Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions (original) (raw)
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Nonlinear Analysis: Real World Applications, 2004
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NoDEA : Nonlinear Differential Equations and Applications, 2002
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Journal of Differential Equations, 2008
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We study the existence and stability of stationary solutions of an integrodifferential model for phase transitions, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. As such, this model is a nonlocal extension of the Allen Cahn equation, which incorporates long-range interactions. We find that the set of stationary solutions for this model is much larger than that of the Allen Cahn equation. Equation (1.2), recently proposed in ref. 3, can model a variety of physical and biological phenomena, e.g., a material whose state is described by an order parameter. Note that (1.2) is an L 2 -gradient flow of the free energy functional E(u)= 1 4 | R n | R n j(x& y)(u(x)&u( y)) 2 dx dy+ | R n W(u(x)) dx (1.3)