On the two-dimensional hyperbolic stochastic sine-Gordon equation

We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary Gaussian multiplicative chaos, we prove local well-posedness of SSG for any value of a parameter beta 2>0in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (Commun Math Phys 341(3):933-989, 2016) and Chandra et al. (The dynamical sine-Gordon model in the full subcritical regime, [math.PR], 2018), where the parameter is restricted to the subcritical range: 0 Elovich > pseudo-second order. The intra-particle diffusion model showed that the rate-controlling step is influenced by pore diffusion and that boundary layer diffusion and the adsorption process is heterogeneous, exothermic, and spontaneous. It can be deduced that mango seed shell is a good precursor in the production of activated due to its high yield and good adsorption capacity and the modification of the activated carbon with nanoparticles increased the precursor adsorption properties.