232 research outputs found
The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor
No full textGrinfeld, M.; Novick-Cohen, A.. (1997). The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/3058
Introduction to the special issue
No full textJamie Davies, Michael Grinfeld and Steven D. Webb provide an introduction to the special issue of theory in bioscience
Steady-state solutions of a mass-conserving bistable equation with a saturating flux
No full textWe consider a mass-conserving bistable equation with a saturating flux on an interval. This is the quasilinear analogue of the Rubinstein–Steinberg equation, suitable for description of order parameter conserving solid–solid phase transitions in the case of large spatial gradients in the order parameter. We discuss stationary solutions and investigate the change in bifurcation diagrams as the mass constraint and the length of the interval are varied
Non-local dispersal and bistability
No full textThe scalar initial value problem [ u_t = ho Du + f(u), ] is a model for dispersal. Here represents the density at point of a compact spatial region and time , and is a function of with values in some function space . is a bounded linear operator and is a bistable nonlinearity for the associated ODE . Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case ). We develop a technique for proving that indeed convergence does hold for small and show by constructing a counter-example that this result does not hold in general for all
Some remarks on stability for a phase-field model with memory
No full textThe phase field system with memory can be viewed as a phenomenological extension of the classical phase equations in which memory effects have been taken into account in both fields. Such memory effects could be important for example during phase transition in polymer melts in the proximity of the glass transition temperature where configurational degrees of freedom in the polymer melt constitute slowly relaxing "internal modes" which are di±cult to model explicitly. They should be relevant in particular to glass-liquid-glass transitions where re-entrance effects have been recently reported [27]. We note that in numerical studies based on sharp interface equations obtained from (PFM), grains have been seen to rotate as they shrink [35, 36]. While further modelling and numerical efforts are now being undertaken, the present manuscript is devoted to strengthening the analytical underpinnings of the model
Well-posedness and stationary solutions
No full textIn this paper we prove existence and uniqueness of variational inequality solutions for a bistable quasilinear parabolic equation arising in the theory of solid-solid phase transitions and discuss its stationary solutions, which can be discontinuous
Bifurcations in the regularized Ericksen bar model
No full textWe consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Muller's conjecture [18] concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of Muller's conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a "loop" structure that characterizes (k, 3k) bifurcations
Finite to infinite steady state solutions, bifurcations of an integro-differential equation
No full textWe consider a bistable integral equation which governs the stationary solutions of a convolution model of solid–solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi–discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem
Assessing lay theories of psychotherapy using the Q-sort method
No full textThis study used the Q-sort method and qualitative interview to gather exploratory data on what potential psychotherapy clients think that therapists should be doing to best help their clients. While current literature suggests that clinical psychologists generally align themselves with either “psychodynamic/humanistic” or “CBT/scientific” views of psychotherapy, little is known regarding whether individuals seeking psychotherapy have similar views of the therapy process. Fifty-eight Q-sort statements addressing what psychotherapy should be like were created in consultation with clinical psychologists. Forty participants, twenty from the general population and twenty seeking therapy at a college counseling center, took part in the study. The procedure involved ranking Q-sort statements according to level of agreement and answering follow-up questions during an interview. Results indicated that participants did have worldviews, or “lay theories” of psychotherapy that corresponded to different approaches to conducting therapy. Factor analysis yielded two distinct groups of participants, with one group endorsing a more “unstructured” type of therapy in which open-ended reflection and exploration were considered important for therapy but having a formal diagnosis was not considered as crucial, and the other characterized by a more “structured” type of therapy in which formal diagnosis, goal setting and problem solving were considered more important. Other significant findings included an overall preference by all participants for therapy to be uniquely tailored to the individual client, for therapists to help the client talk about the past, and for therapists to focus on interpretation of clients’ communications.Psy. D.Includes bibliographical referencesby Lisa Grinfeld Mose
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