Evolution inequalities of high order with complex-valued

H. Ali

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2024), pp. 95-102

Résumé

This article studies the absence of solutions to n-th order evolutionary inequalities with complex values. The relevance of the study lies in the extension of the obtained results from the $n$-dimensional real space to the $n$-dimensional complex space. In the first section of this paper studies the form of $n$-th order evolutionary inequalities in $n$-dimensional complex space, and the second section is devoted to finding the condition for the absence of a solution to $n$-th order Semilinear inequalities with complex-valued and with bounded coefficients. As a result, we obtained conditions for the absence of weak non-trivial solutions to the problem under consideration. It turns out that there are two conditions for the absence of a solution, one of which is associated with the criterion index $q$, and the other with the argument of the complex function, which depends on $q$. The proofs of the results on the absence of solutions in this work are based on the technique proposed by S. I. Pokhozhaev [10] and developed by E. L. Mitidieri and S. I. Pokhozhaev [8], which is based on the method of test functions. Their approach relies heavily primarily on a priori integral estimates for possible solutions to the problem under consideration and on deriving asymptotics for these estimates with respect to some parameter that tends to $ \infty$ or to $0$ depending on the nature of the problem. Finally, the absence of a solution is proven by contradiction. Namely, reaching the zero limit value in the corresponding a priori estimate guarantees that there is no nontrivial solution to this problem.

Détail

Keywords: complex-valued inequalities
Mots-clés : Absence of solutions