Blow-Up Solutions and Global Existence for Quasilinear Parabolic Problems with Robin Boundary Conditions

and Applied Analysis 3 Q t = [g󸀠 (u) u t − αh (t) f (u)] t = [(g (u)) t − h (t) f (u) + (1 − α) h (t) f (u)] t = [∇ ⋅ (a (u, t) b (x) ∇u) + (1 − α) h (t) f (u)] t = a u bu t Δu + a t bΔu + abΔu t + a uu b|∇u|2u t + a ut b|∇u|2 + 2a u b∇u ⋅ ∇u t + a u u t ∇b ⋅ ∇u + a t ∇b ⋅ ∇u + a∇b ⋅ ∇u t + (1 − α) h󸀠f + (1 − α) hf󸀠u t . (15) By (14) and (15), we have ab g ΔQ − Q t = ( abg󸀠󸀠󸀠 g − a uu b) |∇u|2u t + (2 abg󸀠󸀠 g − 2a u b)∇u ⋅ ∇u t + ( abg󸀠󸀠 g − a u b) u t Δu − (α abhf󸀠󸀠 g + a ut b) |∇u|2 − (α abhf󸀠 g + a t b)Δu − a u u t ∇b ⋅ ∇u − a t ∇b ⋅ ∇u − a∇b ⋅ ∇u t + (α − 1) h󸀠f + (α − 1) hf󸀠u t . (16) It follows from (1) that Δu = g󸀠 ab u t − a u a |∇u|2 − 1 b ∇b ⋅ ∇u − hf ab . (17) Next, we substitute (17) into (16) to obtain ab g ΔQ − Q t =( abg󸀠󸀠󸀠 g − a uu b− a u bg󸀠󸀠 g + (a u ) 2 b a )|∇u|2u t + (2 abg󸀠󸀠 g − 2a u b)∇u ⋅ ∇u t + (g󸀠󸀠 − a u g󸀠 a ) (u t ) 2 − ag󸀠󸀠 g u t ∇b ⋅ ∇u + ( a u hf a − fg󸀠󸀠h g − hf󸀠 − a t g󸀠 a )u t +(α a u bhf󸀠 g −α abhf󸀠󸀠 g + a u a t b a −a ut b) ∇u|2 + α ahf󸀠 g ∇b ⋅ ∇u − a∇b ⋅ ∇u t + α h2ff󸀠 g + a t hf a + (α − 1) h󸀠f. (18) With (13), it has ∇u t = 1 g ∇Q − g󸀠󸀠 g u t ∇u + α hf󸀠 g ∇u. (19) Substitute (19) into (18) to get ab g ΔQ + [2b( a g ) u ∇u + a g ∇b] ⋅ ∇Q − Q t = ( abg󸀠󸀠󸀠 g − a uu b + a u bg󸀠󸀠 g + (a u ) 2 b a − 2 ab(g󸀠󸀠) 2

Many authors have studied the blow-up and global solutions of nonlinear parabolic problems (see, for instance, [3][4][5][6][7][8][9][10][11][12][13][14]).Some special cases of the problem (1) have been treated already.Enache [15] investigated the following problem:   = ∇ ⋅ ( () ∇) +  () in  × (0, ) ,   +  = 0 on  × (0, ) ,  (, 0) = ℎ () ≥ 0 in , where  ⊂ R  ( ≥ 2) is a bounded domain with smooth boundary .Some conditions on nonlinearities and the initial data were established to guarantee that (, ) is global existence or blows up at some finite .In addition, an upper bound and a lower bound for  were derived.Zhang [16] dealt with the following problem: where  ⊂ R  ( ≥ 2) is a bounded domain with smooth boundary .By constructing auxiliary functions and using maximum principles, the sufficient conditions characterized by functions , , ℎ, , and  0 were given for the existence of blow-up solution.Ding [17] considered the following problem: where  ⊂ R  ( ≥ 2) is a bounded domain with smooth boundary .By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, the sufficient conditions were specified for the existence of blow-up and global solutions.For the blow-up solution, a lower bound on blow-up time is also obtained.Some authors also discussed blow-up phenomena for parabolic problems with Robin boundary conditions and obtained a lot of interesting results (see [18][19][20][21][22][23][24] and the references cited therein).
As everyone knows, parabolic equation describes the process of heat conduction.Blow-up and global solutions for parabolic equations reflect the unsteady state and steady state of heat conduction process, respectively.In the problems (2) and ( 4), the heat conduction coefficient () depends only on the temperature variable .In the problem (3), the heat conduction coefficient ()() depends on the temperature variable  and space variable .However, in a lot of processes of heat conduction, heat conduction coefficient depends not only on the temperature variable  but also on the space variable  and the time variable .Therefore, in this paper, we study the problem (1).It seems that the method of [15][16][17] is not applicable for the problem (1).In this paper, by constructing completely different auxiliary functions with those in [15][16][17] and technically using maximum principles, we obtain some existence theorems of blow-up solution, an upper bound of "blow-up time, " an upper estimates of "blowup rate, " the existence theorems of global solution, and an upper estimate of the global solution.Our results extend and supplement those obtained in [15][16][17].
We proceed as follows.In Section 2, we study the blow-up solution of (1).Section 3 is devoted to the global solution of (1).A few examples are presented in Section 4 to illustrate the applications of the abstract results.

Blow-Up Solution
The main results for the blow-up solution are Theorems 1-3.For simplicity, we define the constant In Theorems 1-3, the three cases 0 <  < 1,  = 1, and  > 1 are considered, respectively.In the first case, 0 <  < 1, we have the following conclusions.
Theorem 1.Let  be a solution of the problem (1).Suppose the following.
(i) Consider ) Then, the solution  of the problem (1) must blow up in a finite time , and where and  −1 and  −1 are the inverse functions of  and , respectively.
It follows from ( 1) and ( 21) that Next, by using ( 8) and the fact ( 0 ,  0 ) < 0, it follows from (26) that which contradicts inequality (25).Thus, we know that the minimum of  in  × [0, ) is zero.Thus, that is, At the point  * ∈ , where  0 ( * ) =  0 , integrate (29) over [0, ] to get which shows that  must blow up in finite time.In fact, suppose  is a global solution of (1), then, for any  > 0, it follows from (30) that Passing to the limit as  → +∞ in (31) yields which contradicts assumption (9).This shows that  must blow up in a finite time  = .Furthermore, letting  →  in (30), we have lim that is, which implies that Since  is a decreasing function, we have The proof is complete.
In the second case,  = 1, the following two assumptions (i) a and (ii) a can guarantee that inequality (23) holds.Theorem 2. Let  be a solution of the problem (1).Suppose that (i) a and (ii) a hold and assumptions (iii) and (iv) of Theorem 1 hold.Then, the conclusions of Theorem 1 are valid.
In the third case,  > 1, the following two assumptions (i) b and (ii) b imply that inequality (23) holds.

Global Solution
We define the constant The following Theorems 5-7 are the main results for the global solution.In Theorems 5-7, we study the three cases 0 <  < 1,  = 1, and  > 1, respectively.In the first case, 0 <  < 1, we have the following results.
Theorem 5. Let  be a solution of the problem (1).Suppose the following.
(i) Consider Then, the solution  to the problem (1) must be a global solution and where and  −1 is the inverse function of .
Proof.In order to study the global solution by using maximum principles, we construct an auxiliary function Substituting  and  with  and  in (22), respectively, gives Assumptions ( 48) and (49) guarantee that the right side in equality (55) is nonnegative; that is, It follows from (47) that max Replacing  and  with  and  in (26), respectively, we have Combining ( 56)-( 58) with (50) and applying the maximum principles again, it follows that the maximum of  in ×[0, ) is zero.Thus, ()  ()   ≤ ℎ () .
For each fixed  ∈ , integration of (60) from 0 to  yields Since  is an increasing function, we have The proof is complete.
In the second case  = 1 and the third case  > 1, we have the following results.Theorem 6.Let  be a solution of the problem (1).Suppose that assumptions (i) c and (ii) c hold.
In what follows, we present several examples to demonstrate the applications of the abstract results.
a For (, ) ∈ R + × R + , repeating the proof of Theorem 1, we have the following results.