This paper deals with the following IBV problem of nonlinear hyperbolic equations u<sub>tt</sub>- sum from i, j=1 to n a<sub>jj</sub>（u, Du）u<sub>x<sub>i</sub>x<sub>j&...This paper deals with the following IBV problem of nonlinear hyperbolic equations u<sub>tt</sub>- sum from i, j=1 to n a<sub>jj</sub>（u, Du）u<sub>x<sub>i</sub>x<sub>j</sub></sub>=b（u, Du）, t】0, x∈Ω, u（O, x） =u<sup>0</sup>（x）, u<sub>t</sub>（O, x） =u<sup>1</sup>（v）, x∈Ω, u（t, x）=O t】O, x∈（）Ω,where Ωis the exterior domain of a compact set in R<sup>n</sup>, and |a<sub>ij</sub>（y）-δ<sub>ij</sub>|= O（|y|<sup>k</sup>）, |b（y）|=O(|y|<sup>k+1</sup>), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.展开更多
文摘This paper deals with the following IBV problem of nonlinear hyperbolic equations u<sub>tt</sub>- sum from i, j=1 to n a<sub>jj</sub>（u, Du）u<sub>x<sub>i</sub>x<sub>j</sub></sub>=b（u, Du）, t】0, x∈Ω, u（O, x） =u<sup>0</sup>（x）, u<sub>t</sub>（O, x） =u<sup>1</sup>（v）, x∈Ω, u（t, x）=O t】O, x∈（）Ω,where Ωis the exterior domain of a compact set in R<sup>n</sup>, and |a<sub>ij</sub>（y）-δ<sub>ij</sub>|= O（|y|<sup>k</sup>）, |b（y）|=O(|y|<sup>k+1</sup>), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.
