We consider an initial-boundary problem for a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures. Using Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions for the problem in two space dimensions.
Under natural regularity assumptions on the data the powers of regular elliptic boundary value problems (e.b.v.p.) are shown to be higher order regular e.b.v.p.. This result is used in description of the domains of fractional powers of elliptic operators which information is in order important in regularity considerations for solutions of semilinear parabolic equations. Presented approach allows to avoid C∞-smoothness assumption on the data that is typical in many references.
We consider positive solutions of the system ; in a ball or in the whole space, with . 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 . 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...
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].
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...
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...
Propagation of chemical waves at very low temperatures, observed experimentally [V.V. Barelko et al., Advances in Chem. Phys. 74 (1988), 339-384.] at velocities of order 10 cm/s, is due to a very non- standard physical mechanism. The energy liberated by the chemical reaction induces destruction of the material, thereby facilitating the reaction, a process very different from standard combustion. In this work we present recent experimental results and develop a new mathematical model which takes...