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发帖数: 520 | 1 请教parabolic equation收敛到stationary solution 的条件。
Consider parabolic equation
u_t=Lu. (P)
If we know that the second order elliptic equation
Lu=0, (E)
has unique solution v(x), then v(x) is the unique stationary solution of (P).
(We let (P) and (E) have same boundary conditions.)
Under what condition we can say that the solution u(x,t) of (P) converges to
v(x), as t goes to infinity?
In general, if (E) has several solutions v_1(x),...,v_n(x), can we say that
the solution u(x,t) of (P) converges to v_i(x), (1=
initial condition) as t goes to infinity?
请问这类收敛问题有比较普遍的结论吗？
另外对于unbounded domain，如果(E) 似乎只有trivial solution v=0，要证明
nontrivial solution 不存在，除了Pohozaev type identity，还有什么常用技巧？
若u 是vector，即u=(u_1,...,u_m) ，(E) 是含有m 个方程的方程组，通常用什么方
法证明(E) 只有trivial solution？
请大家指点。非常感谢。 |
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