Asymptotic Analysis

In this article, we study the asymptotic behavior of a nonlocal semilinear degenerate heat equation with degenerate past history in a bounded domain. We approximate the degenerate heat equation obtained after the Dafermos transformation with a non-degenerate equation to obtain the existence, uniqueness, and regularity of the solutions of the equation. Then, we obtain the existence of the global attractor of the original equation.

This study delves into the dynamic behavior of a coupled system comprising an Euler-Bernoulli beam equation and a heat equation with memory, governed by different heat conduction laws. These laws, represented by a parameter m include the Coleman-Gurtin and Gurtin-Pipkin laws, which uniquely influence the material’s temperature evolution. By formulating a set of partial differential equations along with appropriate boundary and transmission conditions, we lay the groundwork for analyzing the coupled system’s behavior. Our investigation primarily centers on understanding the asymptotic properties of the solution semigroup within the framework of Dafermos history space. We establish exponential stability under both the Coleman-Gurtin and Gurtin-Pipkin heat conduction models by imposing distinct conditions on the relaxation function. The article concludes with a comparison of the two heat conduction laws and a discussion on potential directions for extending this research.

We investigate the existence of ground state solutions for a ( p , q ) -Laplacian system with p , q > 1 and potential wells on a weighted locally finite graph G = ( V , E ) . By making use of the method of Nehari manifold and the Lagrange multiplier rule, we prove that if the nonlinear term F takes on the super- ( p , q ) -linear growth and the potential functions a ( x ) and b ( x ) satisfy some suitable conditions, then for any fixed parameter λ ≥ 1 , the system is provided with a ground state solution ( u λ , v λ ) . Additionally, we set up the convergence property of the solutions set { ( u λ , v λ ) } when λ → + ∞ .