Solutions u(x, t) of the inequality □u ≥ A|u|p for x ε R3, t ≥ 0 are considered, where □ is the d'Alembertian, and A,p are constants with A > 0, 1 < p < 1 + √2. It is shown that the support of u is compact and contained in the cone 0 ≤ t ≤ t0 -|x - x0|, if the “initial data” u(x, 0), ut(x, 0) have their support in the ball|x - x0| ≤ t0. In particular, “global” solutions of □u = A|u|p with initial data of compact support vanish identically. On the other hand, for A > 0, p > 1 + √2, global solutions of □u = A|u|p exist, if the initial data are of compact support and “sufficiently” small.
展开▼
机译：考虑xεR3，t≥0的不等式□u≥A | u | p 的解u（x，t），其中□是d'Alembertian，而A，p是常数其中A> 0，1 <1 +√2。结果表明，如果“初始数据” u（x，0），ut（，u的支撑是紧凑的，并且包含在圆锥体0≤t≤t0-| x-x 0 |中x，0）在球中得到支撑| x-x 0 | ≤ t 0。尤其是□ u = A | u | p 的“全局”解，带有紧凑支持的初始数据完全消失。另一方面，对于 A p u = A | u的全局解 | p 存在，如果初始数据是紧凑支持的并且“足够”小。
展开▼
