| Abstract: |
| In this paper,
we discuss the Cauchy problem of pseudo-parabolic equation
$$
u_t - \Delta u_t - \Delta u - \nabla\cdot\left(|\nabla u|^{2q}\nabla u\right)
= u^{p},
$$
where $ p \geq 2q + 1$.
Firstly,
under $ p < \frac{n + 2}{n - 2}$,
we prove the local existence of weal solutions
by the Galerkin method.
Secondly,
under $p > 2q + 1$
and certain initial datum,
we establish
the blow-up in finite time
and
the global existence
of solutions
by the potential well method
and
the Poincar\`e inequality.
Thirdly,
we discuss the blow-up in finite time
of solutions under
$p = 2q + 1$
and
the non-positive initial energy.
Fourthly,
we study the blow-up in finite time of solutions
under $p = 2q + 1$, $\Delta u$ vanishing
and
the negative initial energy.
Finally,
we determine a better lower bound for blow-up time
if blow-up does occur. |
|