Blow-up analysis for a reaction-diffusion equation with gradient absorption terms

1.   Introduction

We study the following reaction-diffusion equation with gradient absorption terms:

under nonlinear boundary flux and initial conditions

where $\Omega\subset\mathbb{R}^{N}$$(N\geq2)$ is a bounded star-shaped domain with smooth boundary $\partial\Omega$, $\nu$ is the unit outward normal vector on $\partial\Omega$, and $t^{\ast}$ is a possible blow-up time when blow-up occurs, otherwise $t^{\ast} = +\infty$. Nonlinear functions $f$ and $g$ are assumed to be nonnegative continuous functions and satisfy appropriate conditions. Moreover, initial data $u_{0}(x)$ is a positive $C^{1}$-function and meets an appropriate compatibility condition. Therefore, as it is well-known from the standard parabolic theory, we deduce that the problem (1.1)–(1.3) has unique non-negative classical solutions.

The gradient model (1.1) is often referred to as a viscous Hamilton-Jacobi equation. And it is closely related to the Kardar-Parisi-Zhang equation describing growth and roughening of surfaces in the physical theory, see ^[1,2] and references therein for details. Furthermore, the nonlinear boundary flux (1.2) satisfies the nonlinear radial law from the physical point of view (cf. ^[3,4]).

During the past decades, many scholars have dealt with existence and nonexistence of global solutions, blow-up of solutions, blow-up rates, life span, and asymptotic behavior of the solutions to reaction-diffusion equations (systems), see monographs ^[5,6] and review literature ^[7,8,9]. In particular, the monograph [5, Chapters 2 and 4] illustrates a series of research progresses on reaction-diffusion equations with nonlinear terms $f(u)$ and $f(u, \nabla u)$. Among them, it is important to investigate whether the solution of the reaction-diffusion equation blows up and when blow-up occurs in the sense of appropriate measure.

In this paper, we will investigate bounds for blow-up time of the solution to a gradient diffusion model under nonlinear boundary flux. Levine ^[10] used a variety of methods to study blow-up phenomena and, in many cases, the methods used to show blow-up of solutions often provide an upper bound for blow-up time. However, lower bounds for blow-up time may be harder to be determined. Recently, since researchers, such as Payne, Schaefer and Philippin, made pioneering works on determining lower bounds for blow-up time, there have been many new progresses on the issue of lower bounds for blow-up time in models without gradient term under nonlinear boundary flux. One can refer to papers ^{[11,12,13,14]} for constant coefficients and ^{[15,16,17,18]} for variable coefficients. Note that the lower bounds for the blow-up time are mostly derived in three-dimensional space and the main difficulty lies in determination of Sobolev optimal constant.

However, there are few works on bounds for blow-up time for the gradient diffusion model. The salient feature of the gradient model is that boundary or internal gradient blow-up may or may not occur under some conditions (cf. ^[19,20,21]). In particular, literatures ^[19,20] studied the following reaction-diffusion equation with inner source and gradient absorption terms

under Dirichlet boundary condition. They pointed out that gradient blow-up never occurs, while $L^{\infty}$ blow-up does. Payne and Song ^[22] firstly derived the lower bounds of blow-up time for the gradient damping model (1.4) in three-dimensional space when blow-up occurs. For a high-dimensional case $(N\geq3)$, we refer to ^[23]. Recently, Liu et al. ^[24] studied lower bounds of blow-up time for the reaction-diffusion equation (1.4) with gradient absorption terms in a bounded convex domain in three-dimensional space under nonlinear boundary flux. For the studies on reaction-diffusion equations (systems) with time-dependent or space-dependent coefficients and non-divergence form quasilinear equations with inner gradient terms, one can refer to ^{[25,26,27,28]}.

To the best of our knowledge, no research on blow-up analysis to problem (1.1)–(1.3) with gradient absorption terms under nonlinear boundary flux has been done. The main difficulty lies in finding an effect of the competitive relationship between the inner gradient absorption terms and the nonlinear boundary flux on the blow-up solutions. In particular, comparing with the studies, in the aforementioned literatures, on non-gradient problems under nonlinear boundary flux, we consider the gradient damped model, which can be considered as one of the difficult and interesting research problems. Motivated by these observations, using the auxiliary function method, the technique of modified differential inequality and the method of constructing the sub-solution, we establish some conditions for which the solution of (1.1)–(1.3) exists globally or blows up and derive some bounds for blow-up time in high-dimensional spaces $(N\geq2)$.

The remainder of this paper is organized as follows. In Section 2, we present some conditions on nonlinearities $f$ and $g$ for which the solution of problem (1.1)–(1.3) exists globally. In Section 3, we construct a suitable sub-solution to show the solution blows up at finite time. In Section 4, we are devoted to deriving the lower bounds for blow-up time when blow-up occurs.

2.   The global existence

In this section, we present some conditions on nonlinearities $f(|\nabla\xi|)$ and $g(\xi)$ for which a global solution of problem (1.1)–(1.3) exists. In order to prove our main results, we introduce the following lemma:

Lemma 2.1. Suppose that $\Omega\subset\mathbb{R}^{N}(N\geq2)$ be a bounded domain assumed to be star-shaped and convex in $N-1$ orthogonal directions with smooth boundary $\partial\Omega$. Then for any nonnegative $C^{1}$-function $u$ and constant $l\geq1$, we have the inequality

where

$d = \mathop {\max }\limits_{x \in \bar \Omega } \left| x \right|,$ and $n_0, \; n_1, \; n_2 > 0$ are constants given in the proof.

Proof. Define a function $h_{i}$ on $\bar{\Omega}$ such that

where $\nu_i$ is the unit outward normal on $\partial \Omega$. By divergence theorem, one can have

and

When $N \geq 3$, a similar argument as in the proof of Lemma 4.1 given in ^[18] can be used to obtain the desired result by replacing the integral $\int_{\partial \Omega } {\omega^\sigma } ds$ contained in (4.7) of [18,pp. 9] with (2.1), and hence, we omit the proof. However, we cannot use the argument for the case that $N = 2$ and so we give a detailed proof.

Let $P = \left({\bar x_1, \bar x_2 } \right)$ be an arbitrary point in $\Omega \subset R^{\rm{2}}$, and let $P_k = \left({\xi _k, \bar x_2 } \right)$ and $Q_k = \left({\bar x_1, \eta _k } \right)$ be the points on the boundary $\partial \Omega$ associated with $P$, where $k = 1, 2$, and $\xi _1 < \xi _2$ and $\eta _1 < \eta _2.$ Then we have

and then

Similarly, one can have the inequality

By multiplying (2.2) and (2.3) and integrating the result over $\Omega$, we obtain the inequalities

where $\left({x_k } \right)_m = \mathop {\min }\limits_{\overline \Omega } x_k$, $\left({x_k } \right)_M = \mathop {\max }\limits_{\overline \Omega } x_k$, $k = 1, 2$, and $\nu _i$ is the unit outward normal on $\partial \Omega$, $i = 1, 2.$ We then have the inequality

by inserting (2.1) into (2.4).

Theorem 2.1. Let $\Omega\subset\mathbb{R}^{N}(N\geq2)$ be a bounded star-shaped domain assumed to be convex in $N-1$ orthogonal directions with smooth boundary $\partial\Omega$. Assume that the nonnegative function $f$ and positive function $g$ satisfy the following conditions:

where $a_{1}, a_{2} > 0, \; p, q > 1$, and $2q < p+1$. Then the nonnegative classical solution $u(x, t)$ of problem (1.1)–(1.3) does not blow up; that is, $u(x, t)$ exists for all $t > 0$.

Remark 2.1. Because $p, q > 1$ and $2q < p+1$, it can be easily seen that $p > q$. From a physical point of view, the absorption term is dominant. Therefore, the nonnegative classical solution of problem (1.1)–(1.3) does not blow up.

Proof. Define an auxiliary function

Using (1.1), (1.2), (2.5), and Green's formula, it can be seen that

We begin with estimating the first term on the right side of (2.7). From [12,(2.7)], one can see that

where $\rho _0 = \mathop {\rm min}\limits_{x \in \partial \Omega } \left({x \cdot \nu } \right) > 0$ and $d = \mathop {\max }\limits_{x \in \bar \Omega } \left| x \right|.$ Note that if $\Omega$ is a bounded star-shaped domain containing the origin, then $d$ clearly exists, while if $\Omega$ is a bounded star-shaped domain containing $x_0$ with $x_{0}\neq 0$, we can also have the inequality (2.8) with

by using the technique of translation. It follows from Hölder's and Young's inequalities that

where $\delta_{1}$ is a positive constant to be determined later. Hence, we get the inequality

Next, we estimate the last term on the right side of (2.7). For simplification, we set $v = u^{\frac{{2n + p - 1}}{p}}.$ Then the last term can be written as

Now, we consider the following two cases:

Case 1. $N\geq3$: By Lemma 2.1 and Young's inequality, it can be seen that

where

and $\delta_{2}$ is a positive constant to be determined later. It then follows from (2.12) that

For suitable constants $n_j \left({j = 0, 1, 2} \right)$ and $\delta_{2} > 0$ small enough such that

inequality (2.13) can be reduced to

where

Hence, we obtain the inequality

Case 2. $N = 2$: By Lemma 2.1 and Young's inequality, it can be seen that

where $\delta_{3}$ is a positive constant to be determined later.

For suitable constants $n_j \left({j = 0, 1, 2} \right)$ and $\delta_{3} > 0$ small enough such that

inequality (2.15) can be reduced to

where

We then have the inequality

Setting $B = \min \left\{ {B_1, B_2 } \right\} > 0$ and combining (2.14) with (2.16), we obtain the inequality

Then one can see that

by substituting (2.11) and (2.17) into (2.7). It can be also seen that

by selecting $\delta _1 = \frac{{2\left({2n - 1} \right)}}{{a_2 }} > 0$ such that

We now focus on the first and third terms on the right side of (2.18). From Hölder's inequality, we get

and

Substituting (2.19) and (2.20) into (2.18), one can see that

where

Finally, it follows from (2.20) and (2.21) that

where $\frac{{1 - q}}{n} < 0$ and $\frac{{p - 2q + 1}}{{2n}} > 0.$

We conclude from (2.22) that $\Phi(t)$ remains bounded for all time under the conditions stated in Theorem 2.1. In fact, if $u(x, t)$ blows up at finite time $t^{\ast}$, then $\Phi(t)$ is unbounded near $t^{\ast}$, which implies $\Phi(t)$ is decreasing in some interval $[t_{0}, t^{\ast})$, from (2.22). Hence, we have $\Phi(t)\leq\Phi(t_{0})$ in $[t_{0}, t^{\ast})$, which means that $\Phi(t)$ is bounded in $[t_{0}, t^{\ast})$, which is a contradiction. Therefore, $u(x, t)$ exists for all $t > 0$, which completes the proof.

Remark 2.2. If the boundary is adiabatic; that is, $g(u) = 0.$ From (2.6) and (2.7), we know the energy functional $\Phi(t)$ is decreasing, and hence, the nonnegative classical solution $u(x, t)$ of problem (1.1)–(1.3) exists for all $t > 0$.

Remark 2.3. If we use $L^{2}$-norm $\Phi(t): = \int_\Omega {u^2 } dx,$ the condition $q > 1$ is replaced by $p > q$, and other conditions remain unchanged, then the conclusion of Theorem 2.1 is still valid. In fact, using (2.7), (2.8) and (2.17), we have

We now apply Young's inequality to $\int_\Omega {u^q \left| {\nabla u} \right|} dx$ to obtain the inequality

Choosing $\varsigma = \frac{{(q + 1)a_2 d}}{{2\rho _0 }}$, we obtain the inequality

Since $p > q$ and $2q < p+1$, we can conclude that the nonnegative classical solution $u(x, t)$ of problem (1.1)–(1.3) exists for all $t > 0$. In fact, if $u(x, t)$ blows up at finite time $t^{\ast}$, then $u(x, t)$ is unbounded near $t^{\ast}$. And it is easy to know there exists an interval $[t_{0}, t^{\ast})$, such that

which implies $\Phi(t)$ is decreasing in some interval $[t_{0}, t^{\ast})$. So we have $\Phi(t)\leq\Phi(t_{0})$ in $[t_{0}, t^{\ast})$, which means that $\Phi(t)$ is bounded in $[t_{0}, t^{\ast})$, which is a contradiction. Therefore, $u(x, t)$ exists for all $t > 0$.

3.   Blow-up criterion

In this section, the domain $\Omega$ only needs to be a bounded region with smooth boundary, instead of star-shaped one. We construct a suitable sub-solution to show the solution blows up at finite time. Our result can be summarized as follows:

Theorem 3.1. Let $\Omega\subset\mathbb{R}^{N}(N\geq2)$ be a bounded domain with smooth boundary $\partial\Omega$. Suppose that $u(x, t)$ is a nonnegative classical solution of problem (1.1)–(1.3) and the nonnegative functions $f$ and $g$ are such that

where $2q > p + 1$ and $p > 1.$ Then the solution of (1.1)–(1.3) blows up in a finite time for some suitably large initial data.

Proof. We construct a sub-solution of the form

where $k\geq1, \; \beta > 0, \; \frac{p-1}{2} < \alpha < q-1,$ $A\geq1$ are constants to be determined and $\varphi(x)$ be the positive normalized eigenfunction, i.e., $\mathop {\max }\limits_{x \in \bar \Omega } \varphi \left(x \right) = 1$, corresponding to the first eigenvalue $\lambda _0$ of the problem

It is well known that $\lambda _0 > 0,$ $\varphi \left(x \right) > 0$ in $\Omega$, and $\frac{{\partial \varphi }}{{\partial \nu }} < 0$ on $\partial\Omega$. Moreover, there exist positive constants $R_1, R_2, R_3$ such that $\left| {\nabla \varphi \left(x \right)} \right| \le R_1$ for all $x \in \bar \Omega$ and $R_2 \le - \frac{{\partial \varphi }}{{\partial \nu }} \le R_3$ on $\partial\Omega$.

By direct calculation, one can see that

If $x\in\Omega_{\varepsilon}: = \left\{ {x \in \Omega | {\rm dist}\left({x, \partial \Omega } \right) \ge \varepsilon } \right\}$ for $\varepsilon > 0$, then there exists a positive constant $R_4$ such that $\varphi \left(x \right) \ge R_4$ and

From the inequalities above, it can be seen that

provided that

If $x \in \Omega \backslash \Omega _\varepsilon: = \left\{ {x \in \Omega |dist\left({x, \partial \Omega } \right) < \varepsilon } \right\},$ it is easy to know $|\nabla \varphi \left(x \right)| \ge \frac{{R_2 }}{2}$ and

Hence,

if

In addition, for $\left({x, t} \right) \in \partial \Omega \times \left({0, \frac{1}{\beta }} \right),$

Since $\frac{{p + 1}}{{p - 1}} < \frac{{2q}}{{p - 1}}$ and $\alpha < q - 1,$ we have

provided that

Thanks to $p > 1$ and $\frac{{p - 1}}{2} < \alpha < q - 1,$ then $\alpha > 0$ and $2\alpha - p + 1 > 0.$ The inequalities (3.2)–(3.4) hold for $A$ such that

where $\frac{1}{\theta } = \min \left\{ {\alpha, 2\alpha - p + 1} \right\}.$

Therefore, if we take $u_0 \left(x \right)$ suitably large for which

for every $x \in \Omega,$ then the comparison principle shows that

is a sub-solution of (1.1)–(1.3). Moreover, we easy to see that $\underline{u}$ occurs boundary blow-up in a finite time $t^{\ast} = \frac{1}{\beta}$, and hence, the solution of (1.1)–(1.3) blows up in a finite time $t^{\ast}$ with upper bound $\frac{1}{\beta}$ for suitably large initial data.

4.   Lower bounds for $t^{\ast}$

In this section, we assume some conditions on the nonlinearities $f$ and $g$ to find lower bounds for the blow-up time $t^{\ast}$ in high-dimensional spaces $(N\geq2)$.

4.1.   The case that $N\geq3$

In this subsection, the domain $\Omega \subset R^N \left({N \ge 3} \right)$ is assumed to be a bounded star-shaped domain and convex in $N-1$ orthogonal directions with smooth boundary.

Theorem 4.1. Suppose that $u(x, t)$ is the nonnegative classical solution of problem (1.1)–(1.3), $u(x, t)$ blows up at $t^{\ast}$, and that the nonnegative functions $f$ and $g$ satisfy the following conditions:

where $a_{1}, a_{2} > 0$, $p, q > 1$, and $2q \geq p+1$. Define a function

where

Then the blow-up time $t^{\ast}$ is bounded below, i.e.,

where $\phi(0) = \int_{\Omega}u_{0}^{2n}dx$ and $Q_{1}$–$Q_{3}$ are some positive constants given in the proof.

Proof. Using (1.1), (1.2), (4.1), and Green's formula, we have

By (2.8), we get

Applying Hölder's and Young's inequalities to the terms on the right side of (4.3), respectively, one can see that

where $\epsilon_{1}$ is a positive constant to be determined later.

Next, we estimate the last term on the right side of (4.2). It follows from (2.17) that

where $B > 0$ is the constant given in the proof of Theorem 2.1. By using Hölder's inequality, we have

Substituting (4.3)–(4.7) into (4.2), one can obtain the inequality

We now consider the second term on the right side of (4.8). By using Hölder's and Young's inequalities, we get

where

Substituting (4.9) into (4.8), we obtain the inequality

where

By applying Schwarz's inequality to the second term on the right side of (4.10), we have

To bound $\int_\Omega {\left({u^n } \right)^{\frac{{2N}}{{N - 2}}} dx},$ we use the Sobolev inequality $(N\geq3)$ given in ^[29] and then obtain the inequalities

where $c_{s}$ is a constant depending on $\Omega$ and $N$, and

Substituting (4.12) into (4.11) and using Young's inequality, one can see that

where $\epsilon_{2}$ is a positive constant to be determined later. It follows from Young's inequality that

where

and $\epsilon_{3}$ is a positive constant to be determined later. Substituting (4.13) and (4.14) into (4.10), we obtain the inequality

where

With appropriate constants $\varepsilon_1, \varepsilon_2, \varepsilon _3 > 0$ for which $Q_4 \; {\rm and} \; Q_5 = 0$, inequality (4.15) can be written as

Integrating (4.16) from 0 to $t$, we obtain the inequlity

Letting $t \to t^{ * - }$, we can obtain the desired result

4.2.   The case that $N=2$

In this subsection, the domain $\Omega \subset R^2$ is assumed to be a bounded star-shaped domain with smooth boundary.

Theorem 4.2. Suppose that $u(x, t)$ is the nonnegative classical solution of problem (1.1)–(1.3), $u(x, t)$ blows up at $t^{\ast}$, and that the nonnegative functions $f$ and $g$ satisfy the following conditions:

where $a_{3}, a_{4} > 0, \; \sigma \geq 1, \; p > 1$, and $p \leq \sigma + 1$. Define a function

Then the blow-up time $t^{\ast}$ is bounded below, i.e.,

where

and $K_1, \tilde K_2, K_3$, and $\tilde K_3$ are some positive constants defined in the proof.

Proof. Using (1.1), (1.2), (4.17), Green's formula and an adapted version of (2.8), it can be shown that

Using Hölder's and Young's inequalities to the first term on the right side of (4.18), we have

and

where $\mu_{1}$ is a positive constant to be determined later.

We estimate the last term on the right side of (4.18). From (2.17), one can see that

where $B > 0$ is the constant given in the proof of Theorem 2.1. By Hölder's inequality, one can have the inequality

Substituting (4.19)–(4.22) into (4.18), we obtain the inequality

By applying Hölder's inequality, (2.4) and [12,(2.7)] to the second term on the right side of (4.23), it can be seen that

It follows from Hölder's inequality that

Substituting (4.25) into (4.24) and using the Cauchy inequality, we obtain

where $\mu_{2}$ is a positive constant to be determined later. Substituting (4.26) into (4.23), one can obtain the inequality

where

With $\mu _2 > 0$ such that $K_{4} = 0$, the above inequality becomes

We now consider the following two cases that $p = \sigma +1$ and $p < \sigma + 1$:

Case 1. If $p = \sigma + 1,$ then (4.27) can be written as

where

for $\mu_1 > 0$ small enough. Integrating (4.28) from 0 to $t$, we get the inequality

which implies that

since $\mathop {\lim }\limits_{t \to t^* } \psi \left(t \right) = + \infty$.

Case 2. If $p < \sigma + 1$, we use Young's inequality to obtain

for all $\mu _3 > 0$. Choosing $\mu_3 > 0$ such that $\frac{{K_2 \sigma \mu _3 }}{{2\sigma - p + 1}} - K_5 \left| \Omega \right|^{\frac{{ - p + 1}}{{2\sigma }}} = 0$, one can have the inequality

where

By a similar argument as in Case 1, we obtain

Remark 4.1. If $p > \sigma+1$, then $p + 1 > 2\left({1 + \frac{\sigma }{2}} \right)$, and hence, the nonnegative classical solution $u(x, t)$ of problem (1.1)–(1.3) exists globally by Theorem 2.1.

Remark 4.2. The derivation of (4.24)–(4.27) in the proof of Theorem 4.2 can also adopt the embedded idea, and the lower bound for blow-up time can be obtained. Indeed, using Hölder's inequality, we have

and from [29, Corollary 9.14], one can easily see that $W^{1, 2} \left(\Omega \right) \subset L^5 \left(\Omega \right)$, $N = 2$; that is,

where $C$ is a constant depending on $\Omega$.

Substituting (4.30) into (4.29) and using the inequality $\left({a + b} \right)^p \le a^p + b^p, \left({a, b \ge 0, \; 0 < p \le 1} \right)$ and Young's inequality, one can have

where $\mu_{4}$ is a positive constant to be determined later. Substituting (4.31) into (4.23), we obtain the inequality

where

Choosing appropriate $\mu_4 > 0$ such that $L_{4} = 0$, the above inequality becomes

Similarly, we now consider the following two cases:

Case 1. If $p = \sigma + 1,$ then (4.32) can be written as

where

for $\mu_1 > 0$ small enough. Integrating (4.33) from 0 to $t^{\ast}$, we get the inequality

Case 2. If $p < \sigma + 1$, we use Young's inequality to get

for all $\mu _5 > 0$. Choosing $\mu_5 > 0$ such that $\frac{{5L_2 \sigma \mu _5 }}{{6\sigma - p + 1}} - L_5 \left| \Omega \right|^{\frac{{ - p + 1}}{{2\sigma }}} = 0$, one can have the inequality

where

By a similar argument as in Case 1, we obtain

Remark 4.3. In fact, the results of Theorems 4.1 and 4.2 can be generalized to the following more general divergence form parabolic equations with nonlinear boundary flux:

where $\left({a_{ij} \left(x \right)} \right)_{N \times N}$ is a positive definite matrix; that is, there exists a $\theta > 0$ such that

for all $\eta \in R^N$.

5.   Conclusions

In this paper, by using the modified differential inequality and comparison principle, we study the blow-up phenomena for a reaction-diffusion equation with gradient absorption terms under nonlinear boundary flux. Our results cover the relevant blow-up and life span results of gradient model in existing literature. Meanwhile, its analytical method can be used in other gradient models.

Acknowledgements

The work of Fang was supported by the Fundamental Research Funds for the Central Universities (No.201964008) and the work of Yi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07041879). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.