In this paper, we devote our attention to the singularity analysis of the following semilinear parabolic system:
with nonlocal boundary condition
and initial data
where is a bounded connected domain with smooth boundary and are positive parameters. Most physical settings lead to the default assumption that the functions defined for are nonnegative and continuous, and that the initial data , are nonnegative, which are mathematically convenient and currently followed throughout this paper. We also assume that satisfies the compatibility condition on , and that and for any for the sake of the meaning ofnonlocal boundary.
Over the past few years, a considerable effort has been devoted to studying the blowup properties of solutions to parabolic equations withlocal boundary conditions, say Dirichlet, Neumann, or Robin boundary condition, which can be used to describe heat propagation on the boundary of container (see the survey papers [1, 2]). For example, the system (1.1) and (1.3) with homogeneous Dirichlet boundary condition
has been studied extensively (see [3–5] and references therein), and the following proposition was proved.
Proposition 1.1.

(i)
All solutions are global if , while there exist both global solutions and finite time blowup solutions depending on the size of initial data when (See [4]). (ii) The asymptotic behavior near the blowup time is characterized by
for some (See [3, 5]).
For the more parabolic problems related to the local boundary, we refer to the recent works [6–9] and references therein.
On the other hand, there are a number of important phenomena modeled by parabolic equations coupled with nonlocal boundary condition of form (1.2). In this case, the solution could be used to describe the entropy per volume of the material [10–12]. Over the past decades, some basic results such as the global existence and decay property have been obtained for the nonlocal boundary problem (1.1)–(1.3) in the case of scalar equation (see [13–16]). In particular, for the blowup solution of the single equation
under the assumption that , Seo [15] established the following blowup rate estimate
for any . For the more nonlocal boundary problems, we also mention the recent works [17–22]. In particular, Kong and Wang in [17], by using some ideas of Souplet [23], obtained the blowup conditions and blowup profile of the following system:
subject to nonlocal boundary (1.2), and Zheng and Kong in [22] gave the condition for global existence or nonexistence of solutions to the following similar system:
with nonlocal boundary condition (1.2). The typical characterization of systems (1.8) and (1.9) is the complete couple of the nonlocal sources, which leads to the analysis of simultaneous blowup.
To our surprise, however, it seems that there is no work dealing with singularity analysis of the parabolic system (1.1) with nonlocal boundary condition (1.2) except for the single equation case, although this is a very classical model. Therefore, the basic motivation for the work under consideration was our desire to understand the role of weight function in the blowup properties of that nonlinear system. We first remark by the standard theory [4, 13] that there exist local nonnegative classical solutions to this system.
Our main results read as follows.
Theorem 1.2.
Suppose that . All solutions to (1.1)–(1.3) exist globally.
It follows from Theorem 1.2 and Proposition 1.1(i) that any weight perturbation on the boundary has no influence on the global existence when , while the following theorem shows that it plays an important role when . In particular, Theorem 1.3(ii) is completely different from the case of the local boundary (1.4) (by comparing with Proposition 1.1(i)).
Theorem 1.3.
Suppose that .

(i)
For any nonnegative and , solutions to (1.1)–(1.3) blow up in finite time provided that the initial data are large enough.

(ii)
If , for any , then any solutions to (1.1)–(1.3) with positive initial data blow up in finite time.

(iii)
If , for any , then solutions to (1.1)–(1.3) with small initial data exist globally in time.
Once we have characterized for which exponents and weights the solution to problem (1.1)–(1.3) can or cannot blow up, we want to study the way the blowing up solutions behave as approaching the blowup time. To this purpose, the first step usually consists in deriving a bound for the blowup rate. For this bound estimate, we will use the classical method initially proposed in Friedman and McLeod [24]. The use of the maximum principle in that process forces us to give the following hypothesis technically.
(H)There exists a constant, such that
However, it seems that such an assumption is necessary to obtain the estimates of type (1.5) or (1.10) unless some additional restrictions on parameters are imposed (for the related problem, we refer to the recent work of Matano and Merle [25]).
Here to obtain the precise blowup rates, we shall devote to establishing some relationship between the two components and as our problem involves a system, but we encounter the typical difficulties arising from the integral boundary condition. The following theorem shows that we have partially succeeded in this precise blowup characterization.
Theorem 1.4.
Suppose that , , , and assumption (H) holds. If the solution of (1.1)–(1.3) with positive initial data blows up in finite time , then
where are both positive constants.
Remark 1.5.
If and , then Theorem 1.4 implies that for the blowup solution of problem (1.6), we have the following precise blowup rate estimate:
which improves the estimate (1.7). Moreover, we relax the restriction on .
Remark 1.6.
By comparing with Proposition 1.1(ii), Theorem 1.4 could be explained as the small perturbation of homogeneous Dirichlet boundary, which leads to the appearance of blowup, does not influence the precise asymptotic behavior of solutions near the blowup time and the blowup rate exponents and are just determined by the corresponding ODE system . Similar phenomena are also noticed in our previous work [18], where the single porous medium equation is studied.
The rest of this paper is organized as follows. Section 2 is devoted to some preliminaries, which include the comparison principle related to system (1.1)–(1.3). In Section 3, we will study the conditions for the solution to blow up and exist globally and hence prove Theorems 1.2 and 1.3. Proof of Theorem 1.4 is given in Section 4.