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81.
We investigate the large-time behaviour of solutions to the nonlinear heat-conduction equation with absorption ut = Δ(uσ + 1) − uβ in Q = RN × (0, ∞) (E) with N 1, σ > 0 and critical absorption exponent β = σ + 1 + 2/N; the initial function u(x, 0) = 0 is assumed to be integrable, nonnegative and compactly supported. We prove that u converges as t → ∞ to a unique self-similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut = Δ(uσ + 1), the same for any initial data. The cornerstone of the proof is a result about ω-limits of (infinite-dimensional) asymptotical dynamical systems. Combining this result with an asymptotic evaluation of the mass function as well as typical PDE estimates gives the behaviour of (E) for large times.Similar unusual asymptotic behaviour is obtained for the equation ut = div(¦Du¦σ Du) − uβ with same conditions on σ and u(x, 0) and critical value for β = σ + 1 + (σ + 2)/N. 相似文献
82.
Our first basic model is the fully nonlinear dual porous medium equation with source
for which we consider the Cauchy problem with given nonnegative bounded initial data u0. For the semilinear case m=1, the critical exponent
was obtained by H. Fujita in 1966. For p ∈(1, p0] any nontrivial solution blows up in finite time, while for p > p0 there exist sufficiently small global solutions. During last thirty years such critical exponents were detected for many
semilinear and quasilinear parabolic, hyperbolic and elliptic PDEs and inequalities. Most of efforts were devoted to equations
with differential operators in divergent form, where classical techniques associated with weak solutions and integration by
parts with a variety of test functions can be applied. Using this fully nonlinear equation, we propose and develop new approaches
to calculating critical Fujita exponents in different functional settings.
The second models with a “semi-divergent” diffusion operator is the thin film equation with source
for which the critical exponent is shown to be
相似文献
83.
V. A. Galaktionov 《Studies in Applied Mathematics》2010,124(4):347-381
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. 相似文献
84.
85.
86.
De Marzo C De Palma M Favuzzi C Maggi G Nappi E Posa F Ranieri A Selvaggi G Spinelli P Bamberger A Fuchs M Heck W Loos C Marx R Runge K Skodzek E Weber C Wülker M Zetsche F Artemiev V Galaktionov Y Gordeev A Gorodkov Y Kamyshkov Y Plyaskin V Pojidaev V Shevchenko V Shumilov E Tchudakov V Bunn J Fent J Freund P Gebauer J Glas M Polakos P Pretzl K Schouten T Seyboth P Seyerlein J Vesztergombi G 《Physical review D: Particles and fields》1990,42(3):748-758
87.
Jerry Bebernes Alberto Bressan Victor A. Galaktionov 《NoDEA : Nonlinear Differential Equations and Applications》1996,3(3):269-286
We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns. 相似文献
88.
S. I. Bakholdin D. V. Kovalevskii E. V. Galaktionov E. A. Tropp 《Bulletin of the Russian Academy of Sciences: Physics》2009,73(10):1324-1327
The paper considers the influence of the temperature dependence of the coefficients of thermal expansion of a single crystal
on thermoelastic stresses arising during crystal growth from the melt. We obtain zero stress conditions in an axisymmetric
temperature field of axial symmetry and derive approximate expressions for the components of the stress tensor in a cylindrical
crystal when this dependence is taken into account. 相似文献
89.
90.
A. V. Galaktionov 《Journal of Experimental and Theoretical Physics》1997,84(6):1164-1170
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) 相似文献