Blow-Up Phenomena for Nonlinear Reaction-Diffusion Equations under Nonlinear Boundary Conditions

This paper deals with blow-up and global solutions of the following nonlinear reaction-diffusion equations under nonlinear boundary conditions: (g(u)) t = ∇ ⋅ (a(u)∇u) + f(u) in Ω × (0, T), ∂u/∂n = b(x, u, t) on ∂Ω × (0, T), u(x, 0) = u 0 (x) > 0, in Ω, where Ω ⊂ RN (N ≥ 2) is a bounded domain with smooth boundary ∂Ω. We obtain the conditions under which the solutions either exist globally or blow up in a finite time by constructing auxiliary functions and using maximum principles. Moreover, the upper estimates of the “blow-up time,” the “blow-up rate,” and the global solutions are also given.

Many authors have investigated blow-up and global solutions of nonlinear reaction-diffusion equations under nonlinear boundary conditions and have obtained a lot of interesting results (see, e.g., [12][13][14][15][16][17][18][19][20]).To my knowledge, some special cases of (1) have been studied.Zhang [21] considered the following problem: where Ω ⊂ R  ( ≥ 2) is a bounded domain with smooth boundary Ω.By constructing auxiliary functions and using maximum principles, the existence of blow-up and global solutions were obtained under appropriate assumptions on the functions , , , and  0 .Zhang et al. [22] dealt with the following problem: where Ω ⊂ R  ( ≥ 2) is a bounded domain with smooth boundary Ω.Some conditions on nonlinearities and the initial data were given to ensure that (, ) exists globally or blows up at some finite time .In addition, the upper estimates of the global solution, the "blow-up time," and the "blow-up rate" were also established.
In this paper, we study reaction-diffusion problem (1).It is well known that (), (), (), and (, , ) are nonlinear reaction, nonlinear diffusion, nonlinear convection, and nonlinear boundary flux, respectively.What interactions among the four nonlinear mechanisms result in the blow-up and global solutions of (1) is investigated in this work.We note that the boundary flux function (, , ) depends not only on the concentration variable  but also on the space variable  and the time variable .Hence, it seems that the methods of [21,22] are not applicable for problem (1).In this paper, by constructing completely different auxiliary functions from those in [21,22] and technically using maximum principles, we obtain the existence theorems of the blow-up and global solution.Moreover, the upper estimates of "blow-up time," "blow-up rate," and global solution are also given.Our results can be seen as the extension and supplement of those obtained in [21,22].
The present work is organized as follows.In Section 2, we deal with the blow-up solution of (1).Section 3 is devoted to the global solution of (1).As applications of the obtained results, some examples are presented in Section 4.

Blow-Up Solution
In this section, we discuss what interactions among the four nonlinear mechanisms of (1) result in the blow-up solution.Our main result in this section is the following theorem.
(iii) Consider the following: (iv) Consider the following: Then (, ) blows up in a finite time  and where and  −1 is the inverse function of .
Proof.Introduce an auxiliary function and then we have It follows from ( 12) and ( 13) that The first equation of (1) implies Inserting ( 15) into ( 14), we obtain It follows from (11) that Substituting ( 17) into ( 16), we get By ( 10), we have Now, we insert ( 19) into (18) to deduce Assumptions (4) ensure that the right side in equality (20) is nonnegative; that is, Next, it follows from ( 1) and ( 10) that We note that (6) implies max There, ( 21)-(23), assumption (5) and the maximum principle [10] imply that the maximum of the function  in Ω × [0, T) is zero.In fact, if the function  takes a positive maximum at point ( 0 ,  0 ) ∈ Ω × (0, ), then we have Using assumption ( 5) and the fact that ( 0 ,  0 ) > 0, it follows from (22) that which contradicts the second inequality in (24).Hence, the maximum of the function  in Ω × [0, ) is zero.Now, we have that is, At the point  0 ∈ Ω, where  0 ( 0 ) =  0 , integrating inequality (27) from 0 to , we arrive at Inequality (28) and assumption (7) imply that (, ) blows up in finite time  = .Now, we let  →  in (28) to deduce In the above inequality, letting  → , we obtain We note that  is a strictly decreasing function.Hence, The proof is complete.

Global Solution
In this section, we study what interactions among the four nonlinear mechanisms of (1) result in the global solution of (1).The main results of this section are formulated in the following theorem.
Then (, ) must be a global solution and where and  is the inverse function of .
Proof.We consider an auxiliary function Using the same reasoning process as that of ( 11)-( 20  (47)