Locally Invariant Manifolds for Quasilinear Parabolic Equations

A Note on Quasilinear Parabolic Equations on Manifolds

We prove short time existence, uniqueness and continuous dependence on the initial data of smooth solutions of quasilinear locally parabolic equations of arbitrary even order on closed manifolds. CONTENTS

Quasilinear Parabolic Functional Evolution Equations

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest ...

Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds

Dimension splitting for quasilinear parabolic equations

In the current paper, we derive a rigorous convergence analysis for a broad range of splitting schemes applied to abstract nonlinear evolution equations, including the Lie and Peaceman–Rachford splittings. The analysis is in particular applicable to (possibly degenerate) quasilinear parabolic problems and their dimension splittings. The abstract framework is based on the theory of maximal dissi...

Invariant Properties of the Ansatz of the Hirota Method for Quasilinear Parabolic equations

We propose a new method based on the invariant properties of the ansatz of the Hirota method whcih have been discovered recently. This method allows one to construct new solutions for a certain class the dissipative equations classified by degrees of homogeneity. This algorithm is similar to the method of “dressing” the solutions of integrable equations. A class of new solutions is constructed....