Rearrangements and fourth order equations

Fourth Order Equations In

Travelling Waves for Fourth Order Parabolic Equations

We study travelling wave solutions for a class of fourth order parabolic equations. Travelling wave fronts of the form u(x, t) = U(x + ct), connecting homogeneous states, are proven to exist in various cases: connections between two stable states, as well as connections between an unstable and a stable state, are considered.

Fourth Order Diffusion Equations with Increasing Entropy

The general quasi-linear autonomous fourth order diffusion equation ut = −[G(u)uxxx + h(u, ux, uxx)]x with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may...

Oscillation of Fourth-order Dynamic Equations

In this paper we shall reduce the problem of the oscillation of all solutions of certain nonlinear fourth-order dynamic equations to the problem of oscillation of two second-order dynamic equations, which are discussed intensively in the literature. Further oscillation criteria of fourth-order equations are given and proved using integration and Taylor’s formula on time scales. Some conditions ...

Decay Estimates for Fourth Order Wave Equations

may be thought of as a nonlinear beam equation. In this paper we obtain both L-L estimates and space-time integrability estimates on solutions to the linear equation. We also use these estimates to study the local existence and asymptotic behavior of solutions to the nonlinear equation, for nonlinear terms which grow like a certain power of u. The main L-L estimate (Theorem 2.1) states that sol...