Stabilization of the solution of the Cauchy problem for parabolic equations

Denisov, V. N.; Zhikov, V. V.

doi:10.1007/bf01157682

Cited by 13 publications

(8 citation statements)

References 2 publications

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“…Note that it is possible to deduce Theorem 4 from results obtained in papers [5] (Chap.2), [22], [23]. However, this would require the use of the special terminology of the ergodic theory in Banach modules.…”

Section: Lemma 4 Let the Spectrum Of A Generator A Of A Semigroup T mentioning

confidence: 99%

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Slowly varying at infinity operator semigroups

2014

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Section: Lemma 4 Let the Spectrum Of A Generator A Of A Semigroup T mentioning

confidence: 99%

“…See [3] for a review of papers published before 1961. In [4][5][6][7] one studied the (pointwise and uniform) stabilization of solutions to parabolic equations as t → ∞. See [8] for a review of recent related works.…”

Section: Introductionmentioning

confidence: 99%

Slowly varying at infinity operator semigroups

2014

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“…Remark 3. In the case of Cauchy problem for parabolic equations, assertions similar to Theorems 5-11 were proved in [2].…”

Section: The Case Of Singular Coefficients At Non-linearitiesmentioning

confidence: 99%

“…[1] and references therein) and non-linear (see e.g. [2] and references therein) cases; it means the existence of a finite limit of the solution as t → ∞.…”

Section: Introductionmentioning

confidence: 99%

On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations

Denisov

Muravnik

2003

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We study the Dirichlet problem in half-space for the equation ∆u + g(u)|∇u| 2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as y → ∞, where y denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function.

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“…Asymptotic properties of solutions to parabolic and elliptic equations with KPZ-nonlinearity including singular equations were studied in [17]- [24], but the case of the degeneration at infinity was not studied before.…”

Section: Introductionmentioning

confidence: 99%

On qualitative properties of solutions to quasilinear parabolic equations admitting degenerations at infinity

Muravnik¹

2018

Ufimskii Matematicheskii Zhurnal

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We consider the Cauchy problem for a quasilinear parabolic equations ( ) = Δ + ( )|∇ | 2 , where the positive coefficient degenerates at infinity, while the coefficient either is a continuous function or have singularities of at most first power. These nonlinearities called Kardar-Parisi-Zhang nonlinearities (or KPZ-nonlinearities) arise in various applications (in particular, in modelling directed polymer and interface growth). Also, they are of an independent theoretical interest because they contain the second powers of the first derivatives: this is the greatest exponent such that Bernstein-type conditions for the corresponding elliptic problem ensure apriori ∞ -estimates of first order derivatives of the solution via the ∞ -estimate of the solution itself. Earlier, the asymptotic properties of solutions to parabolic equations with nonlinearities of the specified kind were studied only for the case of uniformly parabolic linear parts. Once the coefficient degenerates (at least at infinity), the nature of the problem changes qualitatively, which is confirmed by the presented study of qualitative properties of (classical) solutions to the specified Cauchy problem. We find conditions for the coefficient and the initial function guaranteeing the following behavior of the specified solutions: there exists a (limit) Lipschitz function ( ) such that, for any positive , the generalized spherical mean of the solution tends to the specified Lipschitz function as the radius of the sphere tends to infinity. The generalized spherical mean is constructed as follows. First, we apply a monotone function to a solution; this monotone function is determined only by the coefficient at the nonlinearity (both in regular and singular cases). Then we compute the mean over the ( − 1)-dimensional sphere centered at the origin (in the linear case, this mean naturally is reduced to a classical spherical mean). To construct the specified monotone function, we use the Bitsadze method allowing us to express solutions of the studied quasilinear equations via solutions of semi-linear equations.

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Stabilization of the solution of the Cauchy problem for parabolic equations

Cited by 13 publications

References 2 publications

Slowly varying at infinity operator semigroups

Slowly varying at infinity operator semigroups

On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations

On qualitative properties of solutions to quasilinear parabolic equations admitting degenerations at infinity

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