Browsing by Author "Zanini, Chiara"

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    An artificial viscosity approach to quasistatic crack growth
    (2006-07-26T10:13:39Z) Toader, Rodica; Zanini, Chiara; Mathematics; Functional Analysis and Applications
    We introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $\epsilon$-gradient flow of the energy functional, as the "viscosity" parameter $\epsilon$ tends to zero.
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    Multiplicity of periodic solutions for differential equations arising in the study of a nerve fiber model
    (2006-07-21T11:14:25Z) Zanini, Chiara; Zanolin, Fabio; Mathematics; Functional Analysis and Applications
    We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a result of existence of two positive solutions, the different situations depending by a threshold parameter related to the integral of the weight function $n(x).$ Here we show that the number of positive periodic solutions may be very large for some special choices of a (large) weight $n.$ We also obtain the existence of subharmonic solutions of any order. The proofs are based on the Poincar\'{e} - Bikhoff fixed point theorem.
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    Quasistatic crack growth for a cohesive zone model with prescribed crack path
    (2005-06-20T12:05:32Z) Dal Maso, Gianni; Zanini, Chiara; Mathematics; Functional Analysis and Applications
    In this paper we study the quasistatic crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.
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    Singular perturbations of finite dimensional gradient flows
    (2006-07-26T08:45:16Z) Zanini, Chiara; Mathematics; Functional Analysis and Applications
    In this paper we give a description of the asymptotic behavior, as $\epsilon\to 0$, of the $\epsilon$-gradient flow in the finite dimensional case. Under very general assumptions we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic solutions of the gradient flow (fast dynamics).