We investigate existence and regularity of weak solutions a 1-dimensional parabolic differential equation with non-constant Hölder diffusion coefficient rough forcing term. Such an appears in studying the Ericksen-Leslie model for nematic liquid crystals where our result applies. The presented here uses continuity which comes from physical background analysis model. Moreover, dependence exponen...

We study the stability of a unique weak solution to certain parabolic systems with nonstandard growth condition, which are additionally dependent on cross-diffusion term. More precisely, we show that two solutions considered system different initial values controlled by their values.

In this article, we study the null-controllability of a heat equation in domain composed two media different constant conductivities. particular, are interested behavior system when conductivity medium on which control does not act goes to infinity, corresponding at limit perfectly conductive medium. that case, and under suitable geometric conditions, obtain uniform result. Our strategy is base...