Existence and Maximal Regularity of Solutions for a Class of Third-Order Differential Equations with an Unbounded Coefficient

This paper investigates a class of third-order partial differential equations with an unbounded lower-order coefficient in the Hilbert space L2(R2). The study is motivated by the wide use of third-order equations, particularly of Korteweg–de Vries type, in mathematical physics and wave theory, as well as by the limited development of the corresponding theory in the presence of unbounded coefficients. The main focus is on the existence, uniqueness, and maximal regularity of solutions. Within a functional-analytic framework, the well-posedness of the problem is established in natural function spaces under minimal assumptions on the coefficients. In particular, a priori estimates ensuring maximal regularity are derived.