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- W261783373 abstract "∂0 = ∂t = ∂/∂t, ∂k = ∂/∂xk (k = 1, 2, 3). We assume f = (fj)1≤j≤N , g = (gj)1≤j≤N ∈ C∞ 0 (R;R). e is a positive and small parameter. For simplicity, we suppose that each Fi (1 ≤ i ≤ N) is a homogeneous polynomial of degree 2 in its arguments (u, ∂u). We say that small data global existence (or (SDGE)) holds if for any f , g ∈ C∞ 0 (R;R), there exists a positive constant e0 such that (1.1) admits a global solution u ∈ C∞ ( [0,∞)× R;R ) for any e ∈ (0, e0]. Let us recall the known results. In general, the solution to (1.1) may blow up in finite time no matter how small e is. In other words, (SDGE) does not hold in general. For example, if we consider a single equation ¤u = F (u, ∂u) with either F = (∂tu) , F = u(∂tu), or F = u , then there exist f and g ∈ C∞ 0 (R) such that the solution u blows up in finite time for any e > 0 (see [4] and [5]). Hence we need some restriction to obtain (SDGE). The null condition introduced by Klainerman [7] is one of sufficient conditions for (SDGE). For Fi = Fi(u, ∂u), we define the reduced nonlinearity F red i by" @default.
- W261783373 created "2016-06-24" @default.
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- W261783373 date "2008-08-26" @default.
- W261783373 modified "2023-09-28" @default.
- W261783373 title "Asymptotic behavior of global solutions to some systems of semilinear wave equations" @default.
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