Initial-boundary value problem of a parabolic–hyperbolic system arising from tumor angiogenesis

Abstract

In this paper, we consider the half-space problem for a parabolic–hyperbolic system arising from tumor angiogenesis. Under a mixed type boundary condition, we construct the Green's function for half-space problem by relating it with fundamental solution for an initial value problem. After differential equation method for the boundary operator and proper estimates for additional exponential factors generated by the asymmetry of characteristics, one reduces the Green's function (for initial-boundary value problem) into fundamental solution (for Cauchy problem) which could be estimated by singularity removal, long wave-short wave decomposition and weighted energy method outside cone. We finally obtain the pointwise estimate for Green's function which results in the pointwise convergence rate for nonlinear problem. The Green's function constructed is precise enough to avoid a priori estimate for derivatives by energy method. The half space problem studied here is a main ingredient for future study of shock profile with presence of boundary.