Solutions of a Fourth Order Degenerate Parabolic Equation with Weak Initial Trace

We show that the nonlinear fourth order degenerate parabolic equation u t + div (u n ru) = 0; n > 0 admits nonnegative solutions to initial data which are a nonnegative Radon measure provided that n < 2. In addition, we prove that the equation has a regularizing eeect in the sense that the solution we construct is in H 1 (R N) for all positive times and in H 2 loc (R N) for almost all positive times. In particular, we give the rst existence results to the Cauchy problem in the case that the initial data are not compactly supported. Hence, it is interesting to note that we can show that the solutions we construct preserve the initial mass. Our results depend on decay estimates in terms of the mass which are known for regularized problems. We also give a counterexample to a decay estimate for 2 < n < 3 and show that the decay estimates are sharp for 0 < n < 2.