Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations

Kalashnikov, A. S.

doi:10.1070/rm1987v042n02abeh001309

Cited by 422 publications

(341 citation statements)

References 141 publications

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“…= (U m (t,.)) xx on (0,f(t)), and m > 1, we can apply a well-known local result (see KALASHNIKOV, 1987) showing that if U(0, x 1 ) > 0 then U(t, x 1 ) > 0 for any t > 0. In particular, we get that U(t, x) > 0, for any x [ [ 0, f(0)) and for any t > 0.…”

Section: Parabolic Casementioning

confidence: 99%

Mathematical Analysis of a Model of River Channel Formation

Díaz

Fowler

Muñoz

et al. 2008

Pure appl. geophys.

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Abstract-The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin.

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Section: Parabolic Casementioning

confidence: 99%

Mathematical Analysis of a Model of River Channel Formation

Díaz

Fowler

Muñoz

et al. 2008

Pure appl. geophys.

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“…Concerning the theory of the boundary value problems in smooth cylindrical domains and interior regularity results for general second-order nonlinear degenerate and singular parabolic equations, we refer to [4][5][6] and to the review article [1]. The well-posedness of the DP to nonlinear diffusion equation ((1.1) with b = 0, m = 1) in a domain Ω ∈ Ᏸ 0,T is accomplished in [2,3].…”

Section: Boundary Value Problemsmentioning

confidence: 99%

“…It is a simple model for various physical, chemical, and biological problems involving diffusion with a source (b < 0) or absorption (b > 0) of energy (see [1]). In this paper, we study the Dirichlet problem (DP) for (1.1) in a general domain Ω ⊂ R N+1 with ∂Ω being a closed N-dimensional manifold.…”

Section: Introductionmentioning

confidence: 99%

Reaction-Diffusion in Nonsmooth and Closed Domains

Abdulla

2007

Boundary Value Problems

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We investigate the Dirichlet problem for the parabolic equationin a nonsmooth and closed domain Ω ⊂ R N+1 , N ≥ 2, possibly formed with irregular surfaces and having a characteristic vertex point. Existence, boundary regularity, uniqueness, and comparison results are established. The main objective of the paper is to express the criteria for the well-posedness in terms of the local modulus of lower semicontinuity of the boundary manifold. The two key problems in that context are the boundary regularity of the weak solution and the question whether any weak solution is at the same time a viscosity solution.

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“…Supposing that the flow is isothermal, the motion may be described by du/dt = div (D grad <p(u)) where u denotes the density of the gas, t time, the coefficient D is determined by experiment, and the function cp is derived from the equation of state relating the pressure and density of the gas at constant temperature. [40]. With m = 1 and p # 1 it is also called the equation of non-Newtonian elastic filtration [40] or more commonly the heat equation with p-Laplacian.…”

Section: ) DXmentioning

confidence: 99%

“…[40]. With m = 1 and p # 1 it is also called the equation of non-Newtonian elastic filtration [40] or more commonly the heat equation with p-Laplacian. For p = 1 and m ^ 1, the equation is widely known as the porous media equation.…”

Section: ) DXmentioning

confidence: 99%

A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

Gilding

Kersner

1996

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A degenerate parabolic partial differential equation with a time derivative and first-and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

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Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations

Cited by 422 publications

References 141 publications

Mathematical Analysis of a Model of River Channel Formation

Mathematical Analysis of a Model of River Channel Formation

Reaction-Diffusion in Nonsmooth and Closed Domains

A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

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