Parabolic systems driven by general differential operators with variable exponents and strong nonlinearities with respect to the gradient

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 .…”

“…In the recent years, a special interest has been given to mathematical analysis of partial differential equations with variable exponent (see e.g 12,21,33 ). These types of problems were generally motivated by some real-world applications such as dynamic fluid, image processing 17,22,34 and take a huge amount of interest in a wide range of models, for example, the epidemiological models and their related predator-prey models (see 1,5,9 ).…”

“…( 1 ) There exists a nonnegative measurable function ∈ In the current literature, great works have been established to answer the often questions about existence, uniqueness, regularities and asymptotic behaviors of the solution to PDEs with variable growth conditions 8,21,23,27,33 . We start by recalling the results of some previous works where the authors have been interested in the particular cases of (1) ( = 1) with an initial condition.…”

“…Thereafter, they based on the sub-and super-solution method to obtain the existence of a weak solution which is the SOLA solution (solution obtained as the limit of approximation). Related to the × parabolic systems with ( )-growth, we refer the readers to see 7,21,35 . Particularly, in 35 Shangerganesh and Balachandran considered a reaction-diffusion systems ( = 3) with ( )-growth conditions modeled a class of the spread of epidemic disease.…”

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

“…In particular, questions of existence, uniqueness, regularity, stability, asymptotic behavior and numerical simulation. For more details, we refer the reader to see the works [1,2,3,4,5,16,17,20,28].…”