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Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition
Boundary Value Problems volume 2020, Article number: 76 (2020)
Abstract
In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain \((\varOmega\in \mathbb{R}^{n}, n\geq 3)\) with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.
1 Introduction
Liu in paper [1] studied the blow-up phenomena for the solution of the following problems:
under the Robin boundary condition
He obtained a lower bound for the blow-up time of the system when the solution blows up.
In paper [2], the authors also studied equations (1.1) and (1.2) subject to either homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. The lower bounds for the blow-up time under the above two boundary conditions were obtained. Equation (1.1) is used in the study of population dynamics (see [3]). For other systems in porous medium, one could see [4]. There have been a lot of papers in the literature on studying the question of blow-up for the solution of parabolic problems under a homogeneous Dirichlet boundary condition and Neumann boundary condition(one can see [5–12]). Some authors have started to consider the blow-up of these problems under Robin boundary conditions (see [13–17]). In papers [18–21], the authors studied the blow-up phenomena for the heat equation under nonlinear boundary conditions. Some new results about the nonlinear evolution equations may be founded in [22–24]. These papers have mainly focused on the bounded convex domain in \({\mathbb{R}^{3}}\). Recently, there have been some papers starting to study the blow-up problems in \({\mathbb{R}^{n}}\) (\(n\geq 3\)) (see [25–29]). We continue the work of [2] for a more general equation. Until now, the authors have not found any paper dealing with lower bound for the blow-up time of a nonlinear nonlocal porous medium equation under nonlinear boundary condition in \({\mathbb{R}^{n}}\) (\(n\geq 3\)). In this sense, the result obtained in this paper is new and interesting. In this paper, we consider the blow-up phenomena of the solution for the following equation:
with the following boundary initial conditions:
where Ω is a bounded convex domain in \({\mathbb{R}^{n}}\), \(n\geq 3\), with sufficiently smooth boundary, △ is the Laplace operator, ∂Ω is the boundary of Ω, and \(t^{*}\) is the possible blow-up time, \(\frac{\partial u}{\partial \nu }\) is the outward normal derivative of u. We assume \(\frac{k_{1}^{\prime }(t)}{k_{1}(t)}\leq \alpha \) and \(\frac{dh(u)}{du}\geq M>0\).
The function \(g(\xi )\) satisfies
where \(s>\max \{\frac{2n}{2n-1},p+q+1-m\}\).
2 Some useful inequalities
We will use the following useful inequalities later in the proof.
Lemma 2.1
We suppose thatuis a nonnegative function andσ, mare positive constants, then we have the result as follows:
where\(\rho _{0} :=\min_{\partial \varOmega } \vert x\cdot \vec{\nu } \vert \), ν⃗is the outward normal vector of∂Ωand\(d:=\max_{\partial \varOmega } \vert x \vert \).
Proof
Applying the divergence definition, we have
Integrating (2.2), we deduce
Applying the divergence theorem, we obtain
Because Ω is a convex domain, we have \(\rho _{0} :=\min_{\partial \varOmega } \vert x\cdot \vec{\nu } \vert >0\). Then we derive
 □
Lemma 2.2
Supposing that\(u\in W^{1,2}(\varOmega )\)and\(n\geq 3\), we have
where\(C=C(n,\varOmega )\)is a Sobolev embedding constant depending onnand Ω.
Proof
In paper [30], we have \(W^{1,2}(\varOmega )\hookrightarrow L^{\frac{2n}{n-2}(\varOmega )}\), \(n\geq 3\). Then we deduce the Sobolev inequality as follows:
that is,
We can get
 □
Remark 2.1
For any nonnegative function u, the following Hölder inequality holds:
where \({n_{1}}\), \({n_{2}}\), \({x_{1}}\), \({x_{2}}\) are positive constants and \({x_{1}}\), \({x_{2}}\) satisfy \({x_{1}}+{x_{2}}=1\).
Remark 2.2
The fundamental inequality
where \(a,b\geq 0 \) and \(0< l\leq 1\), holds.
3 Lower bound for the blow-up time
In this section it is useful in the sequel to define an auxiliary function of the following form:
We will derive a differential inequality for \(\phi (t)\). From the inequality, we can establish the following theorem.
Theorem 3.1
Let\(u(x,t)\)be the classical nonnegative solution of problem (1.4)–(1.7) in a bounded convex domainΩ (\(\varOmega \in R^{n}\) (\(n\geq 3\))). We assume that\(m+s>p+q+1>2\), \(m>3\), \(p>0\), \(q>0\). Then the quantity\(\phi (t) \)defined in (3.1) satisfies the differential inequality
from which it follows that the blow-up time\(t^{*}\)is bounded below. We have
where\(\varTheta ^{-1} \)is the inverse function ofΘ, and\(a(t)\), \(b(t)\)are defined in (3.21), (3.22) respectively.
Proof
Now we prove Theorem 3.1. For convenience, we set \(\phi (t)=\phi \), \(k_{1}(t)=k_{1}\), \(k_{2}(t)=k_{2}\). First we compute
Integrating by parts, we have
Using the result of Lemma 2.1, we obtain
where \(r_{1}=\frac{\sigma m}{M}\frac{n \vert \varOmega \vert }{\rho _{0}}\), \(r_{2}= \frac{\sigma m}{M}\frac{(\sigma +m-2)d}{\rho _{0}}\), \(r_{3}= \frac{\sigma m(\sigma -1)}{M}\frac{4}{(\sigma +m-1)^{2}}\), \(r_{4}= \frac{\sigma \vert \varOmega \vert }{M} \).
Now we estimate the third term of the right-hand side of (3.4). Using Hölder’s inequality, we have
Then we obtain
where \(r_{5}=(k_{1}^{-\frac{ns}{\sigma }} \vert \varOmega \vert ^{ \frac{\sigma -s}{\sigma }}\frac{2}{\sigma +m-1})^{2}\), \(\varepsilon _{1}\) is a positive constant which will be defined later.
From the above deductions, we get
Combining (3.4) and (3.5), we obtain
Using (2.3), (2.4), and (2.5), we obtain
where
Using Hölder’s and Young’s inequalities, we have
where \(r_{7}=\frac{n-2-x_{1}n}{n-2}({\frac{n-2}{x_{1}n}})^{- \frac{x_{1}n}{n-2-x_{1}n}}r_{6}^{\frac{n-2}{n-2-x_{1}n}}\).
By Hölder’s and Young’s inequalities, we get
where \(x_{10}=\frac{m-1}{m+s-2}\), \(n_{10}=\frac{(\sigma +m+s-2)(m-1)}{m+s-2}\), \(x_{20}= \frac{s-1}{m+s-2}\), \(n_{20}=\frac{(s-1)\sigma }{m+s-2}\).
If we choose \(\varepsilon _{2}\) such that \(x_{10}\varepsilon _{2}=\frac{1}{2}\), we have
Combining (3.7)–(3.9), we obtain
Then we can deduce
where \(\varepsilon _{3}\) is a positive constant which will be defined later.
If we choose \(x_{11}=\frac{m-3}{m+s-2}\), \(n_{11}=\frac{(\sigma +m+s-2)(m-3)}{m+s-2}\), \(x_{21}= \frac{s+1}{m+s-2}\), \(n_{21}=\frac{(s+1)\sigma }{m+s-2}\), using (2.4), we get
Then we obtain
Combining (3.10) and (3.12), we have
where \(\varepsilon _{4}\) is a positive constant which will be defined later.
Similarly, if we choose \(x_{12}=\frac{p+q-1}{m+s-2}\), \(n_{12}= \frac{(\sigma +m+s-2)(p+q-1)}{m+s-2}\), \(x_{22}=\frac{m+s-(p+q+1)}{m+s-2}\), \(n_{22}= \frac{\sigma [m+s-(p+q+1)]}{m+s-2}\), using (2.4), we get
Combining (3.10) and (3.14), we obtain
where \(\varepsilon _{5}\) is a positive constant which will be defined later.
Combining (3.6), (3.11), (3.13), and (3.15), we have
If we choose suitable \(\varepsilon _{1}\), \(\varepsilon _{3}\), \(\varepsilon _{4}\), \(\varepsilon _{5}\) such that
Substituting (3.17) into (3.16), we derive
Using Hölder’s and Young’s inequalities, we have
Applying (3.19) to \(\phi ^{1+\frac{2x_{1}}{n-2-x_{1}n}}\), \(\phi ^{1+\frac{2s}{\sigma }}\), \(\phi ^{1+(\frac{2s}{\sigma }+\frac{2x_{1}}{n-2-x_{1}n})}\), \(\phi ^{1+ \frac{2s(n-2)+2x_{1}\sigma }{\sigma (n-2-x_{1}n)}}\) in (3.18), respectively, we obtain
where
and
Multiplying both sides of (3.20) by \(\phi ^{-5}(t)\), we obtain
That is,
Setting \(H(t)=\int _{0}^{t}a(\tau )\,d\tau \), (3.24) can be rewritten as
Integrating (3.25) from 0 to t, we have
That is to say,
where \(\varTheta (t)=\int _{0}^{t}b(\tau )e^{4H(\tau )}\,d\tau \).
Taking the limit to (3.27) as \(t\rightarrow t^{*}\), we get
From the definition of \(\varTheta (t)\), we have \(\frac{d\varTheta (t)}{dt}=b(t)e^{4H(t)}>0\). We get \(\varTheta (t)\) is a strictly increasing function. So we can get
from which we complete the proof of Theorem 3.1. □
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The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions.
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The work was supported by the National Natural Science Foundation of China (Grant ♯ 61907010), Natural Science in Higher Education of Guangdong, China (Grant ♯ 2018KZDXM048), the General Project of Science Research of Guangzhou (Grant ♯ 201707010126), and the Science Foundation of Huashang College Guangdong University of Finance & Economics (Grant ♯ 2019HSDS26).
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Ouyang, B., Lin, Y., Liu, Y. et al. Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition. Bound Value Probl 2020, 76 (2020). https://doi.org/10.1186/s13661-020-01372-x
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DOI: https://doi.org/10.1186/s13661-020-01372-x