A single quenching point for a fractional heat equation based on the Riemann-Liouville fractional derivative with a nonlinear concentrate source
- Wannika Sawangtong^{1, 3} and
- Panumart Sawangtong^{2}Email authorView ORCID ID profile
Received: 28 December 2016
Accepted: 19 June 2017
Published: 29 June 2017
Abstract
This paper aims to study the quenching problem in a fractional heat equation with the Riemann-Liouville fractional derivative. The existence and uniqueness of a solution for the problem are obtained by transforming the problem to an equivalent integral equation. The condition for the quenching occurrence in a finite time is given. Furthermore, the quenching point set is shown.
Keywords
1 Introduction
The study of quenching problems for the initial-boundary value problem of parabolic type began in 1975 with a research paper of Kawarada [1], and since then, this topic has attracted much attention. Recently, many papers have studied quenching problems in PDEs (see [2–4] and [5] etc.).
Definition 1.1
A solution v of problem (1) is said to quench at the point \(x_{0}\) if there exists a positive time \(t_{q}\) such that the function \(v(x_{0},t) \rightarrow1^{-}\) and \(v_{t}(x_{0},t) \rightarrow\infty\) as \(t\rightarrow{t_{q}}\). If \(t_{q}\) is finite, then v is said to quench in a finite time. On the other hand, if \(t_{q}=\infty\), then v is said to quench in an infinite time. Furthermore, the set of all quenching points is called the quenching set.
For the reader’s convenience, we introduce some facts about fractional calculus. For details, one can see [6].
Definition 1.2
Definition 1.3
Definition 1.4
Lemma 1.1
- (1)
\(I_{t}^{\alpha} ( {}_{\mathrm{RL}}D_{t}^{\alpha }f(t) ) =f(t)\),
- (2)
\({}_{\mathrm{C}}D_{t}^{\alpha}f(t)=I_{t}^{1-\alpha}\frac{d}{dt}f(t)\).
In Section 2, we transform problem (1) into the equivalent Volterra integral equation. A unique nonnegative local solution of problem (1) is shown in Section 3. In Section 4, we prove that a solution of problem (1) quenches in a finite time. In the last section, we find the quenching set of problem (1).
2 An integral equation
To analyze equation (5), it is essential to know the properties of \(G_{\alpha}\). Various properties of \(G_{\alpha}\) can be derived from (2). By [9], it is seen that \(G_{\alpha}\) is positive, \(G_{\alpha}\) is continuously differentiable for \(t>0\), \(G_{\alpha}\) is decreasing with respect to t, \(\int_{0}^{t}G_{\alpha}(x,t-\tau;x_{0},0)\,d\tau\) is positive and \(\frac{d}{dt}\int_{0}^{t}G_{\alpha}(x,t-\tau ;x_{0},0)\,d\tau\) is positive.
3 Existence and uniqueness
In this section, we use the Banach fixed point theorem to show that the integral equation (5) has a unique continuous solution on the time interval \([0,t_{1}]\) for some \(t_{1}>0\).
Theorem 3.1
There exists \(t_{1}>0\) such that the integral equation (5) has a unique nonnegative continuous solution v on the interval \([0,t_{1}]\) for any \(x \in[0,l]\).
Proof
- (1)
\(F:S_{R}\rightarrow S_{R}\) and
- (2)
F is a contraction mapping.
Therefore, the Banach fixed point theorem implies that F has a unique fixed point on \(S_{R}\). We can conclude that the integral equation (5) has a unique continuous solution v for any \((x,t)\in[0,l]\times[0,h]\). It follows by the positivity of \(G_{\alpha}\) and f that the solution v is positive. Hence, the proof of this theorem is completed. □
4 Finite time quenching
The following theorem shows that the solution v of the integral equation (5) quenches at the point \(x_{0}\).
Theorem 4.1
Let \([0,t_{q})\) be the maximum time interval such that the integral equation (5) has a continuously differentiable solution \(v(x,t)\) for any \((x,t)\in[0,l]\times[0,t_{q})\). If \(v(x_{0},t)\) converges to 1^{−} as t converges to \(t_{q}\), then \(v_{t}(x_{0},t)\) goes to ∞ as \(t\rightarrow{t_{q}}\).
Proof
We next give the condition that guarantees the occurrence of quenching in a finite time.
Theorem 4.2
There exists a finite time \(t^{*}\) such that the integral equation (5) has no continuous nonnegative solutions v with \(v(x,t)<1\) for every \(x \in[0,l]\) and for \(t>t^{*}\).
Proof
5 Single quenching point
The next theorem shows that the point \(x_{0}\) is the single quenching point.
Theorem 5.1
Assume that the solution \(v(x,t)\) of problem (1) attains, for \(t \in(0,t_{q})\), its maximum at \((x_{0},t)\). If \(v(x_{0},t)\) converges to 1^{−} as t approaches to \(t_{q}\), then v quenches at \(x_{0}\). Moreover, \(x_{0}\) is the single quenching point.
Proof
Declarations
Acknowledgements
This research was funded by The Thailand Research Fund (MRG5980119).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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