Convergence in general periodic parabolic equations in one space dimension. (English) Zbl 0796.35009
By using the Poincaré map, a general convergence theorem for the solution of quasilinear parabolic equations of the type
\[
u_ t = d(t,x,u,u_ x) u_{xx} + f(t,x,u,u_ x),\quad x \in [0,1],
\]
with either of the boundary conditions
\[
u(t,i) = h_ i(t),\;t>0,\;i=0,1, \quad \text{or} \quad u_ x (t,i) = g_ i (t,u(t,i)),\;t>0,\;i=0,1
\]
is presented. The result is a generalization of the one proved by Chen and Matano, and also the one proved by Sandstede. Meanwhile, the proof is simpler than the one in Sandstede’s thesis.
Reviewer: Lu Jiping (Quing das)
MSC:
| 35B10 | Periodic solutions to PDEs |
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
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