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Abstract In this paper we study the degenerate parabolicp-Laplacian, ${\partial }_{t}u-v^{-1}\mathrm{div}\left({\left|\sqrt{Q}\nabla u\right|}^{p-2}Q\nabla u\right)=0${\partial_{t}u-v^{-1}\operatorname{div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=0},where the degeneracy is controlled by a matrixQand a weightv.With mild integrability assumptions onQandv, we prove theexistence and uniqueness of solutions on any interval $[0,T]${[0,T]}. If we further assumethe existence of a degenerate Sobolev inequality with gain, thedegeneracy again controlled byvandQ, then we can prove bothfinite time extinction and ultracontractive bounds. Moreover, weshow that there is equivalence between the existence ofultracontractive bounds and the weighted Sobolev inequality. 

We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest. 

Abstract Variational implicit solvation models (VISMs) have gained extensive popularity in the molecular-level solvation analysis of biological systems due to their cost-effectiveness and satisfactory accuracy. Central in the construction of VISM is an interface separating the solute and the solvent. However, traditional sharp-interface VISMs fall short in adequately representing the inherent randomness of the solute–solvent interface, a consequence of thermodynamic fluctuations within the solute–solvent system. Given that experimentally observable quantities are ensemble averaged, the computation of the ensemble average solvation energy (EASE)–the averaged solvation energy across all thermodynamic microscopic states–emerges as a key metric for reflecting thermodynamic fluctuations during solvation processes. This study introduces a novel approach to calculating the EASE. We devise two diffuse-interface VISMs: one within the classic Poisson–Boltzmann (PB) framework and another within the framework of size-modified PB theory, accounting for the finite-size effects. The construction of these models relies on a new diffuse interface definition $u\left(x\right)$u\left(x), which represents the probability of a point $x$xfound in the solute phase among all microstates. Drawing upon principles of statistical mechanics and geometric measure theory, we rigorously demonstrate that the proposed models effectively capture EASE during the solvation process. Moreover, preliminary analyses indicate that the size-modified EASE functional surpasses its counterpart based on the classic PB theory across various analytic aspects. Our work is the first step toward calculating EASE through the utilization of diffuse-interface VISM.