Globally Lipschitz continuous solutions to a class of quasilinear wave equations

Abstract

This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax's method and generalized Glimm's method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.

Global solutions to a class of nonlinear damped wave operator equations

“…Equation (1.1) appears naturally in the study for liquid crystals [1][2][3][4]. In addition, Chang et al [5], Su [6] and Kian [7] where (x,t) R × R + , u 0 (x),ω 0 (x) R. Furthermore, Nishihara [8] and Hayashi [9] obtained the global solution to one dimension semilinear damped wave equation…”

Global solutions for initial–boundary value problem of quasilinear wave equations

“…consider (1.5)), although we avoid the resonance, but we face the difficulty that the term ∂G/∂U of (1.5) does not satisfy the conditions described in [6,8,15,30]. To overcome the above difficulties, we recently considered the Cauchy problem of (1.1) with more general flux, and proved the existence of globally Lipschitz continuous solution by applying the Lax's method and generalized Glimm's method, see [4]. More precisely, we modify the original Riemann problem into a perturbed Riemann problem.…”

“…Motivated by the work [4], we consider the initial-boundary value problem (1.1) in this article. Unfortunately, due to the boundary condition, the method of [4] for the construction of approximate solutions near the boundary layer cannot be applied directly.…”

“…Unfortunately, due to the boundary condition, the method of [4] for the construction of approximate solutions near the boundary layer cannot be applied directly. To overcome such a difficulty, in Section 3, we consider the initial-boundary Riemann problem of (1.3) near the boundary layer and construct the approximate solutions by using the contraction mapping principle.…”

Global entropy solutions of the general nonlinear hyperbolic balance laws with time-evolution flux and source

“…The method in [18] showed that incorporating the source term as a wave gave sharp time independent bounds for solutions of the initial value problem, while the operator-splitting method gave only time dependent bounds in this nonstrictly hyperbolic setting. Recently, this framework was extended to quasi-linear wave equations [4,14,29],…”