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We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and also continuity of weak solutions in some cases. For example, we consider the family f k,σ k∈N,σ>0 withof infinitely degenerate functions at the origin, and derive conditions on the parameters k and σ under which all weak solutions to the associated infinitely degenerate quasilinear equations of the formwith rough data A and φ, are locally bounded / satisfy a maximum principle / are continuous.As an application we obtain weak hypoellipticity (i.e. smoothness of all weak solutions) of certain infinitely degenerate quasilinear equations ∂u ∂x 2 + f (x, u (x, y)) 2 ∂u ∂y 2 = φ (x, y) , with smooth data f (x, z) ∼ f k,σ (x) and φ (x, y) where f (x, z) has a sufficiently mild nonlinearity and degeneracy.We also consider extensions of these results to R 3 and obtain some limited sharpness. In order to prove these theorems we develop subrepresentation inequalities for these geometries and obtain corresponding Poincaré and Orlicz-Sobolev inequalities. We then apply more abstract results (that hold also in higher dimensional Euclidean space) in which these Poincaré and Orlicz-Sobolev inequalities are assumed to hold. ample in Gilbarg and Trudinger [GiTr] and many other sources. The key breakthrough here was the Hölder apriori estimate of DeGiorgi, and its later generalizations independently by Nash and Moser. The extension of the DeGiorgi-Nash-Moser theory to the subelliptic or finite type setting, was initiated by Franchi [Fr], and then continued by many authors, including one of the present authors with Wheeden [SaWh4].The subject of the present monograph is the extension of DeGiorgi-Moser theory to the infinitely degenerate regime. Our theorems fall into two broad categories. First, there is the abstract theory in all dimensions, in which we assume appropriate Orlicz -Sobolev inequalities and deduce local boundedness and maximum principles for weak subsolutions, and also continuity for weak solutions. This theory is complicated by the fact that the companion Cacciopoli inequalities are now far more difficult to establish for iterates of the Young functions that arise in the Orlicz-Sobolev inequalities. Second, there is the geometric theory in dimensions two and three, in which we establish the required Orlicz-Sobolev inequalities for large families of infinitely degenerate geometries, thereby demonstrating that our abstract theory is not vacuous, and that it does in fact produce new theorems.The results obtained here are of course also in their infancy, leaving many intriguing questions unanswered. For example, our implementation of Moser iteration requires a sufficiently large Orlicz bump, which in turn restricts the conclusions of the method to fall well short of existing counterexamples. It is a major unanswered question as to whether or not this 'Moser gap' is an artificial obstruction to local boundedness. Finally, the contributions of Nash to the classical DeGiorgi-Nas...

The topic of this paper is a semi-linear, energy subcritical, defocusing wave equation ∂ t 2 u − Δ u = − | u | p − 1 u \partial _t^2 u - \Delta u = - |u|^{p -1} u in the 3-dimensional space ( 3 ≤ p > 5 3\leq p>5 ) whose initial data are radial and come with a finite energy. We split the energy into inward and outward energies, then apply the energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: “scattering energy”, which concentrates around the light cone | x | = | t | |x|=|t| and moves to infinity at the light speed, and “retarded energy”, which is at a distance of at least | t | β |t|^\beta behind when | t | |t| is large. Here β \beta is an arbitrary constant smaller than β 0 ( p ) = 2 ( p − 2 ) p + 1 \beta _0(p) = \frac {2(p-2)}{p+1} . A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data ( u 0 , u 1 ) (u_0,u_1) than previously known results. More precisely, we assume \[ ∫ R 3 ( | x | κ + 1 ) ( 1 2 | ∇ u 0 | 2 + 1 2 | u 1 | 2 + 1 p + 1 | u | p + 1 ) d x > + ∞ . \int _{\mathbb {R}^3} (|x|^\kappa +1)\left (\frac {1}{2}|\nabla u_0|^2 + \frac {1}{2}|u_1|^2+\frac {1}{p+1}|u|^{p+1}\right ) dx > +\infty . \] Here κ > κ 0 ( p ) = 1 − β 0 ( p ) = 5 − p p + 1 \kappa >\kappa _0(p) =1-\beta _0(p) = \frac {5-p}{p+1} is a constant. This condition is so weak that the initial data may be outside the critical Sobolev space of this equation. This phenomenon is not covered by previously known scattering theory, as far as the author knows.

In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic spacewhere E is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in H 1/2,1/2 × H 1/2,−1/2 (H n ) if 2 ≤ n ≤ 6 and 1 < p < pc = 1 + 4/(n − 2). As another application we show that a solution to the quintic wave equation ∂ 2 t u − ∆u = −|u| 4 u on R 2 scatters if its initial data are radial and satisfy the conditions |∇u0(x)|, |u1(x)| ≤ A(|x| + 1) −3/2−ε ; |u0(x)| ≤ A(|x|) −1/2−ε ; ε > 0. * The second author is funded in part by NSF DMS 1068815.