Abstract
In this paper, we study a coupled systems of parabolic equations subject to large initial data. By using comparison principle and Kaplan’s method, we get the upper and lower bound for the life span of the solutions.
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1 Introduction
In this paper, we consider the following nonlinear parabolic system:
where \(n>m>1\), \(p>q>1\), \(pm>qn\); Ω is a bounded domain in \(R^{N}\) with a smooth boundary ∂Ω; \(\lambda >0\) is a parameter, φ and ψ are nonnegative continuous functions on Ω̄.
The existence and the uniqueness of local classical solutions to problem of semilinear parabolic systems are well known (see, e.g., [1]). We denote by \(T^{*}_{\lambda }\) the maximal existence time of a classical solution \((u,v)\) of problem (1.1), that is,
and we call \(T^{*}_{\lambda }\) the life span of \((u, v)\). If \(T^{*}_{\lambda }<\infty \), then we have
We are interested in \(T^{*}_{\lambda }\) and aim to give some properties of the \(T^{*}_{\lambda }\).
Since Fujita’s classic work [2], the single equation
has been studied extensively in various directions. Friedman and Lacey [3] gave a result on the life span of solutions of (1.2) in the case of small diffusion. Subsequently, Gui and Wang [4], Lee and Ni [5] obtained the leading term of the expansion of the life span \(T_{\lambda }\) of the solution for (1.2) with the initial data \(\lambda \varphi (x)\), and later, Mizoguchi and Yanagida [6] extended the result and determined the second term of the expansion of \(T_{\lambda }\), the proved that φ attains the maximum at only one point as \(\lambda \to \infty \). Moreover, Mizoguchi and Yanagida [7] extended the result on the life span of solutions of (1.2) in the case of small diffusion. In [8], Sato extended the results to general nonlinearities \(f(u)\) in the case of large initial data. Parabolic systems of the following form:
have also been studied in several directions. In [9], Sato investigated (1.3) with \(f(v)\) and \(g(u)\) replaced by \(v^{p}\) and \(u^{q}\), in this article the life span of \((u,v)\) with large initial data was obtained. The present author studied the case \(f(v)=e^{pv}\), \(g(u)=e^{qu}\) in [10] and got some properties of the life span of the solution when initial data is large enough. For other results on system (1.3), we refer the reader to the survey [11], the monograph [12], as well as [13], and the references therein.
On the other hand, much effort has been devoted to the study of coupled parabolic systems, local and global existence, finite time blowup and blowup rate estimates, etc. We recommend reading the latest results [14, 15]. In [16], Zheng and Zhao considered the radially symmetric solutions for the parabolic system
And Zhang and Zheng in [17] investigated the above system with nonlocal sources \(\lambda \int _{\varOmega }e^{mu+pv}\) and \(\mu \int _{ \varOmega }e^{qu+nv}\). Also the case with localized sources was studied by Li and Wang in [18].
Parabolic equations (1.3) with the nonlinearities \(f(v)=u^{m}e ^{pv}\), \(g(u)=u^{q}e^{nv}\) subject to null Dirichlet boundary conditions were considered in [19] by Liu and Li.
However, to the best of our knowledge, there is little literature on the study of the life span of solutions for problem (1.1). The aim of this paper is to obtain some properties of the life span \(T^{*}_{ \lambda }\) as λ is large enough. We give a quantitative characterization of life span for the solutions. In the following, we denote by \(M_{\varphi }\) and \(M_{\psi }\) the maximum of φ and ψ on Ω̅. Then our main results can be summarized as the following theorem.
Theorem 1.1
Suppose \(\varphi , \psi \in C(\bar{\varOmega })\) satisfy \(\varphi , \psi \geq 0\) in Ω, \(\varphi =\psi =0\) on ∂Ω, \(\varphi +\psi \not \equiv 0\).
-
(i)
If \(p-n>q-m>0\) and \((q-m)M_{\varphi }>(p-n)M_{\psi }\), then we have
$$ \liminf_{\lambda \to \infty }T^{*}_{\lambda }e^{\frac{(pq-mn)M _{\varphi }\lambda }{p-n}} \geq \biggl(\frac{q-m}{p-n} \biggr)^{ \frac{p}{p-n}}\frac{p-n}{pq-mn} . $$(1.4) -
(ii)
If \(q-m>p-n>0\) and \((p-n)M_{\psi }>(q-m)M_{\varphi }\), then we have
$$ \liminf_{\lambda \to \infty }T^{*}_{\lambda }e^{\frac{(pq-mn)M _{\psi }\lambda }{q-m}} \geq \biggl(\frac{p-n}{q-m} \biggr)^{ \frac{q}{q-m}}\frac{p-n}{pq-mn} . $$(1.5)
Theorem 1.2
Suppose \(\varphi , \psi \in C(\bar{\varOmega })\) satisfy \(\varphi , \psi \geq 0\) in Ω, \(\varphi =\psi =0\) on ∂Ω, \(\varphi +\psi \not \equiv 0\).
-
(i)
If \(p-n>q-m>0\) and \((q-m)M_{\varphi }>(p-n)M_{\psi }\), then we have
$$ \limsup_{\lambda \to \infty }T^{*}_{\lambda } \frac{e^{qM_{ \varphi }\lambda }}{\lambda } \leq \frac{qM_{\varphi }-pM_{\psi }}{p}. $$(1.6) -
(ii)
If \(q-m>p-n>0\) and \((p-n)M_{\psi }>(q-m)M_{\varphi }\), then we have
$$ \limsup_{\lambda \to \infty }T^{*}_{\lambda } \frac{e^{pM_{ \psi }\lambda }}{\lambda } \leq \frac{pM_{\psi }-qM_{\varphi }}{q}. $$(1.7)
2 Preliminaries
In this section we first consider the ODE system
where α and β are nonnegative constants.
Here, for constants α and β with \((\alpha , \beta ) \not \equiv (0, 0)\), we define by \((z(t;\alpha ,\beta ),w(t;\alpha ,\beta ))\) the solution for problem (2.1). It is well known that \((z(t;\alpha ,\beta ),w(t;\alpha ,\beta ))\) exists and blows up in finite time. We then give the following lemma.
Lemma 2.1
Suppose that α, β are nonnegative constants and \((\alpha ,\beta )\not \equiv (0,0)\). Then the life span of the solution \((z,w)\) for problem (2.1) is
Proof
Multiplying the first equation in (2.1) by \(e^{qz+nw}\) and the second equation by \(e^{mz+pw}\), we obtain the equality
Integrating this equality over \((0,t)\), we have
Hence we get
Substituting these equalities into the equations of (2.1), we see that \((z,w)\) satisfies the initial-value problem
Integrating equations in (2.3), (2.4) over \((0, t)\) yields
This implies that the life span of \((z,w)\) is
By using the change of variables
we see that
□
3 Proof of main results
We first give a lower bound of life span to the solutions and prove Theorem 1.1.
Proof
We give the proof of (i). It is obvious that the solution \((z(t; \lambda M_{\varphi },\lambda M_{\psi }),w(t; \lambda M_{\varphi }, \lambda M_{\psi }))\) is a supersolution of problem (1.1), so we have
for \(x\in \varOmega \) and \(0< t<\min \{T^{*}_{\lambda M_{\varphi },\lambda M_{\psi }},T^{*}_{\lambda }\}\). This implies
First we assume that \(\varphi \neq 0\). Then by (3.1) and Lemma 2.1, a routine computation shows
and this yields
so we get
One can prove (ii) by using similar arguments. □
Next, we give an upper estimate of \(T^{*}_{\lambda }\) and prove Theorem 1.2.
Proof
We prove by using Kaplan’s method [20]. We only give the proof of (i); case (ii) can be proved similarly. Without loss of generality, we may assume that \(\varphi (0)=M_{\varphi }\). We define by \(\mu _{R}\) the first eigenvalue of −Δ in the ball \(B_{{R}}=B_{{R}}(0)\), \(\phi _{R}\) being the corresponding eigenfunction which satisfies \(\int _{B_{R}}\phi _{R}(x)\,dx=1\). Thus, we have
It is easy to check that
Let R be small enough such that \(B_{{R}}\subset \varOmega \) and set
For \(\varphi , \psi \in C(\bar{\varOmega })\), \(\int _{B_{1}}\phi _{1}(x)\,dx=1\), we have
Multiplying the equations in (1.1) by \(\phi _{{R}}\), integrating by parts and using Jensen’s inequality, we obtain
Hence, we have
Integrating these inequalities over \((0,t)\), we see that
Substituting the second inequality into the first, it follows that
thus we have
We fix \(0<\epsilon <1\) and take \(T_{R}>0\) such that \(e^{-\mu _{{R}}T _{R}}>1-\epsilon \). Then we have
We set
then we have
After a careful computation, we see that
Integrating this inequality over \((0,t)\), it follows that
Dividing the left-hand side by the right-hand side and integrating over \((0,t)\), we obtain
we then take λ large enough such that
Then z blows up at some \(T\leq T_{\epsilon ,R}\), and a careful computation yields
hence we get
Therefore, taking \(R\to 0\) and then \(\epsilon \to 0\), paying attention to \((q-m)M_{\varphi }>(p-n)M_{\psi }\) and \(pm>qn\), it follows that
so we get
which is the inequality in (1.6). By a similar argument, we can prove (1.7), and thus Theorem 1.2 is proved. □
References
Friedman, A., Giga, Y.: A single point blow-up for solutions of semilinear parabolic systems. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 34, 65–79 (1987)
Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_{t}=\Delta u+u^{1+\alpha }\). J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 16, 105–113 (1966)
Friedman, A., Lacey, A.: The blow-up time of solutions of nonlinear heat equation with small diffusion. SIAM J. Math. Anal. 18, 711–721 (1987)
Gui, C.F., Wang, X.F.: Life span of solutions of the Cauchy problem for a semilinear heat equation. J. Differ. Equ. 115, 166–172 (1995)
Lee, T.Y., Ni, W.: Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333, 1434–1446 (1992)
Mizoguchi, N., Yanagida, E.: Life span of solutions with large initial data in a semilinear parabolic equation. Indiana Univ. Math. J. 50(1), 591–610 (2001)
Mizoguchi, N., Yanagida, E.: Life span of solutions for a semilinear parabolic problem with small diffusion. J. Math. Anal. Appl. 261, 350–368 (2001)
Sato, S.: Life span of solutions with large initial data for a superlinear heat equation. J. Math. Anal. Appl. 343, 1061–1074 (2008)
Sato, S.: Life span of solutions with large initial data for a semilinear parabolic system. J. Math. Anal. Appl. 380, 632–641 (2011)
Zhou, S., Yang, Z.: Life span of solutions with large initial data for a semilinear parabolic system. J. Math. Res. Appl. 35, 103–109 (2015)
Deng, K., Levine, H.A.: The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl. 243, 85–126 (2000)
Quittner, P., Souplet, P.: Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States. Birkhäuser, Basel (2007)
Xu, X., Ye, Z.: Life span of solutions with large initial data for a class of coupled parabolic systems. Z. Angew. Math. Phys. 64(3), 705–717 (2013)
Xu, R., Lian, W., Niu, Y.: Global well-posedness of coupled parabolic systems. Sci. China Math. (2019). https://doi.org/10.1007/s11425-017-9280-x
Leng, Y., Nie, Y., Zhou, Q.: Asymptotic behavior of solutions to a class of coupled nonlinear parabolic systems. Bound. Value Probl. 2019, 68 (2019). https://doi.org/10.1186/s13661-019-1181-5
Zheng, S., Zhao, L., Chen, F.: Blow-up rates in a parabolic system of ignition model. Nonlinear Anal. 51, 663–672 (2002)
He, Z., Linghua, K., Sining, Z.: Propagations of singularities in a parabolic system with coupling nonlocal sources. Sci. China Ser. A, Math. 52, 181–194 (2009)
Huiling, L., Mingxing, W.: Blow-up properties for parabolic systems with localized nonlinear sources. Appl. Math. Lett. 17, 771–778 (2004)
Liu, B., Li, F.: Blow-up properties for heat equations coupled via different nonlinearities. J. Math. Anal. Appl. 347, 294–303 (2008)
Kaplan, S.: On the growth of solutions of quasi-linear parabolic equations. Commun. Pure Appl. Math. 16, 305–330 (1963)
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This work is supported by Young Teacher’s Research Funding from College of Science, China Pharmaceutical University, 2018CSYT007.
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Zhou, S. Life span of solutions with large initial data for a semilinear parabolic system coupling exponential reaction terms. Bound Value Probl 2019, 154 (2019). https://doi.org/10.1186/s13661-019-1269-y
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DOI: https://doi.org/10.1186/s13661-019-1269-y