In this paper we deal with some optimal control problems for a solidification phase field model of metallic alloys. The model allows crystallizations of two kinds, each one described by its own phase field. Accordingly, the state is the triplet (τ, u, v), where τ is the temperature and u and v are phase field functions. The optimality conditions for the optimal control problems considered in this work are obtained by using the Dubovitskii-Milyutin formalism.

In this paper we present some theoretical results for a system of nonlinear partial differential equations that provide a phase field model for the solidification/melting of a metallic alloy. It is assumed that two different kinds of crystallization are possible.Consequently, the unknowns are the temperature τ and the phase field functions u and v. The time derivatives u t and v t appear in the equation for τ (the heat equation). On the other hand, the equations for u and v contain nonlinear terms where we find τ .

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This paper is concerned with dynamics of elasticity systems featuring a nonlinear foundation of critical growth and delay effects. The existence of global attractors for such systems has been studied recently. Our main contribution establishes the upper-semicontinuity of attractors with respect to a small parameter multiplying the delay term. Our results are new even for the analogous scalar wave equation.