Instantaneous shrinking of the solution to a nonlinear degenerate equation with variable coefficient and convection

摘要

This paper deals with the Cauchy problem for the equation ut = (um)xx + b(x, t)(up)x, with m > 1, 0 < p < 1, and b(x, t) being positive. First, we establish the local existence and uniqueness results for the problem under appropriate hypotheses. Then we show that instantaneous shrinking of the support of the solution depends on the behavior of b(x, t).