Communications of the
Korean Mathematical Society CKMS

We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $\R^N$: $$ \frac{\partial u}{\partial t} - \text{div}(\sigma (x)\nabla u) + \lambda u+ f(u) = g(x),$$ under a new condition concerning a variable non-negative diffusivity $\sigma(\cdot)$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.

Keywords: semilinear degenerate parabolic equation, weak solution, global attractor, non-compact case, tail estimates method