Improperly Posed Problems for Nonlinear Partial Differential Equations

Abstract

Publisher Summary The study of improperly posed problems has received considerable recent attention in response to the realization that a number of physical situations lead to these kinds of mathematical models. Many classes of improperly posed problems include nonlinear partial differential equations. This chapter provides a review of some of the questions confronted in the investigation of a particular class of such problems—that is, ill-posed Cauchy problems. A boundary or initial value problem for a partial differential equation is understood to be well posed in the sense of Hadamard if it possesses a unique solution that depends continuously on the prescribed data. This definition is made precise by indicating the space to which the solution must belong and the measure in which continuous dependence is desired. A problem is called improperly posed (or ill-posed) if it fails to have a global solution, a unique solution, or a solution that depends continuously on the data. The chapter discusses the questions of uniqueness, existence, and stabilization of solutions to ill-posed problems for nonlinear partial differential equations. The chapter focuses on improperly posed Cauchy problems for nonlinear equations. Because many ill-posed problems for partial differential equations have not usually yielded to standard methods of analysis, a variety of techniques have been developed to study these problems. The chapter focuses on three of these methods because they have proven to be useful in treating nonlinear problems—namely, logarithmic convexity, weighted energy, and concavity. It also describes the quasi-reversibility method.