# Global and blow-up solutions for nonlinear parabolic problems with a gradient term under Robin boundary conditions

## Abstract

In this paper, we study the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:

where $D\subset {\mathbb{R}}^{N}$ ($N\ge 2$) is a bounded domain with smooth boundary ∂D. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for ‘blow-up time’, and an upper estimate of ‘blow-up rate’ are specified under some appropriate assumptions on the functions f, g, b and initial value ${u}_{0}$.

MSC:35K55, 35B05, 35K57.

## 1 Introduction

In this paper, we study the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:

(1.1)

where $q:={|\mathrm{\nabla }u|}^{2}$, $D\subset {\mathbb{R}}^{N}$ ($N\ge 2$) is a bounded domain with smooth boundary ∂D, $\partial /\partial n$ represents the outward normal derivative on ∂D, γ is a positive constant, ${u}_{0}$ is the initial value, T is the maximal existence time of u, and $\overline{D}$ is the closure of D. Set ${\mathbb{R}}^{+}:=\left(0,+\mathrm{\infty }\right)$. We assume, throughout the paper, that $b\left(s\right)$ is a ${C}^{3}\left({\mathbb{R}}^{+}\right)$ function, ${b}^{\prime }\left(s\right)>0$ for any $s\in {\mathbb{R}}^{+}$, $g\left(s\right)$ is a positive ${C}^{2}\left({\mathbb{R}}^{+}\right)$ function, $f\left(x,s,d,t\right)$ is a nonnegative ${C}^{1}\left(\overline{D}×{\mathbb{R}}^{+}×\overline{{\mathbb{R}}^{+}}×{\mathbb{R}}^{+}\right)$ function, and ${u}_{0}\left(x\right)$ is a positive ${C}^{2}\left(\overline{D}\right)$ function. Under the above assumptions, the classical theory [1] of parabolic equation assures that there exists a unique classical solution $u\left(x,t\right)$ with some $T>0$ for problem (1.1) and the solution is positive over $\overline{D}×\left[0,T\right)$. Moreover, the regularity theorem [2] implies $u\left(x,t\right)\in {C}^{3}\left(D×\left(0,T\right)\right)\cap {C}^{2}\left(\overline{D}×\left[0,T\right)\right)$.

Many papers have studied the global and blow-up solutions of parabolic problems with a gradient term (see, for instance, [313]). Some authors have discussed the global and blow-up solutions of parabolic problems under Robin boundary conditions and have got a lot of meaningful results (see [1420] and the references cited therein). Some special cases of problem (1.1) have been treated already. Zhang [21] dealt with the following problem:

where $D\subset {\mathbb{R}}^{N}$ ($N\ge 2$) is a bounded domain with smooth boundary ∂D. By constructing auxiliary functions and using maximum principles, the sufficient conditions characterized by functions f, g and ${u}_{0}$ were given for the existence of a blow-up solution. Zhang [22] investigated the following problem:

where $D\subset {\mathbb{R}}^{N}$ ($N\ge 2$) is a bounded domain with smooth boundary ∂D. By constructing some auxiliary functions and using maximum principles, the sufficient conditions were obtained there for the existence of global and blow-up solutions. Meanwhile, the upper estimate of a global solution, the upper bound of ‘blow-up time’ and the upper estimate of ‘blow-up rate’ were also given. Ding [21] considered the following problem:

where $D\subset {\mathbb{R}}^{N}$ ($N\ge 2$) is a bounded domain with smooth boundary ∂D. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, the sufficient conditions were obtained for the existence of global and blow-up solutions. For the blow-up solution, an upper and a lower bound on blow-up time were also given.

In this paper, we study problem (1.1). Since the function $f\left(x,u,q,t\right)$ contains a gradient term $q={|\mathrm{\nabla }u|}^{2}$, it seems that the methods of [2123] are not applicable for problem (1.1). In this paper, by constructing completely different auxiliary functions with those in [2123] and technically using maximum principles, we obtain some existence theorems of a global solution, an upper estimate of the global solution, the existence theorems of a blow-up solution, an upper bound of ‘blow-up time’, and an upper estimates of ‘blow-up rate’. Our results extend and supplement those obtained [2123].

We proceed as follows. In Section 2 we study the global solution of (1.1). Section 3 is devoted to the blow-up solution of (1.1). A few examples are presented in Section 4 to illustrate the applications of the abstract results.

## 2 Global solution

The main result for the global solution is the following theorem.

Theorem 2.1 Let u be a solution of problem (1.1). Assume that the following conditions (i)-(iv) are satisfied:

1. (i)

for any $s\in {\mathbb{R}}^{+}$,

$\begin{array}{r}{\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\ge 0,\phantom{\rule{2em}{0ex}}s{b}^{\prime }\left(s\right)-{\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\le 0,\phantom{\rule{2em}{0ex}}{\left(\frac{g\left(s\right)}{{b}^{\prime }\left(s\right)}\right)}^{\prime }\le 0,\\ {\left[\frac{1}{g\left(s\right)}{\left(\frac{g\left(s\right)}{{b}^{\prime }\left(s\right)}\right)}^{\prime }+\frac{1}{{b}^{\prime }\left(s\right)}\right]}^{\prime }+\frac{1}{g}{\left(\frac{g\left(s\right)}{{b}^{\prime }\left(s\right)}\right)}^{\prime }+\frac{1}{{b}^{\prime }\left(s\right)}\le 0;\end{array}$
(2.1)
2. (ii)

for any $\left(x,s,d,t\right)\in D×{\mathbb{R}}^{+}×\overline{{\mathbb{R}}^{+}}×{\mathbb{R}}^{+}$,

$\begin{array}{r}{f}_{t}\left(x,s,d,t\right)\le 0,\phantom{\rule{2em}{0ex}}{f}_{d}\left(x,s,d,t\right)\left[{\left(\frac{1}{{b}^{\prime }\left(s\right)}\right)}^{\prime }+\frac{1}{{b}^{\prime }\left(s\right)}\right]\le 0,\\ {\left(\frac{f\left(x,s,d,t\right){b}^{\prime }\left(s\right)}{g\left(s\right)}\right)}_{s}-\frac{f\left(x,s,d,t\right){b}^{\prime }\left(s\right)}{g\left(s\right)}\le 0;\end{array}$
(2.2)
3. (iii)
${\int }_{{m}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=+\mathrm{\infty },\phantom{\rule{1em}{0ex}}{m}_{0}:=\underset{\overline{D}}{min}{u}_{0}\left(x\right);$
(2.3)
4. (iv)
$\alpha :=\underset{\overline{D}}{max}\frac{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)}{{\mathrm{e}}^{{u}_{0}}}>0,\phantom{\rule{1em}{0ex}}{q}_{0}:={|\mathrm{\nabla }{u}_{0}|}^{2}.$
(2.4)

Then the solution u to problem (1.1) must be a global solution and

$u\left(x,t\right)\le {H}^{-1}\left(\alpha t+H\left({u}_{0}\left(x,t\right)\right)\right),\phantom{\rule{1em}{0ex}}\left(x,t\right)\in \overline{D}×\overline{{\mathbb{R}}^{+}},$
(2.5)

where

$H\left(z\right):={\int }_{{m}_{0}}^{z}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\phantom{\rule{1em}{0ex}}z\ge {m}_{0},$
(2.6)

and ${H}^{-1}$ is the inverse function of H.

Proof Consider the auxiliary function

$P\left(x,t\right):={b}^{\prime }\left(u\right){u}_{t}-\alpha {\mathrm{e}}^{u}.$
(2.7)

Now we have

$\mathrm{\nabla }P={b}^{″}{u}_{t}\mathrm{\nabla }u+{b}^{\prime }\mathrm{\nabla }{u}_{t}-\alpha {\mathrm{e}}^{u}\mathrm{\nabla }u,$
(2.8)
$\mathrm{\Delta }P={b}^{‴}{u}_{t}{|\mathrm{\nabla }u|}^{2}+2{b}^{″}\mathrm{\nabla }u\cdot \mathrm{\nabla }{u}_{t}+{b}^{″}{u}_{t}\mathrm{\Delta }u+{b}^{\prime }\mathrm{\Delta }{u}_{t}-\alpha {\mathrm{e}}^{u}{|\mathrm{\nabla }u|}^{2}-\alpha {\mathrm{e}}^{u}\mathrm{\Delta }u,$
(2.9)

and

$\begin{array}{rl}{P}_{t}=& {b}^{″}{\left({u}_{t}\right)}^{2}+{b}^{\prime }{\left({u}_{t}\right)}_{t}-\alpha {\mathrm{e}}^{u}{u}_{t}\\ =& {b}^{″}{\left({u}_{t}\right)}^{2}+{b}^{\prime }{\left(\frac{g}{{b}^{\prime }}\mathrm{\Delta }u+\frac{{g}^{\prime }}{{b}^{\prime }}{|\mathrm{\nabla }u|}^{2}+\frac{f}{{b}^{\prime }}\right)}_{t}-\alpha {\mathrm{e}}^{u}{u}_{t}\\ =& {b}^{″}{\left({u}_{t}\right)}^{2}+\left({g}^{\prime }-\frac{{b}^{″}g}{{b}^{\prime }}\right){u}_{t}\mathrm{\Delta }u+g\mathrm{\Delta }{u}_{t}+\left({g}^{″}-\frac{{b}^{″}{g}^{\prime }}{{b}^{\prime }}\right){u}_{t}{|\mathrm{\nabla }u|}^{2}\\ +\left(2{g}^{\prime }+2{f}_{q}\right)\mathrm{\nabla }u\cdot \mathrm{\nabla }{u}_{t}+\left({f}_{u}-\frac{{b}^{″}f}{{b}^{\prime }}-\alpha {\mathrm{e}}^{u}\right){u}_{t}+{f}_{t}.\end{array}$
(2.10)

It follows from (2.9) and (2.10) that

$\begin{array}{rl}\frac{g}{{b}^{\prime }}\mathrm{\Delta }P-{P}_{t}=& \left(\frac{{b}^{‴}g}{{b}^{\prime }}+\frac{{b}^{″}{g}^{\prime }}{{b}^{\prime }}-{g}^{″}\right){u}_{t}{|\mathrm{\nabla }u|}^{2}+\left(2\frac{{b}^{″}g}{{b}^{\prime }}-2{g}^{\prime }-2{f}_{q}\right)\mathrm{\nabla }u\cdot \mathrm{\nabla }{u}_{t}\\ +\left(2\frac{{b}^{″}g}{{b}^{\prime }}-{g}^{\prime }\right){u}_{t}\mathrm{\Delta }u-\alpha \frac{g}{{b}^{\prime }}{\mathrm{e}}^{u}{|\mathrm{\nabla }u|}^{2}-\alpha \frac{g}{{b}^{\prime }}{\mathrm{e}}^{u}\mathrm{\Delta }u-{b}^{″}{\left({u}_{t}\right)}^{2}\\ +\left(\frac{{b}^{″}f}{{b}^{\prime }}-{f}_{u}+\alpha {\mathrm{e}}^{u}\right){u}_{t}-{f}_{t}.\end{array}$
(2.11)

By (1.1), we have

$\mathrm{\Delta }u=\frac{{b}^{\prime }}{g}{u}_{t}-\frac{{g}^{\prime }}{g}{|\mathrm{\nabla }u|}^{2}-\frac{f}{g}.$
(2.12)

Substitute (2.12) into (2.11), to get

$\begin{array}{rl}\frac{g}{{b}^{\prime }}\mathrm{\Delta }P-{P}_{t}=& \left(\frac{{b}^{‴}g}{{b}^{\prime }}-\frac{{b}^{″}{g}^{\prime }}{{b}^{\prime }}-{g}^{″}+\frac{{\left({g}^{\prime }\right)}^{2}}{g}\right){u}_{t}{|\mathrm{\nabla }u|}^{2}+\left(2\frac{{b}^{″}g}{{b}^{\prime }}-2{g}^{\prime }-2{f}_{q}\right)\mathrm{\nabla }u\cdot \mathrm{\nabla }{u}_{t}\\ -\frac{{\left({b}^{\prime }\right)}^{2}}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }{\left({u}_{t}\right)}^{2}+\left(\frac{f{g}^{\prime }}{g}-\frac{{b}^{″}f}{{b}^{\prime }}-{f}_{u}\right){u}_{t}+\left(\alpha \frac{{g}^{\prime }}{{b}^{\prime }}{\mathrm{e}}^{u}-\alpha \frac{g}{{b}^{\prime }}{\mathrm{e}}^{u}\right){|\mathrm{\nabla }u|}^{2}\\ +\alpha \frac{f}{{b}^{\prime }}{\mathrm{e}}^{u}-{f}_{t}.\end{array}$
(2.13)

With (2.8), we have

$\mathrm{\nabla }{u}_{t}=\frac{1}{{b}^{\prime }}\mathrm{\nabla }P-\frac{{b}^{″}}{{b}^{\prime }}{u}_{t}\mathrm{\nabla }u+\alpha \frac{{\mathrm{e}}^{u}}{{b}^{\prime }}\mathrm{\nabla }u.$
(2.14)

Next, we substitute (2.14) into (2.13) to obtain

$\begin{array}{r}\frac{g}{{b}^{\prime }}\mathrm{\Delta }P+\left[2{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }+2\frac{{f}_{q}}{{b}^{\prime }}\right]\mathrm{\nabla }u\cdot \mathrm{\nabla }P-{P}_{t}\\ \phantom{\rule{1em}{0ex}}=\left(\frac{{b}^{‴}g}{{b}^{\prime }}+\frac{{b}^{″}{g}^{\prime }}{{b}^{\prime }}-{g}^{″}+\frac{{\left({g}^{\prime }\right)}^{2}}{g}-2\frac{{\left({b}^{″}\right)}^{2}g}{{\left({b}^{\prime }\right)}^{2}}+2\frac{{b}^{″}{f}_{q}}{{b}^{\prime }}\right){u}_{t}{|\mathrm{\nabla }u|}^{2}\\ \phantom{\rule{2em}{0ex}}+\left(2\alpha \frac{{b}^{″}g}{{\left({b}^{\prime }\right)}^{2}}{\mathrm{e}}^{u}-\alpha \frac{{g}^{\prime }}{{b}^{\prime }}{\mathrm{e}}^{u}-\alpha \frac{g}{{b}^{\prime }}{\mathrm{e}}^{u}-2\alpha \frac{{f}_{q}}{{b}^{\prime }}{\mathrm{e}}^{u}\right){|\mathrm{\nabla }u|}^{2}\\ \phantom{\rule{2em}{0ex}}-\frac{{\left({b}^{\prime }\right)}^{2}}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }{\left({u}_{t}\right)}^{2}+\left(\frac{f{g}^{\prime }}{g}-\frac{{b}^{″}f}{b}-{f}_{u}\right){u}_{t}+\alpha \frac{f}{{b}^{\prime }}{\mathrm{e}}^{u}-{f}_{t}.\end{array}$
(2.15)

In view of (2.7), we have

${u}_{t}=\frac{1}{{b}^{\prime }}P+\alpha \frac{{\mathrm{e}}^{u}}{{b}^{\prime }}.$
(2.16)

Substituting (2.16) into (2.15), we get

$\begin{array}{r}\frac{g}{{b}^{\prime }}\mathrm{\Delta }P+\left[2{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }+2\frac{{f}_{q}}{{b}^{\prime }}\right]\mathrm{\nabla }u\cdot \mathrm{\nabla }P\\ \phantom{\rule{2em}{0ex}}+\left\{\left[g{\left(\frac{1}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }\right)}^{\prime }+2{f}_{q}{\left(\frac{1}{{b}^{\prime }}\right)}^{\prime }\right]{|\mathrm{\nabla }u|}^{2}+\frac{g}{{\left({b}^{\prime }\right)}^{2}}{\left(\frac{f{b}^{\prime }}{g}\right)}_{u}\right\}P-{P}_{t}\\ \phantom{\rule{1em}{0ex}}=-\alpha {\mathrm{e}}^{u}\left\{g\left[{\left(\frac{1}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{{b}^{\prime }}\right]+2{f}_{q}\left[{\left(\frac{1}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{{b}^{\prime }}\right]\right\}{|\mathrm{\nabla }u|}^{2}\\ \phantom{\rule{2em}{0ex}}-\frac{{\left({b}^{\prime }\right)}^{2}}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }{\left({u}_{t}\right)}^{2}-\alpha \frac{g{\mathrm{e}}^{u}}{{\left({b}^{\prime }\right)}^{2}}\left[{\left(\frac{f{b}^{\prime }}{g}\right)}_{u}-\frac{f{b}^{\prime }}{g}\right]-{f}_{t}.\end{array}$
(2.17)

The assumptions (2.1) and (2.2) guarantee that the right-hand side of (2.17) is nonnegative, i.e.,

(2.18)

By applying the maximum principle [24], it follows from (2.18) that P can attain its nonnegative maximum only for $\overline{D}×\left\{0\right\}$ or $\partial D×\left(0,T\right)$. For $\overline{D}×\left\{0\right\}$, by (2.4), we have

$\begin{array}{rl}\underset{\overline{D}}{max}P\left(x,0\right)& =\underset{\overline{D}}{max}\left\{{b}^{\prime }\left({u}_{0}\right){\left({u}_{0}\right)}_{t}-\alpha {\mathrm{e}}^{{u}_{0}}\right\}=\underset{\overline{D}}{max}\left\{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)-\alpha {\mathrm{e}}^{{u}_{0}}\right\}\\ =\underset{\overline{D}}{max}\left\{{\mathrm{e}}^{{u}_{0}}\left[\frac{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)}{{\mathrm{e}}^{{u}_{0}}}-\alpha \right]\right\}=0.\end{array}$

We claim that P cannot take a positive maximum at any point $\left(x,t\right)\in \partial D×\left(0,T\right)$. In fact, suppose that P takes a positive maximum at a point $\left({x}_{0},{t}_{0}\right)\in \partial D×\left(0,T\right)$, then

$P\left({x}_{0},{t}_{0}\right)>0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{\partial P}{\partial n}{|}_{\left({x}_{0},{t}_{0}\right)}>0.$
(2.19)

With (1.1) and (2.16), we have

(2.20)

Next, by using the fact that ${\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\ge 0$, $s{b}^{\prime }\left(s\right)-{\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\le 0$ for any $s\in {\mathbb{R}}^{+}$, it follows from (2.20) that

$\frac{\partial P}{\partial n}{|}_{\left({x}_{0},{t}_{0}\right)}\le 0,$

which contradicts with inequality (2.19). Thus we know that the maximum of P in $\overline{D}×\left[0,T\right)$ is zero, i.e.,

and

$\frac{{b}^{\prime }\left(u\right)}{{\mathrm{e}}^{u}}{u}_{t}\le \alpha .$
(2.21)

For each fixed $x\in \overline{D}$, integration of (2.21) from 0 to t yields

${\int }_{0}^{t}\frac{{b}^{\prime }\left(u\right)}{{\mathrm{e}}^{u}}{u}_{t}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t={\int }_{{u}_{0}\left(x\right)}^{u\left(x,t\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le \alpha t,$
(2.22)

which implies that u must be a global solution. Actually, if that u blows up at finite time T, then

$\underset{t\to {T}^{-}}{lim}u\left(x,t\right)=+\mathrm{\infty }.$

Passing to the limit as $t\to {T}^{-}$ in (2.22) yields

${\int }_{{u}_{0}\left(x\right)}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le \alpha T$

and

${\int }_{{m}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s={\int }_{{m}_{0}}^{{u}_{0}\left(x\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+{\int }_{{u}_{0}\left(x\right)}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le {\int }_{{m}_{0}}^{{u}_{0}\left(x\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+\alpha T<+\mathrm{\infty },$

which contradicts with assumption (2.3). This shows that u is global. Moreover, it follows from (2.22) that

${\int }_{{u}_{0}\left(x\right)}^{u\left(x,t\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s={\int }_{{m}_{0}}^{u\left(x,t\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s-{\int }_{{m}_{0}}^{{u}_{0}\left(x\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=H\left(u\left(x,t\right)\right)-H\left({u}_{0}\left(x\right)\right)\le \alpha t.$

Since H is an increasing function, we have

$u\left(x,t\right)\le {H}^{-1}\left(\alpha t+H\left({u}_{0}\left(x\right)\right)\right).$

The proof is complete. □

## 3 Blow-up solution

The following theorem is the main result for the blow-up solution.

Theorem 3.1 Let u be a solution of problem (1.1). Assume that the following conditions (i)-(iv) are fulfilled:

1. (i)

for any $s\in {\mathbb{R}}^{+}$,

$\begin{array}{r}{\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\ge 0,\phantom{\rule{2em}{0ex}}s{b}^{\prime }\left(s\right)-{\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\ge 0,\phantom{\rule{2em}{0ex}}{\left(\frac{g\left(s\right)}{{b}^{\prime }\left(s\right)}\right)}^{\prime }\ge 0,\\ {\left[\frac{1}{g\left(s\right)}{\left(\frac{g\left(s\right)}{{b}^{\prime }\left(s\right)}\right)}^{\prime }+\frac{1}{{b}^{\prime }\left(s\right)}\right]}^{\prime }+\frac{1}{g}{\left(\frac{g\left(s\right)}{{b}^{\prime }\left(s\right)}\right)}^{\prime }+\frac{1}{{b}^{\prime }\left(s\right)}\ge 0;\end{array}$
(3.1)
2. (ii)

for any $\left(x,s,d,t\right)\in D×{\mathbb{R}}^{+}×\overline{{\mathbb{R}}^{+}}×{\mathbb{R}}^{+}$,

$\begin{array}{r}{f}_{t}\left(x,s,d,t\right)\ge 0,\phantom{\rule{2em}{0ex}}{f}_{d}\left(x,s,d,t\right)\left[{\left(\frac{1}{{b}^{\prime }\left(s\right)}\right)}^{\prime }+\frac{1}{{b}^{\prime }\left(s\right)}\right]\ge 0,\\ {\left(\frac{f\left(x,s,d,t\right){b}^{\prime }\left(s\right)}{g\left(s\right)}\right)}_{s}-\frac{f\left(x,s,d,t\right){b}^{\prime }\left(s\right)}{g\left(s\right)}\ge 0;\end{array}$
(3.2)
3. (iii)
${\int }_{{M}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s<+\mathrm{\infty },\phantom{\rule{1em}{0ex}}{M}_{0}:=\underset{\overline{D}}{max}{u}_{0}\left(x\right);$
(3.3)
4. (iv)
$\beta :=\underset{\overline{D}}{min}\frac{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)}{{\mathrm{e}}^{{u}_{0}}}>0,\phantom{\rule{1em}{0ex}}{q}_{0}:={|\mathrm{\nabla }{u}_{0}|}^{2}.$
(3.4)

Then the solution u of problem (1.1) must blow up in finite time T, and

$T\le \frac{1}{\beta }{\int }_{{M}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,$
(3.5)
$u\left(x,t\right)\le {G}^{-1}\left(\beta \left(T-t\right)\right),\phantom{\rule{1em}{0ex}}\left(x,t\right)\in \overline{D}×\left[0,T\right),$
(3.6)

where

$G\left(z\right):={\int }_{z}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\phantom{\rule{1em}{0ex}}z>0,$
(3.7)

and ${G}^{-1}$ is the inverse function of G.

Proof Construct the following auxiliary function:

$Q\left(x,t\right):={b}^{\prime }\left(u\right){u}_{t}-\beta {\mathrm{e}}^{u}.$
(3.8)

Replacing P and α with Q and β in (2.17), respectively, we get

$\begin{array}{r}\frac{g}{{b}^{\prime }}\mathrm{\Delta }Q+\left[2{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }+2\frac{{f}_{q}}{{b}^{\prime }}\right]\mathrm{\nabla }u\cdot \mathrm{\nabla }Q\\ \phantom{\rule{2em}{0ex}}+\left\{\left[g{\left(\frac{1}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }\right)}^{\prime }+2{f}_{q}{\left(\frac{1}{{b}^{\prime }}\right)}^{\prime }\right]{|\mathrm{\nabla }u|}^{2}+\frac{g}{{\left({b}^{\prime }\right)}^{2}}{\left(\frac{f{b}^{\prime }}{g}\right)}_{u}\right\}Q-{Q}_{t}\\ \phantom{\rule{1em}{0ex}}=-\beta {\mathrm{e}}^{u}\left\{g\left[{\left(\frac{1}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{{b}^{\prime }}\right]+2{f}_{q}\left[{\left(\frac{1}{{b}^{\prime }}\right)}^{\prime }+\frac{1}{{b}^{\prime }}\right]\right\}{|\mathrm{\nabla }u|}^{2}\\ \phantom{\rule{2em}{0ex}}-\frac{{\left({b}^{\prime }\right)}^{2}}{g}{\left(\frac{g}{{b}^{\prime }}\right)}^{\prime }{\left({u}_{t}\right)}^{2}-\beta \frac{g{\mathrm{e}}^{u}}{{\left({b}^{\prime }\right)}^{2}}\left[{\left(\frac{f{b}^{\prime }}{g}\right)}_{u}-\frac{f{b}^{\prime }}{g}\right]-{f}_{t}.\end{array}$
(3.9)

Assumptions (3.1) and (3.2) imply that the right-hand side in equality (3.9) is nonpositive, i.e.,

(3.10)

With (3.4), we have

$\begin{array}{rl}\underset{\overline{D}}{min}Q\left(x,0\right)& =\underset{\overline{D}}{min}\left\{{b}^{\prime }\left({u}_{0}\right){\left({u}_{0}\right)}_{t}-\beta {\mathrm{e}}^{{u}_{0}}\right\}=\underset{\overline{D}}{min}\left\{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)-\beta {\mathrm{e}}^{{u}_{0}}\right\}\\ =\underset{\overline{D}}{min}\left\{{\mathrm{e}}^{{u}_{0}}\left[\frac{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)}{{\mathrm{e}}^{{u}_{0}}}-\beta \right]\right\}=0.\end{array}$
(3.11)

Substituting P and α with Q and β in (2.20), respectively, we have

(3.12)

Combining (3.10)-(3.12) with the fact that ${\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\ge 0$, $s{b}^{\prime }\left(s\right)-{\left(s{b}^{\prime }\left(s\right)\right)}^{\prime }\ge 0$ for any $s\in {\mathbb{R}}^{+}$, and applying the maximum principles again, it follows that the minimum of Q in $\overline{D}×\left[0,T\right)$ is zero. Thus

and

$\frac{{b}^{\prime }\left(u\right)}{{\mathrm{e}}^{u}}{u}_{t}\ge \beta .$
(3.13)

At the point ${x}^{\ast }\in \overline{D}$, where ${u}_{0}\left({x}^{\ast }\right)={M}_{0}$, integrate (3.13) over $\left[0,t\right]$ to get

${\int }_{0}^{t}\frac{{b}^{\prime }\left(u\right)}{{\mathrm{e}}^{u}}{u}_{t}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t={\int }_{{M}_{0}}^{u\left({x}^{\ast },t\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge \beta t,$
(3.14)

which implies that u must blow up in finite time. Actually, if u is a global solution of (1.1), then for any $t>0$, (3.14) shows

${\int }_{{M}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge {\int }_{{M}_{0}}^{u\left({x}^{\ast },t\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge \beta t.$
(3.15)

Letting $t\to +\mathrm{\infty }$ in (3.15), we have

${\int }_{{M}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=+\mathrm{\infty },$

which contradicts with assumption (3.3). This shows that u must blow up in finite time $t=T$. Furthermore, letting $t\to T$ in (3.14), we get

$T\le \frac{1}{\beta }{\int }_{{M}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.$

By integrating inequality (3.13) over $\left[t,s\right]$ ($0), for each fixed x, we obtain

$\begin{array}{rl}G\left(u\left(x,t\right)\right)& \ge G\left(u\left(x,t\right)\right)-G\left(u\left(x,s\right)\right)={\int }_{u\left(x,t\right)}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s-{\int }_{u\left(x,s\right)}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ ={\int }_{u\left(x,t\right)}^{u\left(x,s\right)}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s={\int }_{t}^{s}\frac{{b}^{\prime }\left(u\right)}{{\mathrm{e}}^{u}}{u}_{t}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\ge \beta \left(s-t\right).\end{array}$

Hence, by letting $s\to T$, we have

$G\left(u\left(x,t\right)\right)\ge \beta \left(T-t\right).$

Since G is a decreasing function, we obtain

$u\left(x,t\right)\le {G}^{-1}\left(\beta \left(T-t\right)\right).$

The proof is complete. □

## 4 Applications

When $b\left(u\right)\equiv u$ and $f\left(x,u,q,t\right)\equiv f\left(u\right)$, the results stated in Theorem 3.1 are valid. When $g\left(u\right)\equiv 1$ and $f\left(x,u,q,t\right)\equiv f\left(u\right)$ or $f\left(x,u,q,t\right)\equiv f\left(u\right)$, the conclusions of Theorems 2.1 and 3.1 still hold true. In this sense, our results extend and supplement the results of [2123].

In what follows, we present several examples to demonstrate the applications of the abstract results.

Example 4.1 Let u be a solution of the following problem:

where $q={|\mathrm{\nabla }u|}^{2}$, $D=\left\{x=\left({x}_{1},{x}_{2},{x}_{3}\right)\mid {|x|}^{2}<1\right\}$ is the unit ball of ${\mathbb{R}}^{3}$. The above problem can be transformed into the following problem:

Now

$\begin{array}{r}b\left(u\right)=u{\mathrm{e}}^{u},\phantom{\rule{2em}{0ex}}g\left(u\right)=\left(1+u\right){\mathrm{e}}^{u},\phantom{\rule{2em}{0ex}}f\left(x,u,q,t\right)=\left({\mathrm{e}}^{-u}+{\mathrm{e}}^{q}\right)\left({\mathrm{e}}^{-t}+{|x|}^{2}\right),\\ {u}_{0}\left(x\right)=2-{|x|}^{2},\phantom{\rule{2em}{0ex}}\gamma =2.\end{array}$

In order to determine the constant α, we assume

$s:={|x|}^{2},$

then $0\le s\le 1$ and

$\begin{array}{rl}\alpha & =\underset{\overline{D}}{max}\frac{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)}{{\mathrm{e}}^{{u}_{0}}}\\ =\underset{\overline{D}}{max}\left\{32{|x|}^{2}-4{|x|}^{4}-18+\left(1+{|x|}^{2}\right)\left[exp\left(-4+2{|x|}^{2}\right)+exp\left(-2+5{|x|}^{2}\right)\right]\right\}\\ =\underset{0\le s\le 1}{max}\left\{32s-4{s}^{2}-18+\left(1+s\right)\left[exp\left(-4+2s\right)+exp\left(-2+5s\right)\right]\right\}\\ =50.4417.\end{array}$

It is easy to check that (2.1)-(2.3) hold. By Theorem 2.1, u must be a global solution, and

$\begin{array}{rl}u\left(x,t\right)& \le {H}^{-1}\left(\alpha t+H\left({u}_{0}\left(x\right)\right)\right)=-1+\sqrt{50.4417t+{\left(1+{u}_{0}\left(x\right)\right)}^{2}}\\ =-1+\sqrt{50.4417t+{\left(3-{|x|}^{2}\right)}^{2}}.\end{array}$

Example 4.2 Let u be a solution of the following problem:

where $q={|\mathrm{\nabla }u|}^{2}$, $D=\left\{x=\left({x}_{1},{x}_{2},{x}_{3}\right)\mid {|x|}^{2}<1\right\}$ is the unit ball of ${\mathbb{R}}^{3}$. The above problem may be turned into the following problem:

Now we have

$\begin{array}{r}b\left(u\right)=u+lnu,\phantom{\rule{2em}{0ex}}g\left(u\right)=1+\frac{1}{u},\phantom{\rule{2em}{0ex}}f\left(x,u,q,t\right)=\left({\mathrm{e}}^{u}-{\mathrm{e}}^{-q}\right)\left(6+t{|x|}^{2}\right),\\ {u}_{0}\left(x\right)=2-{|x|}^{2},\phantom{\rule{2em}{0ex}}\gamma =2.\end{array}$

By setting

$s:={|x|}^{2},$

we have $0\le s\le 1$ and

$\begin{array}{rl}\beta & =\underset{\overline{D}}{min}\frac{\mathrm{\nabla }\cdot \left(g\left({u}_{0}\right)\mathrm{\nabla }{u}_{0}\right)+f\left(x,{u}_{0},{q}_{0},0\right)}{{\mathrm{e}}^{{u}_{0}}}\\ =\underset{\overline{D}}{min}\left\{\frac{-6{|x|}^{4}+26{|x|}^{2}-36}{{\left(2-{|x|}^{2}\right)}^{2}exp\left(2-{|x|}^{2}\right)}+6\left[1-exp\left(-3{|x|}^{2}-2\right)\right]\right\}\\ =\underset{0\le s\le 1}{min}\left\{\frac{-6{s}^{2}+26s-36}{{\left(2-s\right)}^{2}exp\left(2-s\right)}+6\left[1-exp\left(-3s-2\right)\right]\right\}\\ =0.0735.\end{array}$

Again it is easy to check that (3.1)-(3.3) hold. By Theorem 3.1, u must blow up in finite time T, and

$\begin{array}{c}T\le \frac{1}{\beta }{\int }_{{M}_{0}}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\frac{1}{0.0735}{\int }_{2}^{+\mathrm{\infty }}\left(1+\frac{1}{s}\right)\frac{1}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=2.5066,\hfill \\ u\left(x,t\right)\le {G}^{-1}\left(\beta \left(T-t\right)\right)={G}^{-1}\left(0.0735\left(T-t\right)\right),\hfill \end{array}$

where

$G\left(z\right)={\int }_{z}^{+\mathrm{\infty }}\frac{{b}^{\prime }\left(s\right)}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s={\int }_{z}^{+\mathrm{\infty }}\left(1+\frac{1}{s}\right)\frac{1}{{\mathrm{e}}^{s}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\phantom{\rule{1em}{0ex}}z\ge 0,$

and ${G}^{-1}$ is the inverse function of G.

Remark 4.1 We can see from Example 4.1 that when the equation has a gradient term with exponential increase, the functions g and b increase exponentially to ensure that the solution of (1.1) blows up. It follows from Example 4.2 that when the equation has a gradient term with exponential decay, the appropriate assumptions on the functions g and b can guarantee the solution of (1.1) to be global.

## References

1. Amann H: Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Ration. Mech. Anal. 1986, 92: 153-192.

2. Sperb RP: Maximum Principles and Their Applications. Academic Press, New York; 1981.

3. Ding JT, Guo BZ: Global existence and blow-up solutions for quasilinear reaction-diffusion equations with a gradient term. Appl. Math. Lett. 2011, 24: 936-942. 10.1016/j.aml.2010.12.052

4. Tersenov A: The preventive effect of the convection and of the diffusion in the blow-up phenomenon for parabolic equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2004, 21: 533-541. 10.1016/j.anihpc.2003.10.001

5. Zheng SN, Wang W: Effects of reactive gradient term in a multi-nonlinear parabolic problem. J. Differ. Equ. 2009, 247: 1980-1992. 10.1016/j.jde.2009.07.005

6. Payne LE, Song JC: Lower bounds for blow-up time in a nonlinear parabolic problem. J. Math. Anal. Appl. 2009, 354: 394-396. 10.1016/j.jmaa.2009.01.010

7. Chen SH: Global existence and blowup of solutions for a parabolic equation with a gradient term. Proc. Am. Math. Soc. 2001, 129: 975-981. 10.1090/S0002-9939-00-05666-5

8. Chen SH: Global existence and blowup for quasilinear parabolic equations not in divergence form. J. Math. Anal. Appl. 2013, 401: 298-306. 10.1016/j.jmaa.2012.12.028

9. Chipot M, Weissler FB: Some blowup results for a nonlinear parabolic equation with a gradient term. SIAM J. Math. Anal. 1989, 20: 886-907. 10.1137/0520060

10. Fila M: Remarks on blow up for a nonlinear parabolic equation with a gradient term. Proc. Am. Math. Soc. 1991, 111: 795-801. 10.1090/S0002-9939-1991-1052569-9

11. Souplet P, Weissler FB: Poincaré’s inequality and global solutions of a nonlinear parabolic equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1999, 16: 335-371. 10.1016/S0294-1449(99)80017-1

12. Souplet P: Recent results and open problems on parabolic equations with gradient nonlinearities. Electron. J. Differ. Equ. 2001, 2001: 1-19.

13. Souplet P: Finite time blow-up for a non-linear parabolic equation with a gradient term and applications. Math. Methods Appl. Sci. 1996, 19: 1317-1333. 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M

14. Friedman A, Mcleod B: Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 1985, 34: 425-447. 10.1512/iumj.1985.34.34025

15. Enache C: Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition. Appl. Math. Lett. 2011, 24: 288-292. 10.1016/j.aml.2010.10.006

16. Payne LE, Schaefer PW: Blow-up in parabolic problems under Robin boundary conditions. Appl. Anal. 2008, 87: 699-707. 10.1080/00036810802189662

17. Rault JF: The Fujita phenomenon in exterior domains under the Robin boundary conditions. C. R. Math. Acad. Sci. Paris 2011, 349: 1059-1061. 10.1016/j.crma.2011.09.006

18. Li YF, Liu Y, Xiao SZ: Blow-up phenomena for some nonlinear parabolic problems under Robin boundary conditions. Math. Comput. Model. 2011, 54: 3065-3069. 10.1016/j.mcm.2011.07.034

19. Liu Y, Luo SG, Ye YH: Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions. Comput. Math. Appl. 2013, 65: 1194-1199. 10.1016/j.camwa.2013.02.014

20. Li YF, Liu Y, Lin CH: Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions. Nonlinear Anal., Real World Appl. 2010, 11: 3815-3823. 10.1016/j.nonrwa.2010.02.011

21. Zhang LL: Blow-up of solutions for a class of nonlinear parabolic equations. Z. Anal. Anwend. 2006, 25: 479-486.

22. Zhang HL: Blow-up solutions and global solutions for nonlinear parabolic problems. Nonlinear Anal. TMA 2008, 69: 4567-4574. 10.1016/j.na.2007.11.013

23. Ding JT: Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions. Comput. Math. Appl. 2013, 65: 1808-1822. 10.1016/j.camwa.2013.03.013

24. Protter MH, Weinberger HF: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs; 1967.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61074048 and 61174082) and the Research Project Supported by Shanxi Scholarship Council of China (Nos. 2011-011 and 2012-011).

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Ding, J. Global and blow-up solutions for nonlinear parabolic problems with a gradient term under Robin boundary conditions. Bound Value Probl 2013, 237 (2013). https://doi.org/10.1186/1687-2770-2013-237