Blow-Up Solution of a Porous Medium Equation with Nonlocal Boundary Conditions

In this paper, we devote to studying the blow-up phenomena for a porous medium equation under nonlocal boundary conditions. Based on auxiliary functions and differential inequality technique, we derive the sufficient conditions to guarantee the existence of blow-up solutions under different measures and obtain an upper bound for blow-up time. Moreover, we demonstrate the lower bounds for blow-up time under some appropriate measures in R3 and in the higher-dimensional space R(n≥ 3), respectively. At last, two examples are given to illustrate the applications of main results.


Introduction
As is known to all, porous medium equations usually describe processes involving fluid flow, heat transfer, or diffusion.
ere are a lot of applications in physical phenomena, mathematical biology, lubrication, boundary layer theory, and other fields [1,2]. e paper is focused on the following porous medium equation with time-dependent coefficient: (1) subject to nonlocal boundary and initial conditions: where m > 1 and Ω ⊂ R n (n ≥ 3) is a bounded convex domain with smooth boundary zΩ. zu/z] is the outward normal derivative on zΩ , and t * is the maximal existence time of u. Set R + � (0, +∞). roughout the paper, we assume that f, g ∈ C 1 (R + ) are nonnegative, the weight function a ∈ C(Ω) is positive, and k 1 , k 2 ∈ C 1 (R + ) are positive functions. u 0 ∈ C(Ω) is the initial value which satisfies compatibility conditions. By the degenerate parabolic theory in [3], one can deduce that the local classical solution of (1) exists uniquely and is nonnegative.
During the past decades, there have been many works to study the blow-up phenomena of parabolic equations and systems. e related results include the existence of blow-up and global solutions, the bounds of the blow-up time, and blow-up rate. We refer to [4][5][6][7][8][9][10][11][12] and references therein. Porous medium equations as representative examples of parabolic equations have been widely investigated by many scholars [13][14][15][16][17]. In order to research problem (1), the recent papers have aroused our interest. Xiao and Fang [14] considered the following porous medium equation under nonlinear boundary conditions: where m ≥ 1 and Ω ⊂ R n (n ≥ 2) is a bounded star-shaped region with smooth boundary zΩ. Using the auxiliary function method and modified differential inequality technique, they established some conditions on time-dependent coefficient and nonlinear functions to guarantee that the solution u(x, t) exists globally or blows up at some finite time. Moreover, the upper and lower bounds for the blow-up time were derived in the higher-dimensional space.
In [17], the authors dealt with the blow-up problem of the following porous medium equation subject to nonlinear boundary and initial conditions: where m > 1 and Ω ⊂ R n (n ≥ 2) is a bounded convex region with smooth boundary zΩ. Under appropriate assumptions on the data, a criterion was given to guarantee that solution u blows up at finite time, and an upper bound and a lower bound for blow-up time were derived. Recently, there are some papers on the issue of studying reaction-diffusion equations under nonlocal boundary conditions. Marras andVernier Piro [18] studied the following reaction-diffusion equation under nonlocal boundary conditions: where Ω is a bounded convex region in R n (n ≥ 2) with smooth boundary zΩ. Under some conditions on data, the authors showed the solution must blow up at finite time t * . At the same time, they obtained upper bounds of t * . When Ω ⊂ R 3 , lower bounds of t * were derived. In [19], the authors were concerned about the following equations: where Ω is a bounded convex region in R n (n ≥ 2) and the boundary zΩ is smooth. By constructing some auxiliary functions and using differential inequality technique, they derived that the solution blows up at some finite time.
Moreover, upper and lower bounds of the blow-up time were obtained.
In this paper, our main interest lies in blow-up phenomena of problem (1). Obviously, boundary conditions (2) and (3) are different from the one in [14,17], and the main part of equation (1) is different from the problem in [18,19]. erefore, we need to establish the new auxiliary functions to prove our main results. By means of new auxiliary functions and differential inequality technique, we prove the existence of blow-up solutions under different measures and obtain an upper bound for blow-up time. Moreover, we demonstrate the lower bounds for blow-up time under some appropriate measures in R 3 and in the higher-dimensional space R n (n ≥ 3), respectively. e rest of this paper is arranged as follows. In Section , a criterion for the blow-up solutions of problems (1)- (3) is established, and we obtain an upper bound for blow-up time. Section 3 shows that a lower bound is presented under two different measures when blow-up does occur. In Section 4, two examples are given to illustrate our main results.

Blow-Up Solution and Upper Bound for t *
e section shows the blow-up solution for problem (1) in two different measures. And we get the sufficient conditions to ensure an upper bound for the blow-up time under two different measures. Suppose that the functions k 1 , a, f satisfy where α, β, and c are positive constants, and they satisfy To obtain our first result, we define the following auxiliary function: Let λ 1 be the first positive eigenvalue and ω 1 be the corresponding eigenfunction of the following problem: with Ω ω 2 dx � 1.
Using Hölder inequality again, we have We substitute (21) into (20) to get Now, we prove that According to (22) that is, Since k 1 (t) is increasing, we get ere is a contradiction. erefore, (23) holds. Similarly, from (16), we deduce Next, applying (21) to (22), we obtain where (23) and (28) are used. Integrating (29) from 0 to t yields Complexity From (30), we can conclude that the solution u blows up at some finite time t * in the measure Φ(t). In fact, assume that u remains global in the measure Φ(t); then, Φ(t) < + ∞, t > 0, and (32) that is, it is a contradiction. en, the solution u blows up at some finite time t * in the measure Φ(t). Taking the limit as t ⟶ t * in (30), we obtain In the following corollary, we will present the blow-up solution for (1) in the measure Ψ(t), where From the proof similar to eorem 1, we get the following conclusion: □ Corollary 1. Let u be a nonnegative classical solution of problem (1). Assume that (8)- (10) and the following assumptions hold: en, u blows up at some finite time T in the measure Ψ(t), and 3. Lower Bound for t * e section presents lower bounds for blow-up time t * under two different measures when the solution of (1)-(3) blows up at some finite time. Firstly, we state two lemmas from [20] and some inequalities, which play a basic role in the process of our proof.

Lemma 1. Let Ω be a bounded star-shaped domain in
where u is a nonnegative C 1 -function, ρ 0 � min zΩ x · ], d � max Ω |x|, and ] is the unit outward normal vector on zΩ.

Lemma 2.
Let Ω be a bounded domain in R 3 assumed to be star-shaped and convex in two orthogonal directions. en, we have where u is the nonnegative C 1 -function, n ≥ 1. ρ 0 and d have the same meaning as in Lemma 1.
In the proof process of the main results, we need to use the following inequality: and the following Sobolev inequality (n ≥ 3) given in [21]: where C is a constant depending on Ω and n, that is, and Young's inequality yields that For convenience, we suppose that the following assumptions on functions a, f, g are satisfied: In the sequel, we restrict that Ω ⊂ R 3 , and we define the new auxiliary function: where μ, θ are positive constants, and Now, we show our main results.

Theorem 2.
Let u be a nonnegative classical solution of problem (1). Suppose that (44)-(45) hold, and the following conditions are satisfied: (49) Assume that u becomes unbounded in the measure A(t) at some finite time t * . en, t * is bounded below by where where ε i (i � 1, 2, 3, 4) are defined by the following equalities: |Ω| is the volume of Ω, ρ 0 � min zΩ x · ], and d � max Ω |x|.
Proof. With the help of (44), (45), (48) and the divergence theorem, we have In the following, we deal with the second term of the right-hand side of (58). Owing to Lemma 1, we have From Hölder inequality, we have where 0 < (2m − 4/θ) < 1. According to Lemma 2, we have where Hölder inequality is used. en, after inserting (59) into (57), inequality (41) yields anks to Hölder inequality, the last term on the righthand side of (56) satisfies Similarly, from (57)-(60), we can get en, combining (61) and (62), we have where inequality (41) is applied again. According to (60), (63), and Hölder inequality, the second term of the righthand side of (55) is changed into Now, we estimate the last term of the right-hand side of (55). Similarly, from (57)-(60), we get 8 Complexity en, we deal with the terms in (64) and (65) including where ε i (i � 1, 2, . . . , 4) are defined by (56). erefore, we substitute (64)-(69) into (55) and get From (47)-(50), note that where F 1 (t), F 2 (t) are given by (52) and (53). Obviously, we can get Integrating (73) from 0 to t, we obtain at is, Taking the limit as t ⟶ t * in (75), we get Note that t 0 F 2 (s)e 2 s 0 F 1 (τ)dτ ds is a strictly increasing function, and we can conclude that en, we finish the proof. And we assume that Ω ⊂ R n (n ≥ 3) is a smooth bound domain. e functions k 1 and k 2 satisfy where η 1 and η 2 are some positive constants. Now, we define the new auxiliary function: where δ is a positive constant, and it satisfies where where Q i (i � 1, 2, . . . , 7) are defined by the following equalities: 10 Complexity Here, Complexity |Ω| is the volume of Ω, ρ 0 � min zΩ x · ], and d � max Ω |x|.
which is an upper bound for the blow-up time.