CONTINUITY OF LOCAL SOLUTIONS

Abstract

Abstract This chapter addresses a main issue of the theory, namely, the continuity of the solutions for times t > 0. The equation treated in this chapter is a generalized version of the GPME. The question of continuity is introduced in Section 7.1. The precise problem and conditions are stated in Section 7.2; the main result, Theorem 7.1, asserts the uniform equicontinuity of bounded solutions with a definite modulus of continuity that depends only on the bounds on the data and the structural conditions of the equation. The proofs are organized in Sections 7.3 and 7.4. The continuity result is a major component of the theory, and the proof is rather long and difficult. The application to the weak solutions constructed in Chapter 5 and the questions of initial and boundary regularity are discussed in Section 7.5. Once continuity is proved, the natural question is to know how regular the solutions of the PME and related equations are. A first step in that direction is Hölder continuity which is proved for the PME in Section 7.6. A much simpler proof of continuity in the case of one space dimension is presented in Section 7.7. The existence of classical positive solutions is briefly discussed in Section 7.8. Continuity is a typical property of parabolic equations, linear or nonlinear, and even degenerate equations like the PME enjoy this property. But there are limits in the direction of so-called singular coefficients. Examples of those limits are given in Section 7.9.