Abstract
We consider the spectrum associated with the linear operatorobtained when a Cahn--Hilliard system on $\mathbb{R}$ is linearized abouta transition wave solution. In many cases it's possibleto show that the only non-negative eigenvalue is $\lambda = 0$,and so stability depends entirely on the nature of thisneutral eigenvalue. In such cases, we identify a stability conditionbased on an appropriate Evans function, and we verify thiscondition under strong structural conditions on our equations.More generally, we discuss and implement a straightforwardnumerical check of our condition, valid under mild structuralconditions.
Highlights
We consider the spectrum associated with transition wave solutions u(x), u(±∞) = u±, u− = u+, for Cahn–Hilliard systems on R, ut = M (u)(−Γuxx + f (u))x, (1)
We show that the operator L obtained by linearization of this system about the transition front solution u1(x) u2(x)
We look for stationary solutions u(x) for (1) that satisfy u(±∞) = u± ∈ M. (We recall that our notation M is defined in (H1).) Upon substitution of u(x) into (1), and after integrating twice and using f (u±) = 0, we find
Summary
Suppose u(x) denotes a transition front solution of system (14) with m = 2, and the following conditions hold: (i) F (u1, 1 − u1, 0) = F (1 − u1, u1, 0); u1 ∈ [0, 1]; (ii) γ1 = γ2; (iii) The operator Hb := −γ1∂xx + b11(x) + b12(x) is non-negative.
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