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From the introduction: "We continue here the investigation initiated earlier by us [Trans. Amer. Math. Soc. 314 (1989), no. 1, 187–224] concerning the solvability of the Cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation (1) ut − div(|Du| p−2 Du)=0 in ST ≡ RN ×(0,T), 0 2, both in terms of results and techniques. The main difference stems from the fact that, unlike solutions in the case p>2, solutions of (1) are not, in general, locally bounded.''

For weak solutions of equations of the type of nonlinear filtration in RN × (0, T), 0 < T < ∞, we prove precise sup-estimates and local and global Harnack type inequalities. These estimations permit to identify the initial traces and describe the behavior of such solutions as |x| → ∞. The main point is to introduce a new approach, free of the specific features of the porous medium equation such as homogeneity, scaling, quasi-convexity, etc. This approach on one hand allows generalizations to a large variety of equations and on other yields new results on gradient averages.

In questo lavoro si considera il problema di Cauchy per l'equazione di filtrazione ∂u/∂t=∂ 2 ϕ(u)/∂x 2 nella regioneR×(0,T],0 2 si trova in particolare un comportamento holderiano di u(x, t) rispetto a t con l'esponente1/(m−1); viene anche dimostrato che questo esponete e effettivamente assunto da una particolare soluzione, per cui la stima ottenuta e la migliore possibile.

We consider the Cauchy problem (f) \[ { u t − div ⁡ ( | D u | p − 2 D u ) = 0 a m p ; in R N × ( 0 , ∞ ) , p > 2 , u ( x , 0 ) = u 0 ( x ) , a m p ; x ∈ R N , \left \{ {\begin {array}{*{20}{c}} {{u_t} - \operatorname {div}(|Du{|^{p - 2}}Du) = 0} \hfill & {{\text {in}}\;{{\mathbf {R}}^N} \times (0,\infty ),p > 2,} \hfill \\ {u(x,0) = {u_0}(x),} \hfill & {x \in {{\mathbf {R}}^N},} \hfill \\ \end {array} } \right . \] and discuss existence of solutions in some strip S T ≡ R N × ( 0 , T ) {S_T} \equiv {{\mathbf {R}}^N} \times (0,T) , 0 > T ≤ ∞ 0 > T \leq \infty , in terms of the behavior of x → u 0 ( x ) x \to {u_0}(x) as | x | → ∞ |x| \to \infty . The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of L loc 1 ( R N ) L_{{\text {loc}}}^1({{\mathbf {R}}^N}) .

A system of two elliptic p.d.e.'s, which model the conduction-convection problem in a porous medium with change of phase is considered. The first equation describes the heat conduction and holds in a fixed domain. The second takes into account the convective motions and holds in the unknown melted part of the region. The existence of a locally regular weak solution is proved by making use of various compactness arguments.

In his fundamental work on linear elliptic equations, De Giorgi established local bounds and Holder estimates for functions satisfying certain integral inequalities. The main result of this paper is that the Harnack inequality can be proved directly for functions in the De Giorgi classes. This implies that every non-negative Q-minimum (in the terminology of Giaquinta and Giusti) satisfies a Harnack inequality.