Higher Dimensional SLEP Equation and Applications to Morphological Stability in Polymer Problems

• Published in: SIAM journal on mathematical analysis Vol. 36; no. 3; pp. 916 - 966
• Main Authors: Nishiura, Yasumasa, Suzuki, Hiromasa
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2005
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Morphology; Partial differential equations; Sciences and techniques of general use; Matched asymptotic expansion; Patterning; Existence of solution; 35B35; Singular perturbation; reaction diffusion system; Eigenvalue; critical eigenvalues 35B25; Stability criterion; Reaction diffusion equation; 35K57; Mathematical analysis; Diblock copolymer; pattern formation; stability
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/S0036141002420157

Existence and stability of stationary internal layered solutions to a rescaled diblock copolymer equation are studied in higher dimensional space. Rescaling is necessary since the characteristic domain size of any stable pattern eventually vanishes in an appropriate singular limit. A general sufficient condition for the existence of singularly perturbed solutions and the associated stability criterion are given in the form of linear operators acting only on the limiting location of the interface. Applying the results to radially symmetric and planar patterns, we can show, for instance, stability of radially symmetric patterns when one of the components of diblock copolymer dominates the other, and that of the long-striped pattern in a long and narrow domain for the planar case. These results are consistent with the experimental ones. The above existence and stability criterion can be easily extended to a class of reaction diffusion systems of activator-inhibitor type.