Nonnegative solutions to reaction–diffusion system with cross‐diffusion and nonstandard growth conditions

Abstract

We establish the existence of nonnegative weak solutions to nonlinear reaction–diffusion system with cross‐diffusion and nonstandard growth conditions subject to the homogeneous Neumann boundary conditions. We assume that the diffusion operators satisfy certain monotonicity condition and nonstandard growth conditions and prove that the existence of weak solutions using Galerkin's approximation technique.

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 .…”

“…with ℎ = (ℎ 1 , … , ℎ ) is a measurable function belonging in ∞ ( ) . The main tool used to ensure the existence of a weak periodic solution to (7) involves Theorem 1 and the uniqueness result will be established by using the monotony assumption ( 3 ). To be more clear, we shall clarify in which sense we want to solve (7).…”

“…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.…”

An improved nonlinear anisotropic model with p(x)‐growth conditions applied to image restoration and enhancement

“…As well‐known, theoretical analysis of PDEs involving $$$ p(x) $$$‐growth conditions need the use of some complex spaces called Lebesgue and Sobolev spaces with variable exponents (see, e.g., earlier studies [37–47]). Therefore, the variable exponent $$$ p\left(\cdotp \right) $$$ that appears in problem () requires the consideration of these types of spaces.…”

An improved nonlinear anisotropic PDE with p(x)-growth conditions applied to image restoration and enhancement

“…As well known, theoretical analysis of PDEs involving ( )-growth conditions need the use of some complex spaces called Lebesgue and Sobolev spaces with variable exponents (see for example 3,7,16,17,24,32,40,52,53 ). Therefore, the variable exponent (⋅) appears in problem (1) requires the consideration of these types of spaces.…”