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Equilibrium validation in models for pattern formation based on Sobolev embeddings. (English) Zbl 1465.35050
Summary: In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.
MSC:
35B36 Pattern formations in context of PDEs
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35K35 Initial-boundary value problems for higher-order parabolic equations
35K58 Semilinear parabolic equations
37M20 Computational methods for bifurcation problems in dynamical systems
65G20 Algorithms with automatic result verification
65G30 Interval and finite arithmetic
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
Software:
MATCONT; AUTO; INTLAB
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References:
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