Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

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.

Stability of Weak Solutions to Parabolic Problems with Nonstandard Growth and Cross–Diffusion

“…A further important aspect is the effect of a cross-diffusion term, which arises for instance by the modelling of interaction between species [21]. As already mentioned in [1,22], this may lead to unexpected behaviour, e.g., in our case the cross-diffusion term δ∆u, δ ≥ 0 requires that the growth exponent p(x, t) ≥ 2. Only if δ = 0 we may assume that 2n n+2 < p(x, t), n ≥ 2.…”

Blow-up for a viscoelastic von Karman equation with strong damping and variable exponent source terms

“…Later, Park and Kang [19] improved and complemented the result of [15] by obtaining a blow-up result of solution with certain positive initial energy for a wave equation of memory type. We also refer to a recent work [4] for a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. At this point, it is worthwhile to mention that there is little work concerning global nonexistence of solutions for viscoelastic von Karman equations with variable source effect.…”