EXPONENTIAL GROWTH OF SOLUTIONS FOR A VARIABLE-EXPONENT FOURTH-ORDER VISCOELASTIC EQUATION WITH NONLINEAR BOUNDARY FEEDBACK

In this paper we study a variable-exponent fourth-order viscoelastic equation of the form$$|u_{t}|^{\rho(x)}u_{tt}+\Delta[(a+b|\Delta u|^{m(x)-2})\Delta u]-\int_{0}^{t}g(t-s)\Delta^{2}u(s)ds=|u|^{p(x)-2}u,$$in a bounded domain of $R^{n}$. Under suitable conditions on variable exponents and initial data, we prove that the solutions will grow up as an exponential function with positive initial energy level. Our result improves and extends many earlier results in the literature such as the on by Mahdi and Hakem (Ser. Math. Inform. 2020, https://doi.org/10.22190/FUMI2003647M)

Effect of Household Water Tanks on Leakage Reduction for Distribution Networks under Operation

Control and reduction of leakage in water distribution networks are very important due to limitation of drinking water resources. Water distribution networks are currently designed based on hourly peak consumption. Variation of the hourly water consumption causes the pressure variation and increase in water leakage. If household storage tanks are used in buildings, the water networks can be designed based on daily peak consumption, resulting in considerable reduction in the hourly flow variation. In the present study, the effect of domestic water storage tanks on the leakage level of water networks has been studied using water hydraulic simulation. The results showed that the use of household tanks can significantly reduce the leakage level for water networks under operation. For the studied networks, the average leakage reduction level were 31% and 67% for tanks on the roof and in the parking lot, respectively. The results also showed that the branched or looped plans of water network have no significant effect on the amount of leakage reduction. Results of this study can be used in developing new design methods for water distribution networks aiming pressure management and control of the leakage level

Existence of beam-equation solutions with strong damping and p(x)-biharmonic operator

In this paper, we consider a nonlinear beam equation with a strong damping and the p(x)-biharmonic operator. The exponent p(·) of nonlinearity is a given function satisfying some condition to be specified. Using Faedo-Galerkin method, the local and global existence of weak solutions is established with mild assumptions on the variable exponent p(·). This work improves and extends many other results in the literature