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The paper studies the long time behavior of a system that describes the motion of a piece of elastic membrane driven by surface tension and inner air pressure. The system is a degenerate quasilinear hyperbolic one that involves the mean curvature, and also includes a damping term that models the dissipative nature of genuine physical systems. With the presence of damping, a small perturbation of the sphere converges exponentially in time to the sphere, and without the damping the evolution that is $varepsilon$-close to the sphere has life span longer than $varepsilon^{-1/6}$. Both results are proved using an improved Nash-Moser-Hormander theorem.
We consider the problem of the long time dynamics for a diffuse interface model for tumor growth. The model describes the growth of a tumor surrounded by host tissues in the presence of a nutrient and consists in a Cahn-Hilliard-type equation for the
We study the quasilinear non-local Benney System $$left{begin{array}{llll} iu_t+u_{xx}=|u|^2u+buv v_t+a(int_{mathbf{R}^+}v^2dx)v_x=-b(|u|^2)_x,quad (x,t)inmathbf{R}^+times [0,T],, T>0. end{array}right.$$ We establish the existence and uniqueness
In this paper, we consider the nonlinear inhomogeneous compressible elastic waves in three spatial dimensions when the density is a small disturbance around a constant state. In homogeneous case, the almost global existence was established by Klainer
The behavior of solutions to an initial boundary value problem for a hyperbolic system with relaxation is studied when the relaxation parameter is small, by using the method of Fourier Series and the energy method.
We describe the point and contact equivalence groupoids of an important class of two-dimensional quasilinear hyperbolic equations. In particular, we prove that this class is normalized in the usual sense with respect to point transformations, and its