ON NONLINEAR PROBLEM FOR THE THIRD ORDER EQUATION

Abstract

In this paper, we examine the finite-time behavior of the solution to a mixed problem concerning the equation of third-order nonlinearity in its principal part. By employing Levine’s lemma for a function that depends intricately on the solution of the initial-boundary value problem and its derivatives with respect to both x and t, we derive sufficient conditions for the blow-up of this solution within a finite period of time. Our investigation uncovers the nuanced dynamics at play, highlighting the delicate interplay between nonlinearity and time within the context of the problem. Through rigorous analysis, we aim to illuminate the conditions under which solutions may exhibit singular behavior, contributing to a deeper understanding of the complexities inherent in nonlinear differential equations. This exploration not only enriches the theoretical framework surrounding such problems but also sets the stage for potential applications in various fields that grapple with similar mathematical phenomena.