. Soc.93 (1971), no. 2, 265–298.
- M.G. Crandall,The semigroup approach to first order quasilinear equations in several space variables, Israel. J. Math.12 (1972), no. 2, 168–192.
- J.I. Diaz, and L. Veron,Existence theory and qualitative properties of the solutions of some first order quasilinear variational inequalities, Indiana Univ. Math. J.32 (1983), no. 3, 319–361.
- E. Hopf,The partial differential equation u t +uu x =μu xx , Comm. Pure Appl. Math.3 (1950), no. 3, 201–230.
- P.D. Lax,Hyperbolic systems of conservation laws 2, Comm. Pure. Appl. Math.10 (1957), no. 4, 537–566.
- P.D. Lax,Weak solution of nonlinear hyperbolic equations and their numerical computation, Comm. Pure. Appl. Math.7 (1954), no. 1, 159–193.
- J. Smoller,Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
- On the propagation of perturbations for essentially nonautonomous quasilinear first-order equations
Journal of Mathematical Sciences
Volume 75, Issue 3 , pp 1672-1690
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- Industry Sectors