(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this paper, we consider the following second-order abstract Cauchy problem in two variables equation* D_ss^2u(s,t)+D_tt^2u(s,t)+2D_st^2u(s,t)+A[ D_su(s,t)+D_tu(s,t)] =Bu(s,t), equation* where A,B are closed linear operators on a Banach space X such that eqnarray* A &:&Dom(A)X→ X, B &:&Dom(B)X→ X, eqnarray* u(s,t):[ 0,1] × [ 0,1] → X is an unknown twice-continuously partially differentiable function on [ 0,1% ] × [ 0,1] R^2 where Rang(u)Dom(A)Dom(B). Utilizing properties of atomic operators, an atomic solution is obtained.