Incomplete Self‐Similar Blow‐Up in a Semilinear Fourth‐Order Reaction‐Diffusion Equation

Blow‐up behavior for the fourth‐order semilinear reaction‐diffusion equation (1) is studied. For the classic semilinear heat equation from combustion theory (2) various blow‐up patterns were investigated since 1970s, while the case of higher‐order diffusion was studied much less. Blow‐up self‐similar solutions of (1) of the form are constructed. These are shown to admit global similarity extensions for t > T : The continuity at t = T is preserved in the sense that This is in a striking difference with blow‐up for (2) , which is known to be always complete in the sense that the minimal (proper) extension beyond blow‐up is u(x, t) ≡+∞ for t > T . Difficult fourth‐order dynamical systems for extension pairs {f(y), F(y)} are studied by a combination of various analytic, formal, and numerical methods. Other nonsimilarity patterns for (1) with nongeneric complete blow‐up are also discussed.     

Phonon-plasmon interaction in tunneling GaAs/AlAs superlattices

The phonon-plasmon interaction in tunneling GaAs _n /AlAs _m superlattices (m=5and 6≥n≥0.6 monolayers) was studied by Raman scattering spectroscopy. The interaction of optical phonons localized in GaAs and AlAs layers with quasi-three-dimensional plasmons strengthens as the thickness of GaAs quantum wells decreases and the electronic states in the superlattices become delocalized due to tunneling. It is assumed that the plasmons also interact with the TO-like phonon modes localized in quantum islands or in thin ruffled layers.     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

Stimulation effect in anisotropic superconductors

The stimulation of superconductivity in anisotropic superconductors by electromagnetic and acoustic pumping as well as by the injection of a tunnel current at temperatures close to the superconducting transition temperature is studied. The features distinguishing the stimulation effect from the isotropic case are indicated. Zh. éksp. Teor. Fiz. 111, 2134–2146 (June 1997)