Global and blow-up solutions for a nonlinear reaction diffusion equation with Robin boundary conditions

*Correspondence: tyutzll@126.com 1Department of Mathematics, Taiyuan University of Technology, TaiYuan, P.R. China 2State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, P.R. China Abstract In the paper, we investigate global and blow-up solutions for a class of nonlinear reaction diffusion equations with Robin boundary conditions. By using auxiliary functions and a first-order differential inequality technique, we establish conditions on the data to prove the existence of global solution. Moreover, based on maximum principles, we obtain a criterion that guarantees the occurrence of the blow-up. When blow-up occurs, we discuss an upper bound and a lower bound on blow-up time. At last, we apply two examples to illustrate our main results.


Introduction
In recent years, there has been a great deal of literature on the global and blow-up solutions for nonlinear partial differential equations, for instance, in [1][2][3][4][5][6][7][8]. These works have contained a lot of interesting results about the global solutions, blow-up of solutions, bounds for the blow-up time, blow-up rates, and so on. A variety of physical, chemical, and biological applications are discussed in [9,10].
As far as we know, reaction-diffusion equation is an important part of partial differential equation. There have been many research papers devoted to the nonlinear reaction diffusion equations under various boundary conditions, such as Dirichlet boundary conditions [11,12], Neumann boundary conditions [13][14][15][16], nonlocal boundary conditions [17,18], and nonlinear boundary conditions [19,20]. At the same time, the blow-up problems for reaction diffusion equations under Robin boundary conditions have been also studied (see [21][22][23][24]). The authors in [22] considered the following equation: where Ω ⊂ R n (n ≥ 2) is a smooth bounded domain. By making appropriate restrictions and using a differential inequality technique, a lower bound for the blow-up time was investigated in a three-dimensional space if blow-up occurs. When Ω ⊂ R n (n ≥ 2), the authors demonstrated a criterion which guarantees the solution to remain global. [23] was concerned with a more complicated case: g(u) t = ∇ · ρ |∇u| 2 ∇u + k(t)f (u), (x, t) ∈ Ω × (0, T).
According to a differential inequality technique and maximum principles, the authors showed a blow-up or non-blow-up criterion under some appropriate assumptions. Moreover, they were dedicated to the upper and lower bounds for the blow-up time when blowup occurs. In [24], the authors dealt with the blow-up phenomena of the following quasilinear reaction diffusion equations with weighted nonlocal source: where Ω ⊂ R n (n ≥ 2) is a bounded convex domain. They established conditions to guarantee that the solution remains global or blows up in a finite time. Moreover, an upper and lower bounds for blow-up time were derived. These results were obtained by utilizing a differential inequality technique on suitably defined auxiliary functions.
Inspired by the above papers, we are concerned with the following reaction diffusion equations under Robin boundary conditions: (1.1) Here, p > 0 and Ω ⊂ R n (n > 2) is a bounded convex domain and the boundary ∂Ω is smooth. Ω is the closure of Ω. ∂u ∂ν stands for the outward normal derivative on ∂Ω, γ is a positive constant, u 0 denotes the initial value and is a positive C 2 (Ω) function, T is the blow-up time if the blow-up happens, otherwise T = +∞. Set R + = (0, +∞), g is a C 2 (R + ) function with g (s) > 0 for all s > 0, ρ is a positive C 2 (R + ) function with ρ(s) + psρ (s) > 0 for s > 0, k is a positive C 1 (R + ) function, a is a positive C 1 (Ω) function, f is a nonnegative C 1 (R + ) function. The maximum principles imply that the classical solution u of (1.1) is a positive solution in Ω × [0, T).
It is obvious that problem (1.1) is more general than the ones in [22][23][24]. The purpose of the paper is to get a non-blow-up or blow-up criterion for system (1.1). We need to define appropriate auxiliary functions that are different from the ones in [22][23][24]. By using first-order differential inequalities, we show that the solution of (1.1) exists globally under some conditions. Furthermore, combining the first-order differential inequalities with maximum principles, we demonstrate that the solution u(x, t) blows up at some finite time. When blow-up does occur, the upper and lower bounds for the blow-up time are presented.
The rest of the paper is organized as follows. Section 2 shows that solution u(x, t) of (1.1) exists globally in the suitable measure. In Sect. 3, we prove that solution u(x, t) blows up at some finite time and obtain an upper bound for the blow-up time. Section 4 presents a lower bound for the blow-up time when blow-up occurs. The last section gives two examples to illustrate our main results.

Global existence
In the section, under some suitable conditions, we prove the existence of global solution u of problem (1.1). We suppose that u is a classical solution of problem (1.1). In order to obtain our main result, it is required to construct the following notations and functions: Throughout the paper, we need to assume that λ is the first positive eigenvalue of the following problem: In what follows, the main result will be stated.
where l, m, κ, a 1 , q, c, b 1 , b 2 are positive constants satisfying κ > 1, 0 < l < κ + 1 < 2pq + 2, κ + lm < 2pq + 1. Proof Taking the derivation the function Φ(t) in (2.1), we get where the divergence theorem and Robin boundary condition in (1.1) are used. Furthermore, conditions (2.3) and (2.4) imply that Next, we deal with the first term on the right-hand side of (2.7). By virtue of Hölder's inequality, we have the following inequality: (2.8) The general Poincaré inequality yields where the Robin boundary conditions in (1.1) are used. We apply the divergence theorem to derive Then it follows from Hölder's inequality and Young's inequality that After inserting (2.10) into (2.9), we obtain Therefore, (2.7) can be rewritten as follows: Obviously, the following inequality holds: Combining (2.13) and (2.14)-(2.16), we derive An application of Hölder's inequality implies Consequently, (2.17) can be replaced by Assumptions (2.1) and (2.4) imply Hence, From (2.21), we are devoted to proving that u cannot blow up in the measure Φ(t). Now, suppose that u blows up at some finite t * in the measure Φ(t). Then we could find Moreover, Therefore, (2.21) yields that there exists t 1 < t * such that Φ (t) < 0, t ∈ (t 1 , t * ). Namely, We take the limit as t → t *to get This is a contradiction. Then the hypothesis that u blows up at some finite time is false.

Blow-up in finite time and an upper bound for blow-up time
This subsection presents that u blows up in finite time and derives an upper bound for blow-up time under some suitable assumptions of the functions ρ, b, k, f , g. Firstly, we state the following Lemma 3.1 which is used for the proof of the main result. (1.1). And the following assumptions are satisfied:

Lemma 3.1 Suppose that u is a classical solution of problem
Proof Taking a derivative of u t with respect to t, we have For convenience, let ξ = u t , then we get By (3.1) and the properties of the functions a, f , g for any (t, The Robin boundary condition of (1.1) implies According to (3.4)-(3.6) and the maximum principle in [25], we can obtain The proof is complete.
In the sequel, the main result Theorem 3.1 will be stated under Lemma 3.1 and the auxiliary functions defined by (2.1) and (2.2). (1.1). And assume that (3.1)-(3.2) hold and the functions ρ, g, f , k, a satisfy the following conditions:

Theorem 3.1 Let u be a classical solution of problem
where u 0 is the initial value and α is a positive constant. Then u blows up at some finite time T ≤ T * in the measure Φ(t). And Proof Obviously, (2.6) also holds. Then, from assumptions (3.7) and (3.8), we rewrite (2.6) as Differentiating Ψ (t) in (2.2), we obtain where (3.2) and (3.7) are used. Moreover, it follows from Gauss's law and Robin's boundary condition that Therefore, Ψ (t) is a nondecreasing function in t. By virtue of (3.9), it is easy to get that Consequently, (3.10) yields With the help of (3.10)-(3.11), we obtain Therefore, (3.13) can be replaced by namely When α > 0, integrating (3.17) from 0 to t again, we have Hence, it follows from (3.18) that the solution u of (1.1) blows up at some finite time T < T * in the measure Φ(t) and
Thus, if u blows up in the measure A(t) at some finite time t * , and there is a lower bound for blow-up time.

Applications
As applications, the section shows that two examples illustrate our main results.

Conclusion
In this paper, we discussed global and blow-up solutions for a class of nonlinear reaction diffusion equations with Robin boundary conditions. By auxiliary functions and a firstorder differential inequality technique, conditions on the data to prove the existence of global solution are established. And applying maximum principles, we obtain the sufficient conditions that guarantee the occurrence of the blow-up. Moreover, an upper bound and a lower bound on blow-up time are discussed.