Abstract
This paper considers the following semilinear pseudo-parabolic equation with a nonlocal source:
and it explores the characters of blow-up time for solutions, obtaining a lower bound as well as an upper bound for the blow-up time under different conditions, respectively. Also, we investigate a nonblow-up criterion and compute an exact exponential decay.
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1 Introduction
In this paper, we deal with the blow-up problem for the following equation:
where \(\Omega\subset\mathbb{R}^{n}\) (\(n\geq3\)) is a bounded domain with smooth boundary, \(u_{0}(x)\in H_{0}^{1}(\Omega)\), \(T\in(0,\infty]\), and p satisfies
\(k(x,y)\) is an integrable, real valued function satisfying
The blow-up phenomena for problems similar to (1.1) have been extensively researched (see [1–7]). For instance,
and bounds for the blow-up time have been explored [8, 9].
Recently, authors have begun to consider
When \(f(u)=u^{p}\) in (1.4), many results have been studied in [5, 6, 10–12] and the references therein, among which Xu [6] proved finite time blow up for solutions through the so-called potential well method, first introduced in [13]. The method has played an important role in dealing with parabolic and hyperbolic problems since it was discovered. Later on, confirmed by the same conditions as [6] that guarantee the occurrence of blow up, Luo [5] established a lower bound for the blow-up time. Furthermore, when \(f(u)=u^{p}(x,t)\int_{\Omega}k(x,y)u^{p+1}(y,t)\,dy\) with \(0< p\leq \frac{2}{n-2}\) in (1.4), which is a new problem and has not been considered, by means of the potential well method, Yang [14] not only obtained the global existence and asymptotic behavior of solutions with deducing exponential decay, but also got the existence of solutions that blow up in finite time in \(H^{1}_{0}(\Omega)\)-norm with energy \(J(u_{0})\leq d\).
In the last several decades, an increasing number of researchers focused on exploring the upper bounds for the blow-up time. However, the authors have trouble getting lower bounds for the blow-up time, and therefore they received little attention. In this paper, we use the means of a differential inequality technique and present some results on the bounds for the blow-up time to problem (1.1) since little attention is paid to the bounds before we study. Our paper is organized as follows. In Section 2, we come up with the main results: First of all, a blow-up criterion and an upper bound for the blow-up time are determined. Second, we investigate the nonblow-up case. Finally, a lower bound for the blow-up time is obtained.
Before stating our principal theorem, we note that the Fréchet derivative \(f_{u}\) of the nonlinear function \(f(u)=u^{p}(x,t)\int_{\Omega}k(x,y)u^{p+1}(y,t)\,dy\) is
Clearly \(f_{u}\) is symmetric and bounded, so the potential F exists and is given by
Differentiating (1.5) with respect to t, it follows that
where we have used the symmetry of \(k(x,y)\).
To obtain the main results, we introduce the functionals
and
We set out to establish local existence and uniqueness for (1.1).
Theorem 1.1
Assume (1.2) holds. Then for any \(u_{0}\in H_{0}^{1}(\Omega)\), there exists a \(T>0\) for which (1.1) has a unique local solution \(u(t)\in L^{\infty}([0,T);H_{0}^{1}(\Omega))\) with \(u_{t}(t)\in L^{2}([0,T);H_{0}^{1}(\Omega))\), satisfying
for all \(v \in H_{0}^{1}(\Omega)\).
Proof
Choose \(\{\omega_{j}(x)\}\) as the basis functions of \(H^{1}_{0}(\Omega)\). Construct the approximate solutions \(u_{m}(x,t)\) of the problem (1.1)
which satisfy
Multiplying (1.10) by \(g^{\prime}_{s}(t)\), summing over s, and integrating with respect to t from 0 to t, we obtain
In fact
where we have used condition (1.3) and \(u_{0}\in H_{0}^{1}(\Omega)\). Thus for sufficiently large m, we get (1.12). Hence, by (1.12) and
we obtain
We estimate the last term in the right-hand side using the Hölder and Sobolev inequalities (recall \(u_{m}\in C^{1}([0,T],H_{0}^{1}(\Omega))\)):
here \(\kappa= (\int_{\Omega}\int_{\Omega}k^{2}(x,y)\,dx\,dy )^{\frac {1}{2}}<\infty\), and \(C_{\ast}\) is the embedding constant for \(H_{0}^{1}(\Omega)\hookrightarrow L^{2p+2}(\Omega)\). From (1.13) and (1.14) we obtain
On the other hand, by using the Hölder and Sobolev inequalities, here \(q=\frac{2p+2}{2p+1}\), we have
By (1.15) and (1.16), for sufficiently large m, we get
Therefore, by these uniform estimates from (1.17)-(1.19), there exist u and a subsequence still denoted by \(\{u_{m}\}\) such that, as \(m\rightarrow\infty\),
Thus in (1.10), for s fixed, letting \(m\rightarrow\infty\), one has
and
Moreover, (1.11) gives \(u(x,0)=u_{0}(x)\) in \(H^{1}_{0}(\Omega)\). The existence of u solving (1.1) and satisfying (1.9) is so proved. □
2 Main results
I. Upper bound for blow-up time
Here, a condition to ensure blow-up at some finite time as well as an upper bound for the blow-up time is considered.
\(J(u)\) is defined in (1.7) and we introduce
A straightforward computation shows that
Combining (2.1) with (2.2), we obtain
where
for any \(p>0\). We obtain the following by multiplying \(u_{t}(x,t)\) on both sides of (1.1) and integrating by parts:
From (2.5), we have
where we have used the symmetry property of \(k(x,y)\) in the second step in (2.6).
In addition, it is indicated from (2.4) and (2.6) that \(\beta(t)\) is a nondecreasing function of t. So if we assume \(J(u_{0})<0\), then \(\beta(t)> 0\) for all \(t\geq0\). And with (2.3), we have
which becomes
By (2.3), we have
Then
It is obvious that (2.10) implies u blows up at some finite time T. T is given by
The above result is presented in the following theorem.
Theorem 2.1
For any \(p>0\), \(u_{0}\in H_{0}^{1}(\Omega)\cap L^{2p+2}(\Omega)\), \(J(u_{0})< 0\), then the solution \(u(x,t)\) of (1.1) blows up in finite time, and T is bounded by (2.11).
II. Nonblow-up case
In this section, we not only give a criterion which guarantees nonblow up, but also we deduce the exponential decay of \(u(\cdot,t)\) in \(H^{1}_{0}(\Omega)\)-norm when \(u_{0}(x)\) satisfies some conditions.
\(\alpha(t)\) is defined in (2.1).
By a similar calculation to (1.14), we have
By a straightforward computation, we have
The Poincaré inequality gives \(\|\nabla u\|^{2}_{2}\geq\lambda_{1}\| u\|^{2}_{2}\) where \(\lambda_{1}\) is the first eigenvalue of the problem
Then
From (2.13) and (2.15), we know that
Let
If \(\alpha(0)< M\), then we show that \(\alpha(t)< M\).
In fact, if \(\alpha(t)\geq M\), we know that there exists \(t_{0}\) such that \(\alpha(t_{0})=M\) and \(\alpha(t)< M\) for \(0\leq t< t_{0}\). From (2.16), we have \(\alpha^{\prime}(t)<0\) for \(0\leq t< t_{0}\), from which we deduce \(\alpha(0)\geq\alpha(t_{0})=M\). It leads to a contradiction. This shows \(\alpha^{\prime}(t)<0\), and blow up cannot occur in finite time.
Thus (2.16) becomes
then
through variable substitution \(\eta=\gamma^{p}\), (2.18) turns to
and
We state this result in the following theorem.
Theorem 2.2
If \(0< p\leq\frac{2}{n-2}\), \(\|u_{0}\|^{2}_{H_{0}^{1}}< M\), then the solution of (1.1) cannot blow up in finite time in \(H^{1}_{0}(\Omega)\)-norm, and we have the exponential decay estimate
III. Lower bounds for blow-up time
This section is devoted to establishing a lower bound for T if the solution \(u(x,t)\) blows up at \(t=T\) under some conditions.
Let
For the problem (1.1) Yang [14] has proved Theorem 2.3.
Theorem 2.3
(Blow up for \(J(u_{0})< d\)) ([14])
Let p satisfy (1.2), \(u_{0}\in H_{0}^{1}(\Omega)\). Assume that \(J(u_{0})< d\), \(I(u_{0})<0\). Then the weak solution \(u(t)\) of problem (1.1) blows up in finite time.
In view of Theorem 2.3, we see that when \(J(u_{0})< d\) and \(I(u_{0})<0\), the solution \(u(x,t)\) of problem (1.1) blows up in finite time T. To continue our study and estimate the lower bound for the blow-up time T, we assume \(J(u_{0})< d\), \(I(u_{0})<0\) in this section.
We introduce \(\alpha(t)\) in (2.1) and, calculating as (2.13), we have
Here, we claim \(\alpha(t)>0\). In fact, if there exists \(t_{0}\in[0,T)\) such that \(\alpha(t_{0})=0\), then \(\alpha(T)=0\). It leads to a contradiction with the fact that \(u(x,t)\) blows up at T in \(H^{1}_{0}(\Omega)\)-norm. Thus
and
Letting \(t\rightarrow T\) in (2.23), one obtains
We summarize this result in the following theorem.
Theorem 2.4
If \(0< p\leq\frac{2}{n-2}\), \(u_{0}\in H_{0}^{1}(\Omega)\), \(J(u_{0})< d\), \(I(u_{0})<0\), then the solution \(u(x,t)\) of (1.1) blows up in finite time T in \(H^{1}_{0}(\Omega)\)-norm, and T is bounded below by (2.24).
Remark 2.1
We mention that the lower bound \(\frac{\|u_{0}\| ^{-2p}_{H_{0}^{1}}}{2p\kappa C^{2p+2}_{\ast}}\) is smaller than the upper bound \(\frac{\|u_{0}\|^{2}_{H_{0}^{1}}}{-4p(p+1)J(u_{0})}\) under the conditions in Theorem 2.4 and Theorem 2.1.
In fact,
thus
Remark 2.2
Suppose \(p\in(0,\frac{2}{n-2}]\), \(u_{0}\in H_{0}^{1}(\Omega)\), \(J(u_{0})< d\), then it is well known through the results in [14] that problem (1.1) has a sharp condition: the case of \(I(u_{0})>0 \) admits a global weak solution and the case of \(I(u_{0})<0 \) does not admit any global weak solution. However, a more powerful condition \(J(u_{0})< 0\) is required in Theorem 2.1 and we can deduce \(I(u_{0})<0 \) from \(J(u_{0})< 0\). More importantly, we obtain the precise upper bound \(\frac{\|u_{0}\|^{2}_{H_{0}^{1}}}{-4p(p+1)J(u_{0})}\) for the blow-up time.
References
Song, HT, Zhong, CK: Blow up of the nonclassical diffusion equation. J. Math. Phys. 50, 042702 (2009)
Shang, YD: Blow-up of solutions for the nonlinear Sobolev-Galpern equations. Math. Appl. 13, 35-39 (2000) (in Chinese)
Xu, R: Asymptotic behavior and blow up of solutions for the viscous diffusion equation. Appl. Math. Lett. 20, 255-259 (2007)
Yushkov, EV: Investigation of the existence and blow-up of a solution of a pseudoparabolic equation. Differ. Uravn. 47, 291-294 (2011) (in Russian)
Luo, P: Blow-up phenomena for a pseudo-parabolic equation. Math. Methods Appl. Sci. 38, 2636-2641 (2015)
Xu, RZ, Su, J: Global existence and finite blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. 264, 2732-2763 (2013)
Payne, LE, Philippin, GA, Vernier Piro, S: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II. Nonlinear Anal. 73, 971-978 (2010)
Payne, LE, Philippin, GA, Schaefer, PW: Blow-up phenomena for some nonlinear parabolic problems. Nonlinear Anal. 69, 3495-3502 (2008)
Payne, LE, Schaefer, PW: Lower bounds for blow-up time in parabolic problems under Dirichlet conditions. J. Math. Anal. Appl. 328, 1196-1205 (2007)
Showalter, RE, Ting, TW: Pseudo-parabolic partial differential equations. SIAM J. Math. Anal. 1, 1-26 (1970)
Benedetto, ED, Pierre, M: On the maximum principle for pseudo-parabolic equation. Indiana Univ. Math. J. 30, 821-854 (1981)
Liu, Y, Jiang, W, Huang, F: Asymptotic behaviour of solutions to some pseudo-parabolic equations. Appl. Math. Lett. 25, 111-114 (2012)
Payne, LE, Sattinger, DH: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273-303 (1975)
Yang, L, Liang, F, Guo, ZH: Global existence and finite time blow up for a class of semilinear pseudo-parabolic equation with a nonlocal source
Acknowledgements
The authors would like to thank the anonymous referees for their constructive corrections and valuable comments on this paper, which have considerably improved the presentation of this paper. This project is supported in part by China NSF Grant No. 11501442, Natural Science Basic Research Plan in Shaanxi Province of China No. 2016JM1025, the scientific research program funded by Shanxi Provincial Education Department No. 14JK1474, Graduate student self-determined and innovative research funds No. YZZ15070.
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Lu, Y., Fei, L. Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source. J Inequal Appl 2016, 229 (2016). https://doi.org/10.1186/s13660-016-1171-4
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DOI: https://doi.org/10.1186/s13660-016-1171-4