The authors study the global existence and longtime behavior of solutions to the problem \[ \partial_t u = \Delta_{\lambda}u + f(u) \] in a bounded domain \(\Omega \subset\mathbb R^N\) under the following initial-boundary value conditions \( u(x,t) = 0\, \text{for } x\in\partial\Omega,\, t\geq 0 \,\text{and } u(x,0) = u_0(x) \, \text{for } x \in \Omega\), where \(\Delta_{\lambda}u\) is the following degenerate elliptic operator \[ \Delta_{\lambda}u = \sum_{i=1}^N\partial_{x_i}(\lambda_i^2\partial_{x_i}),\qquad \lambda = (\lambda_1,\dots,\lambda_N). \] It is assumed that \(u_0(x)\in \dot W_{\lambda}^{1,2}(\Omega)\) (\(X^{\frac{1}{2}} = \dot W_{\lambda}^{1,2}(\Omega ))\), where \(\dot W_{\lambda}^{1,2}(\Omega)\) is the closure of \(C^1_{0}(\Omega)\) with respect to the norm \[ \|u\|_{\dot W_{\lambda}^{1,2}(\Omega)} = \left(\int_{\Omega}|\nabla_{\lambda}u(x)|^2dx\right)^{1/2}, \] \(f\) is a locally Lipschitz continuous function which satisfies the following growth restriction: There exist nonnegative constants \(c\) and \(\gamma\) such that \(|f(u) - f(v)| \leq c|u - v|(1 + |u|^{\gamma} + |v|^{\gamma})\), for all \(u,v \in\mathbb R\). To show the global existence of solutions the following sign condition \[ \limsup_{|u|\to\infty}f(u)/u < \mu_1 \] is supposed, where \(\mu_1 > 0\) denotes the first eigenvalue of the operator \(-\Delta_{\lambda}\) on \(\Omega\) with the homogeneous Dirichlet boundary conditions. Under the above-mentioned assumptions on the problem data, the authors prove the existence of a unique local weak solution of the problem and then using the technique of Lyapunov functional the global result is given. In fact, the weak solution obtained generates a semigroup \(S_{\lambda}(t)\), \(t \geq 0\) in \(X^{\frac{1}{2}}\). Finally, they prove the existence and finite fractional dimension of the global attractor for \(S_{\lambda}(t)\) in \(X^{\frac{1}{2}}\). Moreover, the convergence of weak solutions to the set of equilibrium solutions is established when time \(t\to\infty\).