Global and blow-up solutions for nonlinear parabolic problems with a gradient term under Robin boundary conditions
© Ding; licensee Springer. 2013
Received: 25 July 2013
Accepted: 25 September 2013
Published: 8 November 2013
In this paper, we study the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:
where () is a bounded domain with smooth boundary ∂D. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for ‘blow-up time’, and an upper estimate of ‘blow-up rate’ are specified under some appropriate assumptions on the functions f, g, b and initial value .
MSC:35K55, 35B05, 35K57.
where , () is a bounded domain with smooth boundary ∂D, represents the outward normal derivative on ∂D, γ is a positive constant, is the initial value, T is the maximal existence time of u, and is the closure of D. Set . We assume, throughout the paper, that is a function, for any , is a positive function, is a nonnegative function, and is a positive function. Under the above assumptions, the classical theory  of parabolic equation assures that there exists a unique classical solution with some for problem (1.1) and the solution is positive over . Moreover, the regularity theorem  implies .
where () is a bounded domain with smooth boundary ∂D. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, the sufficient conditions were obtained for the existence of global and blow-up solutions. For the blow-up solution, an upper and a lower bound on blow-up time were also given.
In this paper, we study problem (1.1). Since the function contains a gradient term , it seems that the methods of [21–23] are not applicable for problem (1.1). In this paper, by constructing completely different auxiliary functions with those in [21–23] and technically using maximum principles, we obtain some existence theorems of a global solution, an upper estimate of the global solution, the existence theorems of a blow-up solution, an upper bound of ‘blow-up time’, and an upper estimates of ‘blow-up rate’. Our results extend and supplement those obtained [21–23].
We proceed as follows. In Section 2 we study the global solution of (1.1). Section 3 is devoted to the blow-up solution of (1.1). A few examples are presented in Section 4 to illustrate the applications of the abstract results.
2 Global solution
The main result for the global solution is the following theorem.
- (i)for any ,(2.1)
- (ii)for any ,(2.2)
and is the inverse function of H.
The proof is complete. □
3 Blow-up solution
The following theorem is the main result for the blow-up solution.
- (i)for any ,(3.1)
- (ii)for any ,(3.2)
and is the inverse function of G.
The proof is complete. □
In what follows, we present several examples to demonstrate the applications of the abstract results.
and is the inverse function of G.
Remark 4.1 We can see from Example 4.1 that when the equation has a gradient term with exponential increase, the functions g and b increase exponentially to ensure that the solution of (1.1) blows up. It follows from Example 4.2 that when the equation has a gradient term with exponential decay, the appropriate assumptions on the functions g and b can guarantee the solution of (1.1) to be global.
This work was supported by the National Natural Science Foundation of China (Nos. 61074048 and 61174082) and the Research Project Supported by Shanxi Scholarship Council of China (Nos. 2011-011 and 2012-011).
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