In this paper, we consider the reaction-diffusion equations with absorption

where $m>1$, $p>0$, $q\ge 1$, $p\ne q$, $\mathrm{\Omega}\subset {\mathbb{R}}^{N}$ is a bounded domain with smooth boundary *∂* Ω, and ${u}_{0}(x)$ is a nontrivial, nonnegative, bounded, and appropriately smooth function. Parabolic equations like (1.1) appear in population dynamics, chemical reactions, heat transfer, and so on. We refer to [2, 8, 9] for details on physical models involving more general reaction-diffusion equations.

The semilinear case ($m=1$) of (1.1) has been investigated by Bedjaoui and Souplet [3]. They obtained that the solutions exist globally if either $p<max\{q,1\}$ or $p=max\{q,1\}$, and the solutions may blow up in finite time for large initial value if $p>max\{q,1\}$. Recently, Xiang *et al.* [11] considered the blow-up rate estimates for nonglobal solutions of (1.1) ($m=1$) with $p>max\{q,1\}$, and obtained that (i) ${max}_{\mathrm{\Omega}}u(x,t)\ge c{(T-t)}^{-\frac{1}{p-1}}$; (ii) ${max}_{\mathrm{\Omega}}u(x,t)\le C{(T-t)}^{-\frac{1}{p-1}}$ if $p\le 1+\frac{2}{N+1}$, where $c,C>0$ are positive constants. Liu *et al.* [7] studied the extinction phenomenon of solutions of (1.1) for the case $0<m<1$ with $q=1$ and obtained some sufficient conditions about the extinction in finite time and decay estimates of solutions in $\mathrm{\Omega}\subset {\mathbb{R}}^{N}$ ($N>2$).

Recently, Zhou

*et al.* [

10] investigated positive solutions of the degenerate parabolic equation not in divergence form

where $p\ge 1$, $q,a,b>0$, $r>1$. They at first gave some conditions about the existence and nonexistence of global solutions to (1.2), and then studied the large time behavior for the global solutions.

Motivated by the above mentioned works, the aim of this paper is threefold. First, we determine optimal conditions for the existence and nonexistence of global solutions to (1.1). Secondly, by using the scaling arguments we establish the exact blow-up rate estimates for solutions which blow up in a finite time. Finally, we prove that every solution to (1.1) is nonextinction.

As it is well known that degenerate equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1).

**Definition 1.1** Let

$T>0$ and

${Q}_{T}=\mathrm{\Omega}\times (0,T)$,

$E=\{u\in {L}^{2p}({Q}_{T})\cap {L}^{2q}({Q}_{T});{u}_{t},\mathrm{\nabla}u\in {L}^{2}({Q}_{T})\}$,

${E}_{0}=\{u\in E;u=0\text{on}\partial \mathrm{\Omega}\}$, a nonnegative function

$u(x,t)\in E$ is called a weak upper (or lower) solution to (1.1) in

${Q}_{T}$ if for any nonnegative function

$\phi \in {E}_{0}$, one has

In particular, $u(x,t)$ is called a weak solution of (1.1) if it is both a weak upper and a weak lower solution. For every $T<\mathrm{\infty}$, if $u(x,t)$ is a weak solution of (1.1) in ${Q}_{T}$, we say that $u(x,t)$ is global. The local in time existence of nonnegative weak solutions have been established (see the survey [1]), and the weak comparison principle is stated and proved in the Appendix in this paper.

The behavior of the weak solutions is determined by the interactions among the multinonlinear mechanisms in the nonlinear diffusion equations in (1.1). We divide the $(m,p,q)$-parameter region into three classes: (i) $p<max\{m,q\}$; (ii) $p=max\{m,q\}$; (iii) $p>max\{m,q\}$.

**Theorem 1.1** *If* $p<max\{m,q\}$, *then all solutions of* (1.1) *are bounded*.

Let

$\varphi (x)$ be the first eigenfunction of

$-\mathrm{\Delta}\varphi (x)=\lambda \varphi (x)\phantom{\rule{1em}{0ex}}\text{in}\mathrm{\Omega},\phantom{\rule{2em}{0ex}}\varphi (x)=0\phantom{\rule{1em}{0ex}}\text{in}\partial \mathrm{\Omega}$

(1.3)

with the first eigenvalue ${\lambda}_{1}$, normalized by ${\parallel \varphi \parallel}_{\mathrm{\infty}}=1$, then ${\lambda}_{1}>0$ and $\varphi >0$ in Ω.

**Theorem 1.2** *Assume that* $p=max\{m,q\}$. *Then all solutions are global if* ${\lambda}_{1}\ge 1$, *and there exist both global and nonglobal solutions if* ${\lambda}_{1}<1$.

**Theorem 1.3** *If* $p>max\{m,q\}$, *then there exist both global and nonglobal solutions to* (1.1).

To obtain the blow-up rate of blow-up solutions to (1.1), we need an extra assumption that $\mathrm{\Omega}={B}_{R}(0)=\{x\in {\mathbb{R}}^{N}:|x|<R\}$ and ${u}_{0}={u}_{0}(r)$, ${u}_{0}^{\prime}(r)\le 0$, here $r=|x|$. By the assumption and comparison principle, we know that *u* is radially decreasing in *r* with ${max}_{\mathrm{\Omega}}u(x,t)=u(0,t)$.

**Theorem 1.4** *Suppose that* $p>max\{m,q\}$.

*If the solution* $u(x,t)$ *of* (1.1)

*blows up in finite time* *T*,

*then there exists a positive constant* *c* *such that* $\underset{\mathrm{\Omega}}{max}u(x,t)\ge c{(T-t)}^{-\frac{1}{p-1}}\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.25em}{0ex}}t\to T.$

*Furthermore*,

*if* $p>m\ge q$,

*then we have also the upper estimate*,

*that is*,

*there exists a positive constant* *C* *such that* $\underset{\mathrm{\Omega}}{max}u(x,t)\le C{(T-t)}^{-\frac{1}{p-1}}\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.25em}{0ex}}t\to T.$

We remark that in

$\mathrm{\Omega}={\mathbb{R}}^{N}$, Liang [

6] studied the blow up rate of blow-up solutions to the following Cauchy problem

${u}_{t}=\mathrm{\Delta}{u}^{m}+{u}^{p},\phantom{\rule{1em}{0ex}}(x,t)\in {\mathbb{R}}^{N}\times (0,T)$

(1.4)

with the bounded initial function, $1<m<p<m\frac{N+2}{{(N-2)}_{+}}$, and obtained that ${\parallel u\parallel}_{{L}^{\mathrm{\infty}}({\mathbb{R}}^{N})}<C{(T-t)}^{-\frac{1}{p-1}}$ for $t\in (0,T)$. By using the same scaling arguments in this paper, we can find that Theorem 1.4 is correct for (1.4) with $p>m$.

Now, we pay attention to the nonextinction property of solutions and have the following result.

**Theorem 1.5** *Any solution of* (1.1) *does not go extinct in finite time for any nontrivial and nonnegative initial value* ${u}_{0}(x)$ *with* $meas\{x\in \mathrm{\Omega};{u}_{0}(x)>0\}>0$.

The rest of this paper is organized as follows. In the next section, we discuss the global existence and nonexistence of solutions, and prove Theorems 1.1-1.3. Subsequently, in Sects. 3 and 4, we consider the estimate of the blow-up rate and study the nonextinction phenomenon for the problem (1.1). The weak comparison principle is stated and proved in the Appendix.