The focus of this paper is an analysis of the long-time behavior of nonnegative solutions of \[ u_t = (u^m)_{xx} + \varepsilon (u^n)_x + au^p,\quad 0 < x < 1, \;t > 0, \tag \(1(varepsilon)\) \]
\[ u(0,t) = u(1,t) = 0, \quad \;t > 0, \quad u(x,0) = u_0 (x) \geq 0,\quad \;0 \leq x \leq 1 \] where \(a\), \(\varepsilon \geq 0\) and \(m,n,p \geq 1\). Herein, we show that the convective term plays an important role in the long time behavior of solutions in the remaining cases \(p > n\). Specifically, our main results are the following.
Theorem 1. Let \(m,n,p \geq 1\). For each \(M > 0\) there exists \(\varepsilon_0 = \varepsilon_0 (M) > 0\) such that any solution of \((1(\varepsilon))\) with \(|u_0 |_\infty \leq M\) and \(\varepsilon\geq\varepsilon_0\) is uniformly bounded on \([0,1] \times [0, \infty)\).
Theorem 2. Let \(m,n,p \geq 1\). If either \(m < p \leq n\) or \(n \leq p < m\), and if \(\varepsilon \geq \varepsilon_1\), then any solution of \((1(\varepsilon))\), \(u(x,t)\), with \(|u (\cdot, 0)|_\infty \leq M\), has \(\lim_{t \to \infty} u(x,t) = 0\) for all \(x \in [0,1]\). Here, \(\varepsilon_1 \equiv \max \{a^{(n - m)/(p - m)}, \varepsilon_0 (M)\}\).
Theorem 3. Suppose \(n \geq p > m \geq 1\). If \(u(x,t)\) is a nonnegative solution of \((1(\varepsilon))\) for some \(\varepsilon > 0\), then \(u(x,t)\) is uniformly bounded on \([0,1] \times [0, \infty)\). Moreover, there exists \(\varepsilon_0 > 0\) such that if \(\varepsilon \geq \varepsilon_0\), then \(\lim_{t \to \infty} u(x,t) = 0\) for all \(x \in [0,1]\). (Here \(\varepsilon_0\) is independent of \(|u (\cdot, 0) |_\infty)\).
For the entire collection see [Zbl 0785.00039].