On the propagation of perturbations for essentially nonautonomous quasilinear firstorder equations
 Yu. G. Rykov
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 firstorder 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 firstorder nonlinear equations, Usp. Mat. Nauk42 (1987), no. 5, 3–40.
 S.N. Kruzhkov, and F. Hidebrand,The Cauchy problem for firstorder 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,Firstorder 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 firstorder 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 firstorder 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 firstorder 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 Nonautonomous FirstOrder Quasilinear Equations, VINITI Akad. Nauk SSSR, no. 485B88 Dep.
 Yu.G. Rykov,On the nature of the perturbation's propagation in the Cauchy problem for firstorder quasilinear equations, Mat. Zametki42 (1987), no. 5, 712–722.
 A.N. Tikhonov, and A.A. Samarskii,On discontinuous solutions of firstorder 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,Largetime 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 quasilinear firstorder 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 ReactionDiffusion Equations, SpringerVerlag, New York, 1983.
 Title
 On the propagation of perturbations for essentially nonautonomous quasilinear firstorder equations
 Journal

Journal of Mathematical Sciences
Volume 75, Issue 3 , pp 16721690
 Cover Date
 19950601
 DOI
 10.1007/BF02368669
 Print ISSN
 10723374
 Online ISSN
 15738795
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Industry Sectors
 Authors