\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation

Abstract Related Papers Cited by
  • The paper deals with the initial-boundary value problem for

    $ u_t=a(x) (\Delta u+\lambda_1 u) \qquad $ (*)

    with zero Dirichlet data in a smoothly bounded domain $\Omega \subset \R^n$, $n\ge 1$. Here $a$ is positive in $\Omega$ and Hölder continuous in $\bar\Omega$, and $\lambda_1>0$ denotes the principal eigenvalue of $-\Delta$ in $\Omega$ with Dirichlet data. It is shown that if $\int_\Omega \frac{(\dist(x,\partial\Omega))^2}{a(x)}dx=\infty$ then there exist initial data in $W^{1,\infty}(\Omega)$ such that the solution of (*) is bounded but not convergent as $t\to\infty$: It has a totally ordered $\omega$-limit set which is not a singleton. Under the above condition, the occurrence of even unbounded ordered $\omega$-limit sets is demonstrated. Conversely, if $\frac{(\dist(x,\partial\Omega))^2}{a(x)}$ is integrable then any solution emanating from initial data in $W^{1,\infty}(\Omega)$ converges to some stationary solution of (*) as time approaches infinity.

    Mathematics Subject Classification: Primary: 35B40, 35B41, 35K65.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(80) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return