Class Of Quasilinear Evolution Equations Research Articles

Quasilinear Evolution Equations in LμP-Spaces with Lower Regular Initial Data

We study the Cauchy problem of the quasilinear evolution equations in Lμp-spaces. Based on the theories of maximal Lp-regularity of sectorial operators, interpolation spaces, and time-weighted Lp-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in Lp,q-spaces with temporal weights.

Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type

Semidiscretization in time is studied for a class of quasi-linear evolution equations in a framework due to Kato, which applies to symmetric first-order hyperbolic systems and to a variety of fluid and wave equations. In the regime where the solution is sufficiently regular, we show stability and optimal-order convergence of the linearly implicit and fully implicit midpoint rules and of higher-order implicit Runge–Kutta methods that are algebraically stable and coercive, such as the collocation methods at Gauss nodes.

Existence and asymptotic behaviour of solutions for a class of quasi‐linear evolution equations with non‐linear damping and source terms

AbstractWe consider a class of quasi‐linear evolution equations with non‐linear damping and source terms arising from the models of non‐linear viscoelasticity. By a Galerkin approximation scheme combined with the potential well method we prove that when m<p, where m(⩾0) and p are, respectively, the growth orders of the non‐linear strain terms and the source term, under appropriate conditions, the initial boundary value problem of the above‐mentioned equations admits global weak solutions and the solutions decay to zero as t→∞. Copyright © 2002 John Wiley & Sons, Ltd.

Blowup of solutions for a class of quasilinear evolution equations

Blowup of solutions for a class of quasilinear evolution equations

Estimates of rate of propagation of perturbations in quasilinear divergent degenerate parabolic equations of high order

A method is given for obtaining integral growth lemmas for solutions of boundary problems for a large class of quasilinear evolution equations. As a possible application a sharp estimate is obtained of the dependence of the support of a solution of a mixed problem and a Cauchy problem for a quasilinear divergent parabolic equation on time.

Tu commutativity for quasilinear evolution equations

A certain class of quasilinear evolution equations is shown to have the Tu commutativity property: two flows commute if they commute with a third flow in the class. The hamiltonian structure of these equations is also discussed.