The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

Abstract

We consider the class of equations ut=f(uxx, ux, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain − ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=ϕ(x−ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=ϕ(x−ct) of ut = F(uxx, uxy, ux, uy, u). Finally, we consider the class of equations ut=f(uxx, ux, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=ϕ(x).