Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems

In this paper, we consider a quasilinear reaction diffusion equation with Neumann boundary conditions in a bounded domain. Basing on Sobolev inequality and differential inequality technique, we obtain upper and lower bounds for the blow-up time of the solution. An example is also given to illustrate the abstract results obtained of this paper.


1.
Introduction. Blow-up problems related to reaction diffusion equations have been widely studied by many authors and numerous interesting results have been obtained in [2,11]. Studies of these references are often concerned with the conditions of global existence and blow-up in finite time, bounds for the blow-up rate, structure of blow-up set, and the asymptotic behavior of the solutions.
In this paper, we assume that h is a C 2 (R + ) function with h (s) > 0 for s > 0, ρ is a positive C 2 (R + ) function satisfying ρ(s) + 2sρ (s) > 0 for s > 0, k is a positive C 1 (R + ) function, and f is a nonnegative C 1 (R + ) function. According to maximum principles [20], we know that the classical solution u of problem (1) is nonnegative in Ω × [0, t * ).
Recently, there are many papers to study the bounds of the blow-up time, especially on the lower bound for the blow-up time. As we all know, lower bounds for blow-up time seem to be more important in applications, due to the explosive nature of the solution. However, the works mentioned above usually derived an upper bound for blow-up time. At meanwhile, we note that lower bounds for blow-up time seem harder to be determined. Since Payne and Scheafer in [17] used a firstorder differential technique and derived a lower bound for blow-up time, the similar idea is also applied in more generalized problems (see [1,[3][4][6][7][8][12][13][14][15][16][18][19]). The direct motivation of this paper comes from [5]. When Ω ⊂ R 3 is a bounded convex domain, Ding and Hu [5] dealt with problem (1) and derived the lower and upper bounds for blow-up time when blow-up occurs. Their main method in [5] is to use a first-order differential inequality technique. Naturally, in this paper, we hope to obtain a lower bound for blow-up time when Ω ⊂ R n (n ≥ 3) is a bounded domain. In addition, we also get an upper bound of blow-up time by restricting bounded domain Ω ⊂ R n (n ≥ 2). Our study is based on the use of Sobolev inequality and differential inequality technique.
The rest of this paper is organized as follows. In the second section, we establish conditions on data to ensure that the solution blows up at some finite time in measure B(t) defined in (2). Moreover, an upper bound for t * is derived. In the third section, under suitable hypotheses on data, we get a lower bound for t * . In the last section, an example is presented to demonstrate the results of this paper.

2.
Upper bound for t * . In this section, we restrict Ω ⊂ R n (n ≥ 2) and define auxiliary functions as follows With the aid of these auxiliary functions, we get an upper bound for blow-up time t * . More precisely we establish the following result.
Theorem 2.1. Let u be a classical solution of (1). Moreover, we suppose the functions k, h, ρ, and f to satisfy where d is a nonnegative constant. Furthermore, initial data are assumed to satisfy Then u must blow up at t * ≤ T in measure B(t) with Proof. We use Green's formula and (5) to derive Differentiating D(t) and using (4), we get It is easy to see that D(t) is a nondecreasing function. In view of (6)-(7), we have B (t) > 0. By Schwarz's inequality, (7) and h (s) > 0 for s > 0, we obtain Integrating by part and using (5), we have We combine (9)-(10) to derive that is Integrating (11) from 0 to t, we obtain In view of (7), we deduce

JUNTANG DING AND XUHUI SHEN
When d > 0, we integrate (12) from 0 to t to get It is obviously that (13) cannot hold for all time. Hence, u must blow up at some finite time t * in measure B(t). Letting t → t * in (13), we have When d = 0, it follows from (12) that which implies T = ∞.
3. Lower bound for t * . In this section, we derive a lower bound for t * by restricting Ω ⊂ R n (n ≥ 3). Here we impose the following constraints on data where a, b 2 , m, l, γ, η, M are positive constants, b 1 is a nonnegative constant, and m > 2l + 1. Two auxiliary functions are defined as follows and r is a parameter to satisfy In this section, we also need to use the following Sobolev inequality (see [9]) for n ≥ 3 where C(n, Ω) is an embedding constant. We state the main result of this section as follows.

BLOW-UP TIME IN REACTION DIFFUSION PROBLEMS 4249
Substituting (26) into (24), we derive Now, we focus on the second term of (27). In view of (16), we have By Hölder inequality and Young inequality, we deduce where ε is given in (20). Using Sobolev inequality (17) to the second term of (28), we derive It follows from (25)-(26) and Young inequality that We substitute (30) into (29) to get Inserting (31) into (27), we obtain where C 1 is given in (18) and . From (14), it follows that Substituting (33) into (32), we get where C 2 is defined in (19). We integrate (34) from 0 to t to obtain Letting t → t * , we get a lower bound for t * t * ≥ > 1 in view of (16).

4.
Application. An example is given in this section to illustrate the abstract results of this paper.
Example 4.1. Let u be a classical solution of the following problem: in Ω, Choosing d = 1, it is easy to verify that (4)-(5) hold. It follows from (2) In addition, from (2)