We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t → +∞. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t → +∞, with the sam...

The objective of this paper is to construct Riemann $k$-wave solutions the general form first-order quasilinear hyperbolic systems partial differential equations geometrically. To end, we adapt and combine elements two approaches construction $k$-waves, namely, symmetry reduction method generalized characteristics. We formulate a geometrical setting for problem discuss in detail conditions exis...

This article studies almost global existence for solutions of quadratically quasilinear systems of wave equations in three space dimensions. The approach here uses only the classical invariance of the wave operator under translations, spatial rotations, and scaling. Using these techniques we can handle wave equations in Minkowski space or Dirichletwave equations in the exterior of a smooth, sta...

In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x)|t ≥ 0, x ≥ 0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the...

Hyperbolic systems in one dimensional space are frequently used modeling of many physical systems. In our recent works, we introduced time independent feedbacks leading to the finite stabilization for optimal homogeneous linear and quasilinear hyperbolic this work, present Lyapunov's functions these use estimates rediscover results.

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large num...

In this paper, for the small initial data in suitably weighted Sobolev spaces on product space $ \mathbb{R}^2\times\mathbb{T} $, we establish global existence of smooth solutions to 3-D quasilinear Klein-Gordon equations with quadratic nonlinearities. It is noted that topic well-posedness nonlinear hyperbolic arises from studies propagation waves along infinite homogeneous waveguides and Kaluza...