The BV Solutions For A Class Of Systems Of Quasilinear Hyperbolic Equations With Measures As Initial Conditions

In this paper we consider the problem for systems of quasilinear hyperbolic equations of the formin Q_T = R × (0, T) with initial conditionfor all x ∈ R, where m > 1, p_i q_i > 1 (i = 1,2), and μ, v is a non-negative finite Radon measure in R, and R = (-∞,+∞), a_i (i = 1,2) is given real number.Clearly,the Cauchy problem (1.1)-(1.2) has no classical solution in the general. Therefore we consider its BV solutions.Definition 1. A nonnegative function (u, v) : Q → R is said to be a solutin of (1.1), ifu ∈ L^∞ (0,+∞ ;L¹(R))∩C(0,+∞ ; L¹(R)) v ∈L^∞ (0, +∞;L¹(R))∩C(0,+∞;L¹(R)) satisfies the following conditions:[H1] For any τ∈ (0, +∞), we have u ∈ L^∞(R × (Ï„,∞ )), u ∈ L^∞(R × (Ï„ , +∞));[H2] For φ ∈ C₀^∞ (Q) and φ≥ 0, we have...