Mathematics – Analysis of PDEs
Scientific paper
2011-10-31
Mathematics
Analysis of PDEs
Scientific paper
Extending investigations of Antman & Malek-Madani, Schecter & Shearer, Slemrod, Barker & Lewicka & Zumbrun, and others, we investigate viscoelasticity models with surface energy effects. We prove the existence of nonconstant planar periodic standing waves in viscoelasticity models with surface energy terms by variational methods and phase-plane analysis, for deformations of arbitrary dimension. For our variational analysis, we require that the mean vector of the unknowns over one period be in the elliptic region with respect to the corresponding inviscid (i.e., elastic) model. For our (1-D) phase-plane analysis, we have no such restriction, obtaining essentially complete information on the existence of nonconstant periodic waves and bounding homoclinic/heteroclinic waves. Our variational framework has implications also for time-evolutionary stability, through the link between the action potential for the traveling-wave ODE and the relative entropy for the time-evolutionary system.
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