Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems

Jüngel, Ansgar; Pinnau, René

doi:10.1137/s0036141099360269

Cited by 77 publications

(138 citation statements)

References 14 publications

Supporting

Mentioning

133

Contrasting

Unclassified

Order By: Relevance

“…Indeed, this conjecture has recently been proved in [9] by different techniques for (1.1) with the boundary conditions introduced in [7,8].…”

Section: Introductionmentioning

confidence: 87%

“…arises as a scaling limit in the study of interface fluctuations in a certain spin system [6], and also models the electron concentration in a quantum semiconductor device with zero temperature and negligible electric field [7]. The initial periodic-boundary value problem for (1.1) was first studied by Bleher, Lebowitz and Speer in [1].…”

Section: Introductionmentioning

confidence: 99%

“…Following the numerical investigation done in [3], one can conjecture exponential convergence towards the constant steady state for global (in time) solutions of the initial-boundary value problem (1.1)-(1.2) with periodic boundary conditions (see also [2,5,9]). Indeed, this conjecture has recently been proved in [9] by different techniques for (1.1) with the boundary conditions introduced in [7,8].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Long-time behavior for a nonlinear fourth-order parabolic equation

Cáceres

Carrillo

Toscani

2004

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We study the asymptotic behavior of solutions of the initialboundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.

show abstract

“…Indeed, this conjecture has recently been proved in [9] by different techniques for (1.1) with the boundary conditions introduced in [7,8].…”

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Long-time behavior for a nonlinear fourth-order parabolic equation

Cáceres

Carrillo

Toscani

2004

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…It has been derived by Derrida et al in [9]; we shall therefore refer to (1.2) as Derrida-Lebowitz-Speer-Spohn (DLSS) equation in the following. It has been first analyzed in [4] for local positive smooth solutions and then in [14] for global nonnegative weak solutions. The existence of weak solutions to the multidimensional equation was proved recently in [10,12].…”

Section: Introductionmentioning

confidence: 99%

Entropies for radially symmetric higher-order nonlinear diffusion equations

Bukal

Jüngel

Matthes

2011

Communications in Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

Abstract. A previously developed algebraic approach to proving entropy production inequalities is extended to deal with radially symmetric solutions for a class of higher-order diffusion equations in multiple space dimensions. In application of the method, novel a priori estimates are derived for the thin-film equation, the fourth-order Derrida-Lebowitz-Speer-Spohn equation, and a sixth-order quantum diffusion equation.Key words. Higher-order diffusion equations, thin-film equation, quantum diffusion model, polynomial decision problem, quantifier elimination.AMS subject classifications. 35B45, 35G25, 35K55, 76A20, 82C70. IntroductionIn the last two decades, there has been a growing interest in the mathematical analysis of fourth and higher-order nonlinear diffusion equations. Such equations arise, for instance, in lubrication theory and as models for the electron transport in semiconductors; below, we will briefly review several specific examples and their origins in physics. Rigorous results about the existence of solutions and their qualitative behavior are typically much harder to obtain than in the context of the well-studied secondorder diffusion equations. One of the principal difficulties is the non-applicability of comparison principles. To substitute for this loss, one has to rely on suitable a priori estimates.In [11], the last two authors have proposed a systematic approach to the derivation of a priori estimates for certain classes of nonlinear evolution equations of even order. This procedure allows one to determine Lyapunov functionals, which we call entropies in the following, and to derive integral bounds from their dissipation, called entropy production inequalities. The developed method has been successfully applied to several equations in one space dimension. The main idea is to translate the procedure of integration by parts -which is the core element in most derivations of a priori estimates -into an algebraic problem about the positivity of polynomials. Roughly speaking, to each evolution equation, a polynomial in the spatial derivatives of the solution is associated, and integration by parts allows one to modify the coefficients of this polynomial. If a suitable change of coefficients can be found that makes the resulting polynomial nonnegative, then this corresponds (formally) to a proof of an a priori estimate on the solutions. The key point is that such polynomial decision

show abstract

“…This idea circumvents the maximum principle which cannot be applied to the present problem. We remark that exponential changes of unknowns have been used in other models to prove the existence of nonnegative or positive solutions to elliptic or parabolic systems and to higher-order equations, see [50,61,68,69]. The second idea is that the cross-diffusion system admits a priori estimates via the functional…”

Section: Cross-diffusion Population Modelsmentioning

confidence: 99%