Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions

Andreu, F.; Igbida, Noureddine; Mazón, José M.; Toledo, Julián

doi:10.1007/978-3-7643-7719-9_2

Cited by 33 publications

(75 citation statements)

References 15 publications

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“…We refer to the works of Andreu, Igbida, Mazón and Toledo [14,15,16] for precise solvability assumptions for the case of Neumann boundary conditions and general nonlinear dynamical boundary conditions for Stefan type problems.…”

Section: Assumptions On the Data And Definition Of Solutionsmentioning

confidence: 99%

“…It remains to characterize the closure of the domain of A, which is a standard task in applications of the nonlinear semigroup theory; in most of the cases, one manages to show that D(A) is dense in L 1 (Ω, j(R)) (see, e.g., [9,14,15]). Then the so constructed mild solution is also the unique integral solution of our abstract evolution problem with initial datum j 0 , see [20,22,17,21].…”

Section: Use Of Integral Solutions and Of Partial Comparison Argumentsmentioning

confidence: 99%

“…Notice that in the parabolicelliptic context and for regular convection flux, one can avoid using entropy solutions and the doubling of variables; then uniqueness results can be obtained for very general nonlinear and dynamical boundary conditions. We refer to Igbida [36], Andreu, Igbida, Mazón and Toledo [14,15,16] and references therein.…”

mentioning

confidence: 99%

“…For references on motivations, results and techniques on the Stefan type equations (1) complementary to those discussed in this paper, we refer to [14,15,16,24,58,37] and the references given therein.…”

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confidence: 99%

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On uniqueness techniques for degenerate convection-diffusion problems

Andreïanov

Igbida

2012

IJDSDE

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Abstract. We survey recent developments and give some new results concerning uniqueness of weak and renormalized solutions for degenerate parabolic problems of the form ut − div (a 0 (∇w) + F (w)) = f , u ∈ β(w) for a maximal monotone graph β, a Leray-Lions type nonlinearity a 0 , a continuous convection flux F , and an initial condition u| t=0 = u 0 . The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space.We avoid the degeneracy that could make the problem hyperbolic in some regions; yet our starting point is the notion of entropy solution, notion that underlies the theory of general hyperbolic-parabolic-elliptic problems. Thus, we focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the "parabolic-elliptic aspects". We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of "going to the boundary" in the Kato inequality for comparison of two solutions; uniqueness for renormalized solutions obtained via reduction to weak solutions. On several occasions, the results are achieved thanks to the notion of integral solution coming from the nonlinear semigroup theory.1. Introduction 1.1. A survey of literature. Study of degenerate parabolic problems has undergone a considerable progress in the last ten years, thanks to the fundamental paper of J. Carrillo [26] in which the Kruzhkov device of doubling of variables was extended to hyperbolic-parabolic-elliptic problems of the form j(v)−div(f (v)+∇ϕ(v)) = 0, and a technique for treating the homogeneous Dirichlet boundary conditions was put forward. In [26], the appropriate notion of entropy solution was established, and this definition (or, sometimes, parts of the uniqueness techniques of [26]) led to many developments; among them, let us mention [2,3,5,4,6,8,9,10,11,12,13,18,19,24,25,27,29,30,34,37,38,39,41,42,45,46,47,48,52,53,57,58,59]. Also numerical aspects of the problem were investigated; see, e.g., [7,32,33,35,40,49].The notion of entropy solution (or, as in the present paper, entropy solutions techniques used on weak solutions) was retained by most of the authors; yet, let us mention the version of Bendahmane and Karlsen [18,19] We thank Safimba Soma for fruitful discussions on the subject of this note, and the anonymous referee for a careful reading and very pertinent remarks.

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Section: Assumptions On the Data And Definition Of Solutionsmentioning

confidence: 99%

Section: Use Of Integral Solutions and Of Partial Comparison Argumentsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On uniqueness techniques for degenerate convection-diffusion problems

Andreïanov

Igbida

2012

IJDSDE

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“…Moreover, since B p is a completely accretive operator in L 1 (Ω) with dense domain satisfying the range condition (see [5]), its closure B p in L 1 (Ω) is an m-completely accretive operator in L 1 (Ω) with dense domain. In [6], it is proved that for any u 0 ∈ L 1 (Ω), the unique entropy solution u(t) of problem N p (u 0 ) (see Theorem 3.1) coincides with the unique mild-solution e tBp u 0 given by the Crandall-Liggett's exponential formula.…”

Section: It Is Not Difficult To See That (24) Is Equivalent Tomentioning

confidence: 99%

A Nonlocalp-Laplacian Evolution Equation with Nonhomogeneous Dirichlet Boundary Conditions

Andreu¹,

Mazón²,

Rossi³

et al. 2009

SIAM J. Math. Anal.

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Abstract. In this paper we study the nonlocal p−Laplacian type diffusion equation,If p > 1, this is the nonlocal analogous problem to the well known local p−Laplacian evolution equation u t = div(|∇u| p−2 ∇u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L ∞ (0, T ; L p (Ω)) to the solution of the p−laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behaviour of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.

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Quasi-linear parabolic problems in L1 with non homogeneous conditions on the boundary

Ammar

Wittbold

2005

J. evol. equ.

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We are interested in parabolic problems with L 1 data of the typeHere, is an open bounded subset of R N with regular boundary ∂ and a: × R N → R N is a Caratheodory function satisfying the classical Leray-Lions conditions and β is a monotone graph in R 2 with closed domain and such that 0 ∈ β(0).We study these evolution problems from the point of view of semi-group theory, then we identify the generalized solution of the associated Cauchy problem with the entropy solution of (P i,j (φ, ψ, β)) in the usual sense introduced in [5].

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Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions

Cited by 33 publications

References 15 publications

On uniqueness techniques for degenerate convection-diffusion problems

On uniqueness techniques for degenerate convection-diffusion problems

A Nonlocalp-Laplacian Evolution Equation with Nonhomogeneous Dirichlet Boundary Conditions

Quasi-linear parabolic problems in L1 with non homogeneous conditions on the boundary

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