, Volume 75, Issue 3, pp 1672-1690

On the propagation of perturbations for essentially nonautonomous quasilinear first-order equations

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Abstract

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.

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 89–117, 1994.