Instability Of Traveling Waves For A Sixth-order Parabolic Equation

In recent years, many math models which are sixth-order parabolic equation are used to describe semiconductor physics, epitaxial thin film growth, image process-ing and so on. So many mathematician have dedicated to study sixth-order parabolic equations. In this paper, we consider the instability of traveling waves for the following sixth-order parabolic equation where g(u)=a(1-u2), a>0 is nonlinear function. It is derived from the quantum theory, and u(x, t) is surface slope. First, we use direct integral method to get the exact solutions of (1) which satisfy certain conditions. Then we give the important lemma which W. Strauss and G. Wang had used in the problem of evolution equation where L is a linear operator that generates a strongly continuous semigroup etL on a Banach space X, and F is a strongly continuous operator such that F(0)=0.Lemma Assume the following(i) X, Z are two Banach spaces with X c Z and‖u‖Z≤c1‖u‖X for u∈X.(ii) L generates a strongly continuous semigroup etL on the space Z, and the semigroup etL maps Z into X for t>0 and∫01‖etL‖Z→Xdt=C4<∞.(ⅲ) The spectrum of L on X meets the right half-plane, {Reλ>0}.(ⅳ) F:X→Z is continuous and (?)ρ0>0,C3>0,a>1 such that‖F(u)‖Z< C0‖a‖xa,for‖u‖X<(?)0.Then the zero solution of (2) is nonlinearly unstable in the space X.We used the lemma to translate the stability of traveling wave solutions into the stability of the zero solutions, and according to the problems we proved the conditions in the lemma, and then we got the proof of stability of traveling wave solutions of (1).