Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order

We consider the equation u _tt + A(u _t ) + B(u) = 0, where A and B are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the minimum growth of any nonzero solution of the mixed problem at infinity.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 239–249, February, 2005.     

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

Kinetics and Mechanism of the Photoinduced Addition of Water and Methanol to the Double Bond of 2,2,4-Trimethyl-8-Methoxy-1,2-Dihydroquinoline

The mechanism of the photoinduced Markovnikov addition of water and methanol to the double bond of 2,2,4-trimethyl-8-methoxy-1,2-dihydroquinoline was studied by flash photolysis over a wide range of acid and alkali concentrations. In neutral MeOH solutions, the successive formation of two active intermediates was observed. These intermediates have absorption bands at _max = 420 and 500 nm, respectively, and lifetimes of a few milliseconds. The rate constants of decay for these species (k ₁ and k ₂) were measured over the temperature range from 10 to 45°C, and the activation energies E ₁ (3.7 ± 0.4) and E ₂ (6.9 ± 0.7) kcal/mol were determined in MeOH. The addition of an acid accelerates the conversion of the former intermediate into the latter. The rate constant for the reaction of the former intermediate with the acid is (2.0 ± 0.4) × 10⁸ l mol^–1 s^–1. The latter intermediate was proposed to be a corresponding carbocation. In H₂O at pH < 9, the formation of the carbocation (_max = 500 nm) was observed on a millisecond timescale, with the lifetime being an order of magnitude higher than that in MeOH. In alkaline media, only the former intermediate (_max = 420 nm) was observed with a lifetime of 150 or 400 ms in MeOH or H₂O, respectively, at [NaOH] > 2 × 10^–3 mol/l. The quantum yields of the product and the quantum yields of fluorescence were measured as functions of acidity. Possible mechanisms of the reaction are discussed.     

Diffusion versus absorption in semilinear elliptic equations

We study the limit behaviour of a sequence of singular solutions of a nonlinear elliptic equation with a strongly degenerate absorption term at the boundary of the domain. We give sharp conditions on the rate of degeneracy in terms of the distance to the boundary in order the pointwise singularity not to propagate along the boundary.     

Experimental investigation of laval nozzles with gas blown asymmetrically

An experimental determination was made of the transverse forces resulting from asymmetric blowing of a transverse gas jet into the supersonic part of a Laval nozzle. The experimental data are generalized on the basis of the analogy between blowing and flow over an equivalent body and using the generalized theory of one-dimensional flows. An approximating dependence is obtained for determining the gain as a result of blowing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 188–193, March–April, 1981.