Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

and Applied Analysis 3 Then, we have ∇P = b 󸀠󸀠 u t ∇u + b 󸀠 ∇u t − αe u ∇u, (12) ΔP = b 󸀠󸀠󸀠 u t |∇u| 2 + 2b 󸀠󸀠 ∇u ⋅ ∇u t + b 󸀠󸀠 u t Δu + b 󸀠 Δu t − αe u |∇u| 2 − αe u Δu. (13)


Introduction
The study of global and blow-up solutions for nonlinear parabolic equations has received a lot of attention in the past several decades (see [1][2][3][4]).In most works, so far, a variety of approaches have been developed in dealing with different nonlinear parabolic problems, such as the existence of global solution, blow-up solution, an upper bound for "blow-up time, " an upper estimate of "blow-up rate, " or global solution.So far, some applications in physics, chemistry, and biology are relevant to blow-up phenomena which can be found in [5][6][7][8][9][10][11].In this paper, we consider the global and blow-up solutions of the following nonlinear parabolic equation with Robin boundary condition: ( ())  = ∇ ⋅ (ℎ ()  ()  () ∇) +  (, , , ) , in  × (0, ) ,   +  = 0, on  × (0, ) ,  (, 0) =  0 () > 0, in , where  = |∇| 2 ,  ⊂ R  ( ≥ 2) is a bounded domain with smooth boundary , / represents the outward normal derivative on ,  is positive constant,  0 is the initial value,  is the maximal existence time of , and  is the closure of .Set R + = (0, +∞).We assume, throughout the paper, that () is a positive  3 (R + ) function,   () > 0 for any  ∈ R + , () is a positive  2 (R + ) function, () is a positive  1 () function, ℎ() is a positive  1 (R + ) function, (, , , ) is a nonnegative  1 ( × R + × R + × R + ) function, and  0 () is a positive  2 () function.Under these assumptions, the classical parabolic equation theory [12] ensures that there exists a unique classical solution (, ) with some  > 0 for the problem (1), and the solution is positive over  × [0, ).Moreover, by the regularity theorem [13], The problems of the global and blow-up solutions for nonlinear parabolic equations have been investigated extensively by many authors and have got a lot of meaningful results.Some special cases of problem (1) have been treated already.Ding [14] deals with the following problem: where  is a bounded domain of R  ( ≥ 2) with smooth boundary .By constructing auxiliary functions and using a first-order differential inequality technique, Ding derives conditions on the data, which guarantee the existence of blow-up or global solution.The following problem is investigated by Enache in [15]: where  is a bounded domain of R  ( ≥ 2) with smooth boundary .By constructing auxiliary functions and firstorder differential inequality technique, Enache establishes some conditions on nonlinearities and the initial date to guarantee that (, ) exists for all times  > 0 or blows up at some finite time .Besides, the following problem is investigated by Zhang in [16]: where  is a bounded domain in R  ( ≥ 2) with smooth boundary.Under appropriate assumptions on the functions , , and ℎ, Zhang obtains the conditions under which the solutions may exist globally or blow up in a finite time.Moreover, upper estimates of the "blow-up time, " blow-up rate, and global solutions are obtained also.
In this paper, we obtain the existence theorem of global and blow-up solution by constructing completely different auxiliary functions and technically using maximum principles.As a result, the sufficient conditions for the existence of a global solution and an upper estimate of the global solution and 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 assumption on the functions , ℎ, , , and  and initial value  0 .Our results extend and supplement those obtained in [14][15][16].
The content of this paper is organized as follows.In Section 2, we study the existence of the global solution of (1).In Section 3, we investigate the blow-up solution of (1).In Section 4, we will give a few examples to explain our results.

Global Solution
Our main result for the global solution is the following Theorem 1.

Theorem 1.
Let  be a solution of (1).Suppose that the following conditions ()-() are satisfied. ( (iii) Consider the integration (iv) Then the solution  to problem (1) must be a global solution and where and  −1 is the inverse function of .

Blow-Up Solution
The following theorem is the main result for the blow-up solution of (1).
Theorem 2. Let  be a solution of problem (1).Assume that the following conditions (i)-(iv) are satisfied.
(i) For any  ∈ R + , (iii) Consider the integration Then the solution  of problem (1) must blow up in finite time , and where By applying maximum principle (see [17]), it follows from (47) that  can attain its nonpositive minimum only for  × {0} or  × (0, ).