Global existence and blow-up analysis for parabolic equations with nonlocal source and nonlinear boundary conditions

We investigate the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions: {(g(u))t=∑i,j=1N(aij(x)uxi)xj+γ1um(∫Duldx)p−γ2urin D×(0,t∗),∑i,j=1Naij(x)uxiνj=h(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} (g(u) )_{t} =\sum_{i,j=1}^{N} (a^{ij}(x)u_{x_{i}} ) _{x_{j}}+\gamma _{1}u^{m} (\int _{D} u^{l}{\,\mathrm{d}}x ) ^{p}-\gamma _{2}u^{r}& \mbox{in } D\times (0,t^{*}), \\ \sum_{i,j=1}^{N}a^{ij}(x)u_{x_{i}}\nu _{j}=h(u) & \mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq 0 &\mbox{in } \overline{D}, \end{cases} $$\end{document} where p and γ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma _{1}$\end{document} are some nonnegative constants, m, l, γ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma _{2}$\end{document}, and r are some positive constants, D⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D\subset \mathbb{R}^{N}$\end{document} (N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 2$\end{document}) is a bounded convex region with smooth boundary ∂D. By making use of differential inequality technique and the embedding theorems in Sobolev spaces and constructing some auxiliary functions, we obtain a criterion to guarantee the global existence of the solution and a criterion to ensure that the solution blows up in finite time. Furthermore, an upper bound and a lower bound for the blow-up time are obtained. Finally, some examples are given to illustrate the results in this paper.


Introduction
There has been a vast amount of literature to discuss the global solutions, the blow-up solutions, and the bounds for the blow-up time for nonlinear parabolic equations. We refer the readers to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The global and blow-up solutions for the parabolic problems with nonlinear boundary conditions have been studied in [1][2][3][4][5][6][7]. Recently, some authors have already considered the blow-up phenomena of the parabolic equations with nonlocal source. We recommend the literature [8][9][10] and the references therein. In the present work, we study the following nonlinear parabolic equations with nonlocal source and non-linear boundary conditions: where p and γ 1 are some nonnegative constants, m, l, γ 2 , and r are some positive constants, D ⊂ R N (N ≥ 2) is a bounded convex region, the boundary ∂D is smooth, t * is the blow-up time when blow-up occurs, or t * = +∞, and ν = (ν 1 , ν 2 , . . . , ν N ) is the unit outward normal vector on ∂D. Moreover, (a ij (x)) N×N is a differentiable positive definite matrix; that is, for all z = (z 1 , z 2 , . . . , z N ) ∈ R N , there exists a constant θ > 0 such that a ij z i z j ≥ θ |z| 2 . (1.2) Set R + = (0, +∞). We assume that g is a C 2 (R + ) function which satisfies g (s) > 0 for all s ∈ R + , h is a nonnegative C 1 (R + ) function, and u 0 (x) is a nonnegative C 1 (D) function which satisfies the compatibility condition. Our study is motivated by the following three papers. Li et al. [1], Baghaei et al. [11], and Zhang et al. [2] studied the following parabolic equations with local source and nonlinear boundary conditions: u(x, 0) = u 0 (x) ≥ 0 i n D, where D is a bounded convex region in R N (N ≥ 2), and the boundary ∂D is smooth. When the function k(t) ≡ -1, Li et al. [1] obtained the conditions to ensure that the global solution exists and the solution blows up in finite time t * . Moreover, they derived upper bounds of the blow-up time in D ⊂ R N (N ≥ 2). Using the three-dimensional Sobolev type inequality that Payne and Schaefer had proven in [7], they got lower bounds in D ⊂ R 3 . When k(t) ≡ -1, in [11], Baghaei et al. used the embedding theorems in Sobolev spaces to get a lower bound of the blow-up time in D ⊂ R N (N ≥ 3). Recently, Zhang et al. [2] considered problem (1.3) when k(t) > 0. Their main contribution was to extend the threedimensional Sobolev type inequality obtained by Payne and Schaefer in [7] to the higher dimensional one. They used this Sobolev inequality in multidimensional space to derive a lower bound of the blow-up time in D ⊂ R N (N ≥ 3). In addition, they obtained the global existence of the solution and an upper bound of the blow-up time when blow-up occurs in D ⊂ R N (N ≥ 2). In the present paper, we study problem (1.1). In the process of getting lower bounds of the blow-up time, the key of our work is to deal with the nonlocal source. We discuss the blow-up problems of (1.1) by constructing some suitable auxiliary functions and making use of the embedding theorems in Sobolev spaces and differential inequality technique. The present work is organized as follows. In Sect. 2, we establish some conditions on the data g and h to obtain that the solution u(x, t) exists globally in D ⊂ R N (N ≥ 2). In Sect. 3, we develop conditions on the data of (1.1) to guarantee the blow-up of the solution and derive an upper bound of blow-up time in D ⊂ R N (N ≥ 2). When blow-up does occur, we derive a lower bound of t * for D ⊂ R N (N ≥ 3) in Sect. 4 and a lower bound of t * for D ⊂ R 2 in Sect. 5. In Sect. 6, some examples are presented to illustrate the results in this paper.

The global solution
In this section, we establish some conditions on the data of (1.1) to ensure that the global solution exists. We define the auxiliary functions Now we state our main results as follows.
where ζ , γ , and q are some positive constants. Assume Then u(x, t) exists for all t > 0 in the measure Φ(t).

It follows from (2.12) and (2.4) that
, we obtain that Φ (t) < 0 in some interval (t 0 , t * ). So, for any t ∈ [t 0 , t * ), we get Φ(t) ≤ Φ(t 0 ). We take the limit as t → t *to get This is a contradiction.

Upper bounds of the blow-up time t *
In this section, we set up some conditions on g and k to guarantee that the solution of (1.1) blows up in finite time. An upper bound of the blow-up time t * is obtained in D ⊂ R N (N ≥ 2). The auxiliary functions of this section are defined as follows: Our main result is the following Theorem 3.1.

Theorem 3.1 Let u(x, t) be a nonnegative classical solution of problem (1.1). Suppose that the function g satisfies
Then the solution u(x, t) must blow up in the measure Ψ (t) in finite time t * and Proof Making use of the divergence theorem, we get By the Hölder inequality and (3.3), we can compute Substituting (3.6) into (3.5) and applying the Young inequality and (3.2)-(3.4), we deduce (3.7) We know that (3.3) implies In fact, if (3.8) does not hold, we let From (3.9), we have It follows from (3.7) and (3.10) that Ψ (t) > 0, 0 < t < t , and Ψ (t ) > Ψ (0). By (3.3), we derive which contradicts with (3.9). Integrating (3.7) over [0, t], we obtain which implies that the solution u must blow up at some finite time t * in the measure Ψ (t).
In fact, if u does not blow up at t * in the measure Ψ (t), we get Letting t → +∞, we have +∞ Hence, u blows up at t * in the measure Ψ (t). We pass to the limit as t → t *in (3.11) to derive 4 Lower bounds of the blow-up time t * in D ⊂ R N (N ≥ 3) In this section, we impose restriction D ⊂ R N (N ≥ 3). Assume that the functions h and g satisfy where σ , q, and ξ are some positive constants and It follows from [17, Corollary 9.14] that from which we have the following Sobolev inequality: We define auxiliary functions as follows: Now we present our main results in Theorem 4.1.

Theorem 4.1 Let u(x, t) be a nonnegative classical solution of
Suppose that (4.1)-(4.2) hold and If u(x, t) becomes unbounded at some finite time t * in the measure A(t), then we conclude t * is bounded from below by , (4.8) (4.10)

Applications
In what follows, three examples are given to demonstrate the results of Theorems 2.1-5.1 obtained in this paper.

Conclusion
In this paper, we derive the global existence and bounds for the blow-up time of nonlinear parabolic problem (1.1) with nonlocal source. To deal with nonlocal source, we must establish some new auxiliary functions different from those in [1,2] and [11]. Furthermore, to obtain the lower bound of the blow-up time in D ⊂ R N (N ≥ 3) and D ⊂ R 2 , we need to use the embedding theorems in Sobolev spaces W 1,2 → L 2N N-2 , N ≥ 3 and W 1,2 → L 4 , N = 2, respectively. Applying these auxiliary functions, the embedding theorems in Sobolev spaces, and the differential inequality technique, we complete our study with the blow-up and global solutions of problem (1.1).