THE ILL-POSEDNESS OF A MIXED PROBLEM IN A CYLINDRICAL DOMAIN FOR MULTIDIMENSIONAL HYPERBOLIC-ELLIPTIC EQUATIONS IN CANCER MODELING

Abstract

Abstract. This paper investigates a mixed boundary value problem in a cylindrical domain for a class of multidimensional hyperbolic-elliptic equations arising in mathematical models of malignant tumor growth. In particular, axisymmetric approximations and local models of tumor spread in brain tissues lead to problem formulations in cylindrical geometry. Under assumptions, the problem leads to a class of multidimensional hyperbolic-elliptic equations, i.e., mixed-type equations whose properties may change between hyperbolic and elliptic in different subdomains or according to the parameters and coefficients. Idealized cylindrical geometry is used as a convenient framework for rigorous mathematical analysis.  The main results establish the ambiguity of the solutions to the mixed problem and provide an explicit representation of its classical solutions. These results contribute to the analysis of multidimensional mixed-type equations in bounded domains and may support further computational investigations of tumor growth models.
Keywords: ill-posedness, mixed problem, hyperbolic-elliptic equations, cylindrical domain, glioma growth modeling.