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A concept of solution and numerical experiments for forward-backward diffusion equations. (English) Zbl 1105.35007
The authors study the gradient flow associated with the functional F φ (u):=1 2 I φ(u x )dx, where φ is non convex, and with its singular perturbation F φ x (u):=1 2 I (ε 2 (u xx ) 2 +φ(u x ))dx. With the support of numerical simulations, various aspects of the global dynamics of solutions u ε of the singularly perturbed equation u t =-ε 2 u xxxx +1 2φ '' (u x )u xx for small values of ε>0 are discussed. Their analysis leads to a reinterpretation of the unperturbed equation u tt =1 2(φ ' (u x )) x , and to a well defined notion of a solution. Examine the conjecture that this solution coincides with the limit of u ε as ε0 + is given.
MSC:
35B25Singular perturbations (PDE)
35K55Nonlinear parabolic equations
34E13Multiple scale methods (ODE)
49L25Viscosity solutions (infinite-dimensional problems)
35A15Variational methods (PDE)