On fractional p-Laplacian parabolic problem with general data

Abstract

Abstract. In this article the problem to be studied is the followingwhere Ω is a bounded domain, and (−∆ s p ) is the fractional p-Laplacian operator defined by|x − y| N+ps dy with 1 < p < N , s ∈ (0, 1) and f, u 0 are measurable functions. The main goal of this work is to prove, problem (P ) has a weak solution with suitable regularity. In addition, if f 0 , u 0 are nonnegative, we show that the problem above has a nonnegative entropy solution.In the case of nonnegative data, we give also some quantitative and qualitative properties of the solution according the values of p.

Neumann fractionalp-Laplacian: Eigenvalues and existence results

“…In this case, we will prove conservation of the mass and monotony of the associated energy, as in [10]. Investigations on parabolic equations in presence of the fraction p−Laplacian have started in recent years, but only in presence of Dirichlet boundary conditions, see for instance [1], [15], [26], [27]. On the other hand, [10] is the first paper where linear parabolic problems with the associated Neumann boundary condition are considered, and, in this direction, we intend to introduce the nonlinear case with the associated nonlinear Neumann conditions.…”

“…After these preliminary, but natural, properties, we will consider problem (1) first with a given source, just to treat the easy case. Then, we will study (1) in presence of a general nonlinear term which doesn't satisfy the usual Ambrosetti-Rabinowitz condition, showing the existence of two solutions, one being positive in the whole of R N , and the other being negative.…”

Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations

“…Noting that the de nition of Λ J(u ) , we obtain ω(u ) ∩ N = ∅. Then ω(u ) = { }, which contradicts dist( , N−) > in Lemma 3.8 (1). Therefore, ω(u ) = ∅, T(u ) < ∞.…”

“…When equation (13) is coupled with the Neumann boundary condition and the Cauchy initial condition, the existence, uniqueness and asymptotic behavior of strong solutions are obtained by the semigroup methods in [19]. Very recently, authors in [1] established the existence and suitable regularity of the weak solution and the entropy solution with general data. They also gave some properties of the solution, such as extinction, non nite speed of propagation, according to the values of p. Some regularities are also established in [12].…”

A class of diffusion problem of Kirchhoff type with viscoelastic term involving the fractional Laplacian

“…So the density u satisfies (1.3). For recent references on nonlocal diffusion problems, see [5,1,29]. If we consider the effects of total population, then equation (13) becomes…”