Some results on the existence of weak periodic solutions for quasilinear parabolic systems with L1 data

Abstract

The aim of this paper is to prove the existence of weak periodic solution and super solution for M×M reaction diffusion system with L1 data and nonlinearity on the gradient. The existence is proved by the technique of sub and super solution and Schauder fixed point theorem.

Time periodic solutions for strongly nonlinear parabolic systems with p(x)-growth conditions

“…Among the novelty of our work, we establish an interesting compactness result which will be applied in the studies of periodic parabolic systems with nonstandard growth conditions. As can be viewed the results of our paper generalized the existing work in the literature not only on the constant exponent case 2,3,4,11,13,14,15,18 but also these involving variable exponent with particular assumptions on the nonlinearities 7,12,21,35,38 .…”

“…At the same time, the literature of periodic PDEs is very vast and for more clarification, we refer the readers to see the works 6,11,13,14,15,16,20,25 . Particularly, in 6 Amann studied a periodic parabolic equation with an additionnal nonlinearity and subquadratic growth on the gradient.…”

“…Under the assumption that a coupled of ordered functions ( , ) are known, the authors shown the existence of a weak periodic solution belonging to the space (0, ; 1, 0 (Ω)) ∩ ([0, ]; 2 (Ω)) ∩ ∞ ( ). Alaa et al in 15 are interested by the equation (1) ( = 1) in the case ( ) = 2 and belongs to 1 ( ). By applying Schauder fixed point theorem, they demonstrated the existence of a weak periodic solution ∈ 1 (0, ; 1,1 0 (Ω)) ∩ ([0, ]; 1 (Ω)).…”

Periodic parabolic problem with discontinuous coefficients: Mathematical analysis and numerical simulation

“…A large literature exists on periodic equations and several researchers have been interested in the subject, we quote in particular [8,6,11,12,13,15,14,18,19,23,24]. We start by referring the readers to see the book [24] for a major and comprehensive introduction to periodic parabolic equations with regular data.…”

“…They established the existence and regularity property of a weak solution by using the techniques of sub-and super-solutions. The work [18] was generalized by Alaa et al in their paper [11], the authors examined the existence of a weak solution to a nonlinear parabolic equation with L 1 data. They combined the truncation method with the sub-and super-solution techniques to obatin SOLA solution (Solution Obtained as the Limit of Approximation).…”