Large-time structure of solutions of scalar conservation laws without convexity in the presence of a linear source field
type: Item
instance of: scholarly article
title: Large-time structure of solutions of scalar conservation laws without convexity in the presence of a linear source field
zbMATH Open document ID: 0757.35014
DOI: 10.1016/0022-0396(92)90026-J
author: Athanasios N. Lyberopoulos
published in: Journal of Differential Equations
publication date: 1993-01-16T00:00:00Z
review text: The problem studied here is the scalar law \(u_ t+[f(u)]_ x+g(u)=0\), \(x\in R^ 1\), \(u(x,0)=u_ 0(x)\), \(u_ 0\) \(L\)-periodic, with mean \(0:\int_ 0^ L u_ 0(x)dx=0\). The model in mind has \(f(u)=| u|^{p-1}u\), \(p>2\). The author proves that, as \(t\to\infty\), the solution remains in a bounded set whose size is independent of the size of the initial data. Further, for sufficiently large times, the number of shocks per period will attain a finite value which then remains a constant number. The proofs involve a detailed use of the theory of generalized characteristics via the use of Fillipov's theory of differential inclusions.
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 93320
zbMATH Keywords: admissible weak solutions
zbMATH Keywords: Cauchy data
zbMATH Keywords: left-contact discontinuities
zbMATH Keywords: nonlinear system of functional and delay-differential equations
zbMATH Keywords: generalized characteristics
zbMATH Keywords: Fillipov's theory of differential inclusions
reviewed by: Emmanuel N. Barron
MaRDI profile type: Publication
cites work: Asymptotic behavior for a hyperbolic conservation law with periodic initial data
cites work: Generalized characteristics in hyperbolic systems of conservation laws
cites work: Q4157095
cites work: Q3915362
cites work: Regularity and large time behaviour of solutions of a conservation law without convexity
cites work: Q3320768
cites work: Q4067483
cites work: Differential equations with discontinuous right-hand side
cites work: Decay of periodic solutions of \(\partial u/\partial t + \partial f(u)/\partial x= 0\)
cites work: The Cauchy problem for some classes of quasi-linear parabolic equations
cites work: Hyperbolic systems of conservation laws II
cites work: Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws
cites work: Quasilinear hyperbolic systems
cites work: Asymptotic oscillations of solutions of scalar conservation laws with convexity under the action of a linear excitation
cites work: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation
cites work: THE SPACES $ BV$ AND QUASILINEAR EQUATIONS
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description: scientific article; zbMATH DE number 93320