Asymptotic expansion of solutions of quasilinear parabolic problems in perforated domains

An asymptotic expansion is constructed for solutions of quasilinear parabolic problems with Dirichlet boundary conditions in domains with a fine-grained boundary. It is proved that the sequence of remainders of this expansion in the space W ₂ ^1.1/2 strongly converges to zero.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1542–1566, November, 1993.     

Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|^σ?1 u is considered in the exterior of a ball in ? ⁿ , _n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.     

Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities

We study solutions of the Dirichlet problem for a second-order parabolic equation with variable coefficients in domains with nonsmooth lateral surface. The asymptotic expansion of the solution in powers of the parabolic distance is obtained in a neighborhood of a singular point of the boundary. The exponents in this expansion are poles of the resolvent of an operator pencil associated with the model problem obtained by freezing the coefficients at the singular point. The main point of the paper is in proving that the resolvent is meromorphic and in estimating it. In the one-dimensional case, the poles of the resolvent satisfy a transcendental equation and can be expressed via parabolic cylinder functions.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 12–23, January, 1996.This work was partially supported by the Russian Foundation for Basic Research under grant No. 242 93-01-16035.     

Asymptotic behavior of solutions of equations of the Emden-Fowler type at infinity

We obtain an asymptotic representation of solutions of equations of the Emden-Fowler type with “supercritical” exponent and prove the existence of solutions with a given asymptotics. The methods used include the construction of supersolutions for deriving a priori estimates and the use of Kondrat’ev’s results for weighted spaces. The existence of solutions is proved by the Leray-Schauder method.     

Asymptotic approximations of solutions to parabolic boundary value problems in thin perforated domains of rapidly varying thickness

We consider initial boundary value problems for parabolic differential equations with rapidly oscillating coefficients in thin perforated domains of rapidly varying thickness. Under certain symmetry conditions on the domain and coefficients, we construct an asymptotic expansion of a solution to the problem with homogeneous third kind conditions on the exterior boundary and the boundary of cavities. In the case of inhomogeneous Neumann conditions, we construct an asymptotic solution without symmetry assumptions and prove an asymptotic estimate in the corresponding Sobolev space. Bibliography: 27 titles. Illustrations: 1 figure.     

Asymptotic behavior of solutions of degenerating parabolic equations

Translated from Issledovaniya po Prikladnoi Matematike, No. 17, pp. 166–173, 1990.     

Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes of order      

Asymptotic behavior of the solutions of certain parabolic equations

Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) \documentclass{article}\pagestyle{empty}\begin{document}$$\frac{{\partial u}}{{\partial t}}=\frac{\partial}{{\partial x_j }}(\eta \frac{{\partial u}}{{ax_j}}-vju),$$\end{document} j≤m, where η and v_j are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x, 0) = u_o(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § R^m the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u₀(x) ≥ 0 we derive bounds d₁u₁ ≤ u(x, t) ≤ d₂u₂, Where d_i, i = 1, 2, depend on bounds for η, v and G, and the u_i are bounds on the initial condition u₀.     

Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfies a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, that is, all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e., solutions of the corresponding elliptic problem) and time periodic solutions.     

Asymptotic properties of solutions of the emden-fowler equation at infinity

We prove a number of theorems on asymptotic properties of solutions of the equation y″+x ^a y ^σ = 0, σ < 0. First, we prove the absence of solutions on (x ₁, +∞) for some values of the parameters a and σ; after that, we obtain asymptotic formulas for solutions defined on (x ₀, +∞).     

Asymptotic behavior of solutions to a quasilinear nonuniform parabolic system modelling chemotaxis

In this paper, we study a quasilinear nonuniform parabolic system modelling chemotaxis and taking the volume-filling effect into account. The results on the existence of a unique global classical solution was obtained in Cie?lak (2007) [4]. However, the convergence to equilibrium was not considered in that paper. In this paper, we first obtain the crucial uniform boundedness of the solution. Then with the help of a suitable non-smooth Simon-?ojasiewicz approach we obtain the results on convergence to equilibrium and the decay rate.     

Asymptotic behavior in time periodic parabolic problems with unbounded coefficients

We study asymptotic behavior in a class of nonautonomous second order parabolic equations with time periodic unbounded coefficients in R×Rd. Our results generalize and improve asymptotic behavior results for Markov semigroups having an invariant measure. We also study spectral properties of the realization of the parabolic operator u?A(t)u−ut in suitable Lp spaces.