Abstract
We investigate the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions:
where p and \(\gamma _{1}\) are some nonnegative constants, m, l, \(\gamma _{2}\), and r are some positive constants, \(D\subset \mathbb{R}^{N}\) (\(N\geq 2\)) is a bounded convex region with smooth boundary ∂D. By making use of differential inequality technique and the embedding theorems in Sobolev spaces and constructing some auxiliary functions, we obtain a criterion to guarantee the global existence of the solution and a criterion to ensure that the solution blows up in finite time. Furthermore, an upper bound and a lower bound for the blow-up time are obtained. Finally, some examples are given to illustrate the results in this paper.
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1 Introduction
There has been a vast amount of literature to discuss the global solutions, the blow-up solutions, and the bounds for the blow-up time for nonlinear parabolic equations. We refer the readers to [1–16]. The global and blow-up solutions for the parabolic problems with nonlinear boundary conditions have been studied in [1–7]. Recently, some authors have already considered the blow-up phenomena of the parabolic equations with nonlocal source. We recommend the literature [8–10] and the references therein. In the present work, we study the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions:
where p and \(\gamma _{1}\) are some nonnegative constants, m, l, \(\gamma _{2}\), and r are some positive constants, \(D\subset \mathbb{R}^{N}\) (\(N\geq 2\)) is a bounded convex region, the boundary ∂D is smooth, \(t^{*}\) is the blow-up time when blow-up occurs, or \(t^{*}=+\infty \), and \(\nu =(\nu _{1},\nu _{2}, \ldots ,\nu _{N})\) is the unit outward normal vector on ∂D. Moreover, \((a^{ij}(x))_{N\times N}\) is a differentiable positive definite matrix; that is, for all \(z=(z_{1},z_{2},\ldots ,z_{N}) \in \mathbb{R}^{N}\), there exists a constant \(\theta >0\) such that
Set \(\mathbb{R}_{+}=(0,+\infty )\). We assume that g is a \(C^{2}(\overline{ \mathbb{R}_{+}})\) function which satisfies \(g'(s)>0\) for all \(s \in \overline{\mathbb{R}_{+}}\), h is a nonnegative \(C^{1}(\overline{ \mathbb{R}_{+}})\) function, and \(u_{0}(x)\) is a nonnegative \(C^{1}(\overline{D})\) function which satisfies the compatibility condition. Our study is motivated by the following three papers. Li et al. [1], Baghaei et al. [11], and Zhang et al. [2] studied the following parabolic equations with local source and nonlinear boundary conditions:
where D is a bounded convex region in \(\mathbb{R}^{N}\) (\(N\geq 2\)), and the boundary ∂D is smooth. When the function \(k(t)\equiv -1\), Li et al. [1] obtained the conditions to ensure that the global solution exists and the solution blows up in finite time \(t^{*}\). Moreover, they derived upper bounds of the blow-up time in \(D\subset \mathbb{R}^{N}\) (\(N\geq 2\)). Using the three-dimensional Sobolev type inequality that Payne and Schaefer had proven in [7], they got lower bounds in \(D\subset \mathbb{R}^{3}\). When \(k(t)\equiv -1\), in [11], Baghaei et al. used the embedding theorems in Sobolev spaces to get a lower bound of the blow-up time in \(D\subset \mathbb{R}^{N}\) (\(N\geq 3\)). Recently, Zhang et al. [2] considered problem (1.3) when \(k(t)>0\). Their main contribution was to extend the three-dimensional Sobolev type inequality obtained by Payne and Schaefer in [7] to the higher dimensional one. They used this Sobolev inequality in multidimensional space to derive a lower bound of the blow-up time in \(D\subset \mathbb{R}^{N}\) (\(N\geq 3\)). In addition, they obtained the global existence of the solution and an upper bound of the blow-up time when blow-up occurs in \(D\subset \mathbb{R}^{N}\) (\(N\geq 2\)).
In the present paper, we study problem (1.1). In the process of getting lower bounds of the blow-up time, the key of our work is to deal with the nonlocal source. We discuss the blow-up problems of (1.1) by constructing some suitable auxiliary functions and making use of the embedding theorems in Sobolev spaces and differential inequality technique. The present work is organized as follows. In Sect. 2, we establish some conditions on the data g and h to obtain that the solution \(u(x,t)\) exists globally in \(D\subset \mathbb{R}^{N}\) (\(N \geq 2\)). In Sect. 3, we develop conditions on the data of (1.1) to guarantee the blow-up of the solution and derive an upper bound of blow-up time in \(D\subset \mathbb{R}^{N}\) (\(N\geq 2\)). When blow-up does occur, we derive a lower bound of \(t^{*}\) for \(D\subset \mathbb{R} ^{N}\) (\(N\geq 3\)) in Sect. 4 and a lower bound of \(t^{*}\) for \(D\subset \mathbb{R}^{2}\) in Sect. 5. In Sect. 6, some examples are presented to illustrate the results in this paper.
2 The global solution
In this section, we establish some conditions on the data of (1.1) to ensure that the global solution exists. We define the auxiliary functions
Now we state our main results as follows.
Theorem 2.1
Letube a nonnegative classical solution of problem (1.1). Suppose that the functionsgandhsatisfy
whereζ, γ, andqare some positive constants. Assume
Then\(u(x,t)\)exists for all\(t>0\)in the measure\(\varPhi (t)\).
Proof
It follows from (2.2) that
that is,
Using (1.2), (2.2), and the divergence theorem, we have
By the divergence theorem, we get the inequality (see [5])
where
Applying the Hölder inequality and the Young inequality to the right-hand side of (2.6) yields
and
with
Hence, substituting (2.7) and (2.8) into (2.6), we get
Inserting (2.10) into (2.5) and using (2.9), we obtain
Since (2.3) implies the fact that \(0<\frac{l}{l+2q-2}<1\), \(0< \frac{l+m-1}{l+2q-2}<1\), and \(0<\frac{l+2q-2}{l+s-1}<1\), it follows from the Hölder inequality that
and
where \(|D|\) is the volume of D. By virtue of (2.14), we have
We insert (2.12)–(2.13) and (2.15) into (2.11) to derive
It follows from (2.12) and (2.4) that
By (2.3), we know \(\frac{2(1-q)}{l+2q-2}<0\), \(\frac{m-2q+1+lp}{l+2q-2}<0\), and \(\frac{r-2q+1}{l+2q-2}>0\). Substituting (2.17) into (2.16), we deduce
where \(C_{1}=l\gamma (\frac{N^{2}}{2\rho _{0}^{2}}+\frac{(l+q-1)^{2}d ^{2}}{2\varepsilon \rho _{0}^{2}} )\), \(C_{2}=\frac{l\gamma }{2} \zeta ^{\frac{2(q-1)}{l}}|D|^{\frac{2(q-1)}{l}}\), \(C_{3}=l\gamma _{1} \zeta ^{\frac{2q-m-1}{l}-p}|D|^{\frac{2q-m-1}{l}}\), and \(C_{4}=l \gamma _{2}\zeta ^{\frac{2q-r-1}{l}}|D|^{\frac{2q-r-1}{l}}\).
Thus, (2.18) implies that \(u(t)\) cannot blow up for all time \(t>0\) in the measure \(\varPhi (t)\). In fact, if \(u(x,t)\) blows up at finite time \(t^{*}\) in the measure \(\varPhi (t)\), we have
It follows from (2.3) that \(\frac{2-2q}{l}<0\), \({p+\frac{m-2q+1}{l}}<0\), and \(\frac{r-2q+1}{l}>0\). From (2.18), we obtain that \(\varPhi '(t)<0\) in some interval \((t_{0}, t^{*})\). So, for any \(t\in [t_{0}, t^{*})\), we get \(\varPhi (t)\leq \varPhi (t_{0})\). We take the limit as \(t\rightarrow {t^{*-}}\) to get
This is a contradiction. □
3 Upper bounds of the blow-up time \(t^{*}\)
In this section, we set up some conditions on g and k to guarantee that the solution of (1.1) blows up in finite time. An upper bound of the blow-up time \(t^{*}\) is obtained in \(D\subset \mathbb{R}^{N}\) (\(N \geq 2\)). The auxiliary functions of this section are defined as follows:
Our main result is the following Theorem 3.1.
Theorem 3.1
Let\(u(x,t)\)be a nonnegative classical solution of problem (1.1). Suppose that the functiongsatisfies
Assume
with
Then the solution\(u(x,t)\)must blow up in the measure\(\varPsi (t)\)in finite time\(t^{*}\)and
Proof
Making use of the divergence theorem, we get
By the Hölder inequality and (3.3), we can compute
that is,
Substituting (3.6) into (3.5) and applying the Young inequality and (3.2)–(3.4), we deduce
We know that (3.3) implies
In fact, if (3.8) does not hold, we let
From (3.9), we have
It follows from (3.7) and (3.10) that \(\varPsi '(t)>0\), \(0< t< t'\), and \(\varPsi (t')>\varPsi (0)\). By (3.3), we derive
which contradicts with (3.9).
Integrating (3.7) over \([0,t]\), we obtain
which implies that the solution u must blow up at some finite time \(t^{*}\) in the measure \(\varPsi (t)\). In fact, if u does not blow up at \(t^{*}\) in the measure \(\varPsi (t)\), we get
Therefore,
Letting \(t\rightarrow +\infty \), we have
which contradicts with
Hence, u blows up at \(t^{*}\) in the measure \(\varPsi (t)\). We pass to the limit as \(t\rightarrow t^{*-}\) in (3.11) to derive
□
4 Lower bounds of the blow-up time \(t^{*}\) in \(D\subset \mathbb{R}^{N}\) (\(N\geq 3\))
In this section, we impose restriction \(D\subset \mathbb{R}^{N}\)\((N\geq 3)\). Assume that the functions h and g satisfy
where σ, q, and ξ are some positive constants and
It follows from [17, Corollary 9.14] that
from which we have the following Sobolev inequality:
where \(C=C(N,D)\) is a Sobolev embedding constant depending on N (\(N\geq 3\)) and D. In order to derive a lower bound of the blow-up time \(t^{*}\), we use inequality (4.3) and
We define auxiliary functions as follows:
Now we present our main results in Theorem 4.1.
Theorem 4.1
Let\(u({x},t)\)be a nonnegative classical solution of (1.1) in\(D\subset \mathbb{R}^{N}\) (\(N\geq 3\)). Suppose that (4.1)–(4.2) hold and
If\(u({x},t)\)becomes unbounded at some finite time\(t^{*}\)in the measure\(A(t)\), then we conclude\(t^{*}\)is bounded from below by
where
andθis defined in (1.2).
Proof
We use (1.2), (4.1), and the divergence theorem to obtain
Repeating the calculation process of (2.6)–(2.8), we have
where \(\rho _{0}\) and d are given in (4.11) and \(\varepsilon _{1}=\frac{ \theta (l-1)}{\sigma }\). Substituting (4.13) into (4.12), we derive
First we take into account the second term of (4.14). Assumptions (4.2) and (4.6) imply \(0<\frac{(N-2)(q-1)}{l}<1\) and \(0<\frac{N(q-1)}{l}<1\). Now we apply the Hölder inequality, the Young inequality, and inequalities (4.3)–(4.4) to deduce
where \(\varepsilon _{2}=\frac{\theta (l-1)}{I_{1}N(q-1)}\), \(I_{1}=l \sigma C^{\frac{2N(q-1)}{l}} (\frac{N^{2}}{2\rho _{0}^{2}}+\frac{(l+q-1)^{2}d ^{2}}{2\varepsilon _{1}\rho _{0}^{2}} ) \).
Then we deal with the fourth term of (4.14). Due to (4.6), we get \(0<\frac{(m-1)(N-2)}{2l}<1\), \(0<\frac{N(m-1)}{2l}<1\). Using the Hölder inequality, the Young inequality, and (4.3)–(4.4), we derive
where \(\varepsilon _{3}=\frac{\gamma _{2}(l+r-1)}{l\gamma _{1}}C^{ \frac{N(1-m)}{l}}|D|^{\frac{1-r}{l}}\), \(\varepsilon _{4}=\frac{2\theta (l-1)}{l\gamma _{1}N(m-1)}C^{\frac{N(1-m)}{l}}\).
Finally we compute the last term of (4.14). Applying the Hölder inequality, we deduce
that is,
Substituting (4.15)–(4.17) into (4.14), we have
By (4.1), we obtain
which implies
Now, inserting (4.19) into (4.18), we derive
where \(J_{1}\), \(J_{2}\), \(J_{3}\), \(J_{4}\), and \(J_{5}\) are defined in (4.7)–(4.10). Since \(\lim_{t\to t^{*-}}A(t)=+\infty \), we integrate (4.20) from 0 to \(t^{*}\) to get
□
5 Lower bounds of the blow-up time \(t^{*}\) in \(D\subset \mathbb{R}^{2}\)
In this section, we search for a lower bound of \(t^{*}\) in \(D\subset \mathbb{R}^{2}\). Suppose that the functions h and g satisfy
where σ̃, q̃, and ξ̃ are some positive constants and
By [17, Corollary 9.14], we have
which implies the following Sobolev inequality:
where C̃ is a Sobolev embedding constant depending on D. We need to use inequality (5.3) to get a lower bound of \(t^{*}\). Now, we define auxiliary functions as follows:
The main result of this section is the following Theorem 5.1.
Theorem 5.1
Let\(u({x},t)\)be a nonnegative classical solution of (1.1) in\(D\subset \mathbb{R}^{2}\). Assume that (5.1)–(5.2) hold and
In addition, we assume that\(u({x},t)\)blows up at some finite time\(t^{*}\)in the measure\(\tilde{A}(t)\). Then\(t^{*}\)is bounded below by
where
andθis defined in (1.2).
Proof
Repeating the calculation process of (4.12)–(4.14) and noting \(N=2\) in this section, we get
where \(\tilde{\rho }_{0}\) and d̃ are defined in (5.10) and \(\tilde{\varepsilon }_{1}=\frac{\theta (l-1)}{\tilde{\sigma }}\).
First, we focus on the second term of (5.11). By (5.2) and (5.5), we get \(0<\frac{2(\tilde{q}-1)}{l}<1\) and \(0< \frac{4(\tilde{q}-1)}{l}<1\). Now we apply the Hölder inequality, the Young inequality, (4.4), and (5.3) to deduce
where \(\tilde{\varepsilon }_{2}=\frac{\theta (l-1)}{4\tilde{I}_{1}( \tilde{q}-1)}\), \(\tilde{I}_{1}=l\tilde{\sigma }\tilde{C}^{\frac{8( \tilde{q}-1)}{l}} (\frac{2}{\tilde{\rho }_{0}^{2}} +\frac{(l+ \tilde{q}-1)^{2}\tilde{d}^{2}}{2\tilde{\varepsilon }_{1}\tilde{\rho } _{0}^{2}} ) \).
Next, we deal with the fourth term of (5.11). Due to (5.5), we get \(0<\frac{m-1}{l}<1\), \(0<\frac{2(m-1)}{l}<1\). Using the Hölder inequality, the Young inequality, (4.4), and (5.3), we derive
where \(\tilde{\varepsilon }_{3}=\frac{\gamma _{2}(l+r-1)}{l\gamma _{1}} \tilde{C}^{\frac{4(1-m)}{l}}|D|^{\frac{1-r}{l}}\), \(\tilde{\varepsilon }_{4}=\frac{\theta (l-1)}{2l\gamma _{1}(m-1)} \tilde{C}^{\frac{4(1-m)}{l}}\).
Finally, we consider the last term of (5.11). Repeating the calculation process of (4.17), we have
Substituting (5.12)–(5.14) into (5.11), we deduce
We repeat the calculation process of (4.19) to obtain
Now, inserting (5.16) into (5.15), we derive
where \(\tilde{J}_{1}\), \(\tilde{J}_{2}\), \(\tilde{J}_{3}\), \(\tilde{J} _{4}\), and \(\tilde{J}_{5}\) are defined in (5.6)–(5.9). Since \(\lim_{t\to t^{*-}}\tilde{A}(t)=+\infty \), we integrate (5.17) from 0 to \(t^{*}\) to get
□
6 Applications
In what follows, three examples are given to demonstrate the results of Theorems 2.1–5.1 obtained in this paper.
Example 6.1
Let \(u(x,t)\) be a nonnegative classical solution of the following problem:
where \(D= \{ x=({x}_{1}, {x}_{2}, {x}_{3}) \mid |x|^{2}=\sum_{i=1}^{3}{x}_{i}^{2} <\frac{1}{4} \} \) is a ball of \(\mathbb{R}^{3}\). Here,
Now, \(N=3\), \(\gamma _{1}=1\), \(\gamma _{2}=1\), \(m=\frac{1}{2}\), \(l=3\), \(p= \frac{1}{3}\), and \(r=\frac{5}{2}\). Choosing \(\zeta =2\), \(\gamma =\frac{2\sqrt{5}}{5}\), and \(q=\frac{3}{2}\), we can verify that (2.2)–(2.3) are satisfied. Hence, Theorem 2.1 implies that u exists for all time in \(\varPhi (t)\) with
Example 6.2
Let \(u(x,t)\) be a nonnegative classical solution of the following problem:
where \(D= \{ x=({x}_{1}, {x}_{2}, {x}_{3}) \vert |x|^{2}=\sum_{i=1}^{3}{x}_{i}^{2} . <\frac{9}{16} \} \) is a ball of \(\mathbb{R}^{3}\). Now we have
Here, \(N=3\), \(\gamma _{1}=\frac{9}{4}\), \(\gamma _{2}=\frac{7}{4}\), \(m= \frac{3}{2}\), \(l=3\), \(p=\frac{1}{10}\), \(r=\frac{7}{5}\), and \(|D|=\frac{9 \pi }{16}\). Now, we compute
It follows from (3.4) that \(\delta =0.7436\), \(M_{1}=0.7552\), and \(M_{2}=1.0969\). It is easy to check that (3.2)–(3.3) hold. By Theorem 3.1, we deduce that u must blow up in the measure \(\varPsi (t)\) at some finite time \(t^{*}\) and
We note that (6.1) gives an upper bound of \(t^{*}\). Next, we use Theorem 4.1 to obtain a lower bound of \(t^{*}\). Now we select \(q=\frac{5}{4}\), \(\sigma =\frac{3\sqrt{5}}{5}\), \(\xi =\frac{1}{2}\), and \(\theta =1\). By (4.11), we derive \(\rho _{0}=d=\frac{3}{4}\). Noting \(N=3\), we can verify that (4.1)–(4.2) and (4.6) hold. Moreover, according to Theorems 2.1 and 3.2 in [18], we obtain the Sobolev embedding constant \(C=7.5931\). Substituting above constants into (4.7)–(4.10), it is easy to get \(J_{1}= 4.0249\), \(J_{2}=287.4\), \(J_{3}=814.14\), \(J_{4}=95979\), and \(J_{5}=68.205\). By (3.6), we obtain
and
Blow-up of u in the measure \(\varPsi (t)\) at \(t^{*}\) means that u blows up at \(t^{*}\). Hence u must also blow up in the measure \(A(t)\) at \(t^{*}\). By Theorem 4.1, we get a lower bound of the blow-up time \(t^{*}\) as follows:
Combining (6.1) and (6.2), we have
Example 6.3
Let \(u(x,t)\) be a nonnegative classical solution of the following problem:
where \(D= \{ x=({x}_{1}, {x}_{2}) \mid |x|^{2}=\sum_{i=1}^{2} {x}_{i}^{2} <\frac{9}{16} \} \) is a circular region of \(\mathbb{R}^{2}\). Now
Here, \(N=2\), \(\gamma _{1}=2\), \(\gamma _{2}=2\), \(m=\frac{3}{2}\), \(l=3\), \(p= \frac{1}{10}\), \(r=\frac{7}{5}\), and \(|D|=\frac{9\pi }{16}\). By (3.1), we have
By (3.4), we obtain \(\delta =0.6506\), \(M_{1}=0.7552\), and \(M_{2}=2.0004\). After some simple calculations, we know that (3.2)–(3.3) hold. From Theorem 3.1, it follows that u blows up in the measure \(\varPsi (t)\) at \(t^{*}\) and
Next, we apply Theorem 5.1 to get a lower bound of \(t^{*}\). Now we choose \(\tilde{q}=\frac{5}{4}\), \(\tilde{\sigma }=\frac{3\sqrt{5}}{5}\), \(\tilde{\xi }=1\), and \(\theta =1\). It follows from (5.10) that \(\tilde{\rho }_{0}=\tilde{d}=\frac{3}{4}\). Noting \(N=2\), it is easy to verify that (5.1)–(5.2) and (5.5) hold. Moreover, using Theorems 2.1 and 3.2 in [18], we derive the Sobolev embedding constant \(\tilde{C}=10.887\). Inserting the above constants into (5.6)–(5.9), we deduce \(\tilde{J}_{1}= 2.0125\), \(\tilde{J}_{2}=140.34\), \(\tilde{J}_{3}=783.76\), \(\tilde{J}_{4}=997648\), and \(\tilde{J}_{5}=90\). By (5.4), we have
and
Since u blows up in the measure \(\varPsi (t)\) at \(t^{*}\), u must blow up in the measure \(\tilde{A}(t)\) at \(t^{*}\). By Theorem 5.1, we get
From (6.3) and (6.4), it follows that
7 Conclusion
In this paper, we derive the global existence and bounds for the blow-up time of nonlinear parabolic problem (1.1) with nonlocal source. To deal with nonlocal source, we must establish some new auxiliary functions different from those in [1, 2] and [11]. Furthermore, to obtain the lower bound of the blow-up time in \(D\subset \mathbb{R}^{N}\) (\(N\geq 3\)) and \(D\subset \mathbb{R} ^{2}\), we need to use the embedding theorems in Sobolev spaces \(W^{1, 2}\hookrightarrow L^{\frac{2N}{N-2}}\), \(N\geq 3 \) and \(W^{1, 2}\hookrightarrow L^{4}\), \(N=2\), respectively. Applying these auxiliary functions, the embedding theorems in Sobolev spaces, and the differential inequality technique, we complete our study with the blow-up and global solutions of problem (1.1).
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Kou, W., Ding, J. Global existence and blow-up analysis for parabolic equations with nonlocal source and nonlinear boundary conditions. Bound Value Probl 2020, 37 (2020). https://doi.org/10.1186/s13661-020-01340-5
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DOI: https://doi.org/10.1186/s13661-020-01340-5