The global existence or nonexistence and the blow-up in finite time of solutions to semilinear or quasilinear parabolic equations and systems have received a lot of attention. Payne and Schaefer [15] considered the following problem of a semilinear heat equation:
$$\begin{aligned} u_{t}=\Delta u+f(u) \end{aligned}$$
under homogeneous Dirichlet boundary conditions and appropriate constraints on the nonlinearity \(f(u)\). By using a differential inequality technique, a lower bound on the blow-up time was determined if blow-up occurs.
Grillo et al. [9] considered the nonlinear evolution problem of the form
$$\begin{aligned} u_{t}=\Delta u^{m}+u^{p}, \end{aligned}$$
in an N-dimensional complete, simply connected Riemannian manifold with nonpositive sectional curvatures (namely a Cartan–Hadamard manifold). Under some appropriate constraints on \(p,m\) and the initial data, they proved that the problem has a global in time solution or the solution of the problem blows up at a finite time.
Yang et al. [21] considered local quasilinear parabolic equation with a potential term
$$\begin{aligned} u_{t}=\Delta u^{m}-V(x)u^{m}+u^{p}. \end{aligned}$$
By using the test function method and constructing a supersolution technique, they proved that every nontrivial solution blows up in finite time if \(1< p\leq p_{c}\) and there are both global and nonglobal solutions if \(p>p_{c}\). For more results, see [5, 12, 17, 20].
In this paper, we consider a more interesting system of quasilinear parabolic equation
$$\begin{aligned} &u_{t}=\Delta u^{m}-V(x)u+ \vert x \vert ^{\alpha }u^{p} \biggl( \int _{\Omega }\beta (x)u^{q}\,dx \biggr)^{\frac{r}{q}}, \quad \text{in }\Omega \times (0, T), \end{aligned}$$
(1)
$$\begin{aligned} &u=0,\quad \text{on }\partial \Omega \times (0, T), \end{aligned}$$
(2)
$$\begin{aligned} & u(x,0)=u_{0}(x),\quad \text{in }\Omega, \end{aligned}$$
(3)
where \(\alpha, p, q, r>0, m>1\), \(u_{0}\in C_{0}(\Omega )\) is nonnegative, \(V(x)\) is a positive function satisfying \(V(x)\sim |x|^{-\sigma },\sigma >0\), \(\beta (x)\) is positive. The model (1)–(3) describes the diffusion of concentration of some Newtonian fluids through a porous medium or the density of some biological species in many physical phenomena and biological species theories (see [2, 7]). Many methods (e.g., the Fourier coefficient method, the supersolution technique, the Green function method, the test function method, weighted energy arguments, the comparison method, and the concavity method) used to determine an upper bound for the blow-up time. The lower bound for the blow-up time is equally important and may be more difficult to obtain. In this paper, we first use the Sobolev inequalities to prove the existence of global solution. Our main results can be written as follows.
Theorem 2.1
Letting \(u(x,t)\)be a nonnegative solution of problem (1)–(3) in Ω, where Ω is a simply connected, bounded domain in \(R^{N}\ (N\geq 2)\). Then, if \(m+3>4(p+r)\), then problem (1)–(3) has a solution that is global in time whether \(N=2\)or \(N>2\).
Furthermore, if \(m+3\leq 4(p+r)\), the solution of (1)–(3) maybe blows up in some finite time. In this case, it is necessary to derive the lower bound of blow-up time. Whether the blow-up occurs or not, such a lower bound is still meaningful. We can obtain the following result.
Theorem 2.2
Letting \(u(x,t)\)be a nonnegative solution of problem (1)–(3) in Ω, where Ω is a simply connected, bounded domain in \(R^{N}\ (N\geq 2)\). If \(u(x,t)\)blows up at some finite time \(t^{*}\), then \(t^{*}\)can be bounded from below.
More precisely,
if \(2< N<\frac{4(p+r)}{4(p+r)-(m+3)}\)and \(2(p+r)< m+3<4(p+r)\), then
$$\begin{aligned} t^{*}\geq C_{4}\frac{2(m+3)-4(p+r)}{4(p+r)-(m+3)} \bigl[\varphi (0) \bigr]^{\frac{(m+3)-4(p+r)}{2(m+3)-4(p+r)}}, \end{aligned}$$
where \(C_{4}\)is a positive constant and \(\varphi (0)=\int _{\Omega }u_{0}^{2}\,dx\).
If \(N=2\)and \(2(p+r)< m+1, m+3<4(p+r)\), then
$$\begin{aligned} t^{*}\geq \frac{(m+1)-(p+r)}{p+r} C_{6} \bigl[\varphi (0) \bigr]^{- \frac{p+r}{(m+1)-(p+r)}}, \end{aligned}$$
where \(C_{6}\)is a positive constant.