Asymptotic Analysis

This work examines the physics of free-surface flow and groundwater flow within a coupled model. Coupled models for such phenomena are not clearly justified, and there is a lack of precision in the derivation of such models. The primary objective of this work is to derive a coupled model of the shallow water equations (SWE) and Richards’ equation (RE) using asymptotic considerations. The numerical coupling approach chosen for the unified model will be described as a parallel coupling. Additionally, numerical considerations regarding how to solve this model using the discontinuous Galerkin (DG) methods will be provided. Furthermore, the exchange of information between the two models, which are time-synchronized, will be explained. The solution of RE coupled with SWE, following the described procedure and implemented using the DG formulation, is integrated into RIVAGE (an in-house numerical code based on the DG method). This implementation is then tested on a numerical problem and validated against an experimental benchmark.

In this paper, we study the following nonlocal system exhibiting singular nonlinearity and weighted singular terms: { ( − Δ ) p s u + v u p − 1 = h ( x ) u − α , u > 0 in  Ω ; u = 0 , in  R N ∖ Ω , ( − Δ ) p s v = u p , v > 0 in  Ω ; v = 0 , in  R N ∖ Ω , where 0 < s < 1 , p > 1 , α > 0 , and Ω ⊂ R N , with N > s p , is an open bounded domain with C 1 , 1 boundary ∂ Ω . The function h : Ω → R + exhibits growth of negative powers of the distance function d ( x ) := dist ( x , ∂ Ω ) near the boundary, that is, h ( x ) ∼ d − β ( x ) for some β ≥ 0 , when x is close to the boundary ∂ Ω . For β < s p , we discuss the existence of a positive weak solution ( u , v ) ∈ W loc s , p ( Ω ) × W loc s , p ( Ω ) using the classical method of regularization and the fixed point theorem together. Indeed, we found some essential uniform a priori estimates for the approximating sequence before proceeding to the limits. Moreover, we address the uniqueness of finite energy solutions, that is, ( u , v ) ∈ W 0 s , p ( Ω ) × W 0 s , p ( Ω ) , and demonstrate that this solution pair is a saddle point of a suitable functional when α < 1 . We also provide the boundary behavior of the weak solutions in terms of the distance function. Finally, we establish the nonexistence of a weak solution for the case where β ≥ s p .

This paper investigates a Kirchhoff wave equation with a memory term and a weak damping. First, we establish the global well-posedness of the dynamical system using the semigroup method. Next, we apply the energy method to derive a bounded absorbing set and show that the system is asymptotically smooth. Furthermore, we establish a stabilizability like inequality, which leads to the existence of a global attractor with finite fractal and Hausdorff dimensions.

In this article, we are concerned with the eigenvalue problem driven by the mixed local and nonlocal p -Laplacian operator having the interpolated Hardy term T ( u ) := − Δ p u + ( − Δ p ) s u − μ | u | p − 2 u | x | p θ , where 0 < s < 1 < p < N , θ ∈ [ s , 1 ] , and μ ∈ ( 0 , μ 0 ( θ ) ) . First, we establish a mixed interpolated Hardy inequality and then show the existence of eigenvalues and their properties. We also investigate the Fučík spectrum, the existence of the first nontrivial curve in the Fučík spectrum, and prove some of its properties. Moreover, we study the shape optimization of the domain with respect to the first two eigenvalues, the regularity of the eigenfunctions, the Faber–Krahn inequality, and a variational characterization of the second eigenvalue.