Global existence and blow-up for a class of nonlinear reaction diffusion problems
© Ding; licensee Springer 2014
Received: 8 April 2014
Accepted: 25 June 2014
Published: 25 September 2014
This paper deals with the global existence and blow-up of the solution for a class of nonlinear reaction diffusion problems. The purpose of this paper is to establish conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified. Finally, as applications of the obtained results, some examples are presented.
MSC: 35K57, 35K55, 35B05.
where () is a bounded domain with smooth boundary ∂D, is the closure of D, is the outward normal derivative on ∂D, T is the maximal existence time of u. Set . We assume, throughout this paper, that is a positive function, is a positive function, is a positive function, is a function, for any , and is a positive function. Under the above assumptions, it is well known from the classical parabolic equation theory  and maximum principle  that there exists a unique local positive solution for problem (1.1). Moreover, by the regularity theorem , .
where () is a bounded domain with smooth boundary ∂D. The sufficient conditions were developed for the existence of global and blow-up solutions. Meanwhile, the upper estimate of the global solution, the upper bound of the ‘blow-up time’, and the upper estimate of the ‘blow-up rate’ were also given.
In this paper, we study reaction diffusion problem (1.1). Note that , and are nonlinear reaction, nonlinear diffusion and nonlinear convection, respectively. Since the diffusion function depends not only on the concentration variable u but also on the space variable x, it seems that the methods of ,  are not applicable for the problem (1.1). In this paper, by constructing completely different auxiliary functions from those in ,  and technically using maximum principles, we obtain the conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also given. Our results extend and supplement those obtained in , .
We proceed as follows. In Section 2 we study the blow-up solution of (1.1). Section 3 is devoted to the global solution of (1.1). A few examples are given in Section 4 to illustrate the applications of the obtained results.
2 Blow-up solution
In this section we establish sufficient conditions on the data of the problem (1.1) to produce a blow-up of the solution at some finite time T and under these conditions we derive an explicit upper bound for T and an explicit upper estimate of the ‘blow-up rate’. The main result of this section is formulated in the following theorem.
- (i)for any ,(2.1)
- (ii)the constant(2.2)
- (iii)the integration(2.3)
andis the inverse function of Φ.
The proof is complete. □
3 Global solution
In this section we establish sufficient conditions on the data of the problem (1.1) in order to ensure that the solution has global existence. Under these conditions, we derive an explicit upper estimate of the global solution. The main results of this section are the following theorem.
- (i)for any ,(3.1)
- (ii)the constant(3.2)
- (iii)the integration(3.3)
andis the inverse function of Ψ.
The proof is complete. □
In what follows, we present several examples to demonstrate the applications of the obtained results.
All results belong to Juntang Ding.
This work was supported by the National Natural Science Foundation of China (Nos. 61074048 and 61174082), the Research Project Supported by Shanxi Scholarship Council of China (Nos. 2011-011 and 2012-011), and the Higher School ‘131’ Leading Talent Project of Shanxi Province.
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