Abstract
This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.
Highlights
We study a nonlinear parabolic system with nonstandard growth condition, where the solution satisfies a homogeneous Neumann boundary condition, which is motivated by several issues and numerous applications
To derive our existence result, we will need the following Poincaré type estimate, which is a modification of the Poincaré type estimate from [22, Lemma 3.9]
The proof of the stability estimate of Lemma 2.3 is very similar to the proof of the uniqueness Theorem 2.2
Summary
We study a nonlinear parabolic system with nonstandard growth condition, where the (weak) solution satisfies a homogeneous Neumann boundary condition, which is motivated by several issues and numerous applications. In [26] the existence, uniqueness and stability of a weak solution to the equation ut − div(a(x, t, ∇u)) = − |u|p(x,t)−2u, where ≥ 0 and the vector-field a(x, t, ⋅) satisfies certain p(x, t)-growth and monotonicity conditions, cf [25], was shown, see [14] for p =constant. Very recently the existence of weak solutions to a homogeneous Dirichlet problem of a nonlinear diffusion equation involving anisotropic variable exponents and convection was studied in [39]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
