FINITE-DIFFERENCE APPROXIMATIONSTO A CLASS OF STRONGLY DEGENERATEPARABOLIC EQUATIONS

摘要

In this paper a new notion of generalized solution to the initial bound-ary value problem for a nonlinear strongly degenerate parabolic equation of the form ut + V · A(x, t, u) + B(x, t, u) = Δβ(u) is treated. This type of solution is called a BV-entropy solution. Since equations of this form are linear conbinations of time-dependent conservation laws and porous medium type equations, it is interesting to investigate interactions between singularities of solutions associated with the two different kinds of nonlinearities. Therefore the part of conservation laws and that of porous medium type diffusion term have to be treated in a separate way. In fact, for the convective term, method for the existence and uniqueness of the entropy solutions is employed in the sense of Kruzkov and so the initial-boundary-value problem is formulated in the space BV of functions of bounded variation. This observation leads us to the new notion of BV-entropy solution. Our objective here is to establish unique existence of such BV-entropy solutions under the homogeneous Neumann boundary conditions.