Connecting orbits for nonlinear parabolic equations

On Connecting Orbits of Semilinear Parabolic Equations on S

It is well-known that any bounded orbit of semilinear parabolic equations of the form ut = uxx + f(u, ux), x ∈ S 1 = R/Z, t > 0, converges to steady states or rotating waves (non-constant solutions of the form U(x − ct)) under suitable conditions on f . Let S be the set of steady states and rotating waves (up to shift). Introducing new concepts — the clusters and the structure of S —, we clarif...

Heteroclinic orbits of semilinear parabolic equations

Linearly implicit methods for nonlinear parabolic equations

We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes...

On Normal Stability for Nonlinear Parabolic Equations

We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional C-manifold which is normally stable.

Renormalization group methods for nonlinear parabolic equations

Renormalization group (RG) methods are described for determining the key exponents related to the decay of solutions to nonlinear parabolic differential equations. Higher order (in the small coefficient of the nonlinearity) methods are developed. Exact solutions and theorems in some special cases confirm the RG results.