Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations

摘要

Equations (1.1) arise in many applications, e.g., heat flow in materials with temperature dependent conductivity, flow in a porous medium, the Stefan problem, and the boundary layer theory (see ref. [1]). Clearly equations (1.1) might have discontinuous solution, since a(u) ≥ 0. Many papers have been devoted to the uniqueness of generalized solution for Cauchy problem (1.1)-(1.2). The paper by Volpert and Hudjaev was the first to be devoted to the solvability of (1.1)-(1.2). So far the uniqueness of solution for (1.1)-(1.2) in one-space variable case has been deeply investigated (see refs. [2—4]). As for the multi-space variables, the uniqueness of solution similar to one-space variable case has not been obtained.