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Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

We consider positive solutions of the system u t - Δ u = v p ; v t - Δ v = u q in a ball or in the whole space, with p , q > 1 . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case p = q . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for the final...

Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order

V. Vougalter, V. Volpert (2012)

Mathematical Modelling of Natural Phenomena

We obtain solvability conditions in H6(ℝ3) for a sixth order partial differential equation which is the linearized Cahn-Hilliard problem using the results derived for a Schrödinger type operator without Fredholm property in our preceding article [18].

Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

Luise Blank, Martin Butz, Harald Garcke (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading...

Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

Luise Blank, Martin Butz, Harald Garcke (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space...

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