A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity

n/

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 r\Deltau) = 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 effect 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 first 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. AMS Subject Classification. 35K55, 35K65, 35K30, 35B30, 76D08 1

On A Fourth-Order Degenerate Parabolic Equation: Global Entropy Estimates, Existence, And Qualitative Behaviour Of Solutions

By means of energy and entropy estimates, we prove existence and positivity results in higher space dimensions for degenerate parabolic equations of fourth order with nonnegative initial values. We discuss their asymptotic behaviour for t !1 and give a counterexample to uniqueness. 0

The Thin Viscous Flow Equation In Higher Space Dimensions

We prove local integral (entropy) estimates for nonnegative solutions of the fourth order degenerate parabolic equation u t + div(u n r\Deltau) = 0 in space dimensions two and three. These estimates enable us to show that solutions have finite speed of propagation if n 2 ( 1 8 ; 2) and that the support cannot shrink if the growth exponent n is larger than 3=2. In addition, we prove decay estimates for solutions of the Cauchy problem and a growth estimate for their support. 1

A system of degenerate parabolic nonlinear PDE's: a new free boundary problem

Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 , Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal