Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 76, pp. 1-30.

Title: Mild solutions to Love-type equations on R^2

Authors: Bui Duc Nam (Ho Chi Minh City Univ. of Industry and Trade, Vietnam)
         Bui Dai Nghia (Univ. of Science, Ho Chi Minh City, Vietnam)
         Nguyen Anh Tuan (Van Lang Univ., Ho Chi Minh City, Vietnam)

Abstract:
In this article, we study a non-local Love problem on unbounded domains where
the non-locality in the main equation is interpreted as a fractional Laplacian operator.
With various assumptions on the initial conditions, we derive several estimates for
mild solutions for the homogeneous source scenario. For the nonlinear problem,
we show the existence and uniqueness of a global mild solution.
In two cases, we obtain convergence results.
The first one states that the solution to the fractional Love equation converges
to the mild solution of the fractional wave equation according to a cross-section
radius parameter. The second result shows that solutions of the fractional
Love equation incorporating the fractional Laplacian operator  converge to those of
the classical problem, involving the usual Laplacian, as the fractional orders
approach 1. This work is the first that we are aware of that deals with mild
solutions of Love equations on unbounded domains.

Submitted March 19, 2025. Published July 22, 2025.
Math Subject Classifications: 35L05, 35Q74, 35B40.
Key Words: Love type equation; global solution; regularity; behavior of solutions;
 convergence