On a class of nonlocal evolution equations with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si8.svg"><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math>-Laplace operator

Antont︠s︡ev, S. N.; Shmarev, Sergey

doi:10.1016/j.nonrwa.2020.103165

Cited by 9 publications

(1 citation statement)

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“…The corresponding parabolic version of () has been also studied. Antontsev and Shmarev 14 studied the homogeneous Dirichlet problem for the parabolic equation

u_{t} - div ({|\nabla u|}^{p [u] - 2} \nabla u) = f, in Q_{T} = Ω \times] 0, T [,

where

normalΩ \subset ℝ^{N}, N \geq 2,

is a smooth domain,

p false[u false] = p false(l false(u false) false), p

is a given differentiable function such that (2 N / N + 2) < p − ≤ p + < 2, and

\sup_{s \in ℝ} (||, p^{'} false(s false)) < + \infty; l false(u false) = \int_{normalΩ} {(||, u false(x, t false))}^{α} d x, α \in false[1,2 false],

and

f \in L^{false(p^{-} false)'} false(Q_{T} false) .

A result of existence and uniqueness of a solution

u \in C^{0} ((), false[0, T false]; L^{2} false(normalΩ false)), {(||, \nabla u)}^{p false[u false]} \in L^{\infty} ((), 0, T; L^{1} false(normalΩ false)), u_{t} \in L^{2} false(Q_{T} false)

has been proved. This result has been extended in Antontsev and Shmarev 15 to the case when the source f is replaced by the nonlinear term f (( x , t ...…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

On some equation defined on the whole euclidean space ℝN and involving the p(u)‐Laplacian

Aouaoui

Bahrouni

2021

Math Methods in App Sciences

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In this paper, we deal with some new kinds of problems defined on the whole euclidean space R N , N ≥ 1, involving an operator with variable exponent which depends on the unknown solution u. In the first part of this work, we treat a local second-order partial differential equation, that is, when the exponent depends on the variable x ∈ R N through the unknown solution u. In the second part, we study the nonlocal version of the first problem; more precisely, we are interested in the situation where the exponent depends on a scalar function of u. A suitable approximation scheme is performed, and the process of passage to the limit is completed using some sophisticated arguments. These results are immediately extended to some fourth-order problem with variable exponents depending on the unknown function u as well as one or many of its partial derivatives 𝜕u/𝜕x j , 1 ≤ j ≤ N.

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