16 citations · 41 across the 12 of their papers we have counts for
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Self-similar blow-up in parabolic equations of Monge--Ampère type
C. R. Budd, V. A. Galaktionov
We propose nonlinear parabolic equations of Monge--Ampére (M--A) type that admit regional, single poin, and global blow-up of similarity type. A similar model is derived for a four…
On blow-up shock waves for a nonlinear PDE associated with Euler equations
V. A. Galaktionov
A second-order PDE is derived from Euler's equaitons under certain assumptions. It is shown that this PDE admits shock and rarefaction waves, and that a single point gradient blow-…
Shock waves and compactons for fifth-order nonlinear dispersion equaitons
V. A. Galaktionov
Fifth-order 1D nonlinear dispersion equations are shown to admit blow-up formation of shock waves as well as rarefaction waves. The concepts of smooth deformations are applied to d…
Incomplete self-similar blow-up in a semilinear fourth-order reaction-diffusion equation
V. A. Galaktionov
It is shown that self-similar blow-up for a fourth-order reaction-diffusion equation is incomplete in the sense that, in general, there exists a self-similar extension of solutions…
On convergence in smooth gradient systems with branching of equilibria
V. A. Galaktionov, S. I. Pohozaev, A. E. Shishkov
T.I. Zelenyak's ideas and method of 1968 are shown to apply to N-dimensional second- and higher-order parabolic equtions and ensure a fast exponential convergence to degenerate equ…
Nonlinear dispersion equations: smooth deformations, compactons, and extensions to higher orders
V. A. Galaktionov
Third-order nonlinear dispersion equations (NDEs) are shown to admit both shock and rarefaction waves (as weak solutions), which are distinguished by a smooth deformation approach.…