On uniqueness for a family of nonstandard problems

Abstract

In this study we investigate the uniqueness of solutions of the nonstandard problemin the general case where we do not assume the positivity of the operator A. We prove that whenever α = −β with |α| = 1 we have always uniqueness of solutions. We also obtain some families of the parameters α, β where uniqueness fails. It is worth noting that the intersection of these families of parameters α, β with the families obtained in [L.E. Payne, P.W. Schaefer, Energy bounds for some nonstandard problems in partial differential equations, J. Math. Anal. Appl. 273 (2002) 75-92] of parameters α, β where the uniqueness holds is not empty. Thus the assumption of positivity of the operator A assumed in [L.E. Payne, P.W. Schaefer, Energy bounds for some nonstandard problems in partial differential equations, J. Math. Anal. Appl. 273 (2002) 75-92] plays a relevant role. We end this note by giving sufficient conditions for guaranteeing the uniqueness of solutions for two concrete problems.

On spatial evolution of the solution of a non-standard problem in the bending theory of elastic plates

“…Using the semigroup theory, an unified way to treat existence and uniqueness of the solutions of this type of nonstandard problems was given by Quintanilla (2005). The uniqueness of the solution of the considered type of nonstandard was also considered in Quintanilla (2008).…”