Global weak entropy solutions to quasilinear wave equations of Klein-Gordon and Sine-Gordon type

摘要

In this paper we establish the existence of global Lipschitz continuous solutions to the Cauchy problem for the one-dimensional quasilinear wave equation (1.1) Partial deriv_t~2 w - Partial deriv_xσ(Partial deriv_xw)+f(w)=0, for all (x,t) ∈ R x (0,∞), with initial conditions (1.2) w(x,0) = w_0(x), Partial deriv _t w(x,0) = W_1(x), for all x ∈ R. Here f is a smooth function with f(0) = 0 and σ is a given smooth function such that σ′(u) ≥ γ > 0 (γ > 0) and uσ″(u) > 0 for u ≠ 0; w_0 and w_1 are bounded functions with compact support, w_0 is also Lipschitz continuous.