Blow-up analysis for parabolic p-Laplacian equations with a gradient source term

where p > 2, the spatial domainΩ ⊂ RN (N ≥ 2) is bounded, and the boundary ∂Ω is smooth. Our research relies on the creation of some suitable auxiliary functions and the use of the differential inequality techniques and parabolic maximum principles. We give sufficient conditions to ensure that the solution blows up at a finite time t∗. The upper bounds of the blow-up time t∗ and the upper estimates of the blow-up rate are also obtained.

As far as I know, there are many papers on the blow-up problems of parabolic equations with a gradient term (see, for instance, [11][12][13][14][15][16][17][18][19][20][21][22][23]). For the sake of research (1.1), we mainly focus on the papers [12,22]. In [22], Zhang et al. researched the following problems: (1.2) In (1.2), p ≥ 2, the spatial domain Ω ⊂ R N (N ≥ 2) is bounded, and the boundary ∂Ω is smooth. They derived some conditions which ensure the solution of (1.2) blows up in a finite t * or exists globally. Moreover, an upper bound and a lower bound of the blow-up time were also specified when the blow-up occurs. Their research relied on using differential inequality techniques. In [12], Ding et al. studied the following problems: is bounded, and the boundary ∂Ω is smooth.
By combining the parabolic maximum principles and the differential inequality techniques, they obtained sufficient conditions for the existence of the blow-up solution and the global solution of (1.3). In addition, the upper bounds of the blow-up time t * and the upper estimates of the blow-up rate were also given.
Inspired by the above two research works, in this paper we study the blow-up solutions of (1.1). Since the source term of the equation in (1.1) contains a gradient term, and the source term of the equation in (1.2) does not, the research methods in [22] cannot be used to research (1.1). In this paper, we use the method in [12] to study (1.1). In other words, we combine the parabolic maximum principles with the differential inequality techniques to study (1.1). The difficulty in using this research method is that some suitable auxiliary functions need to be constructed. We note that although the source terms in equations of (1.1) and (1.3) are the same, the boundary conditions are different. Therefore, the auxiliary functions in the paper [12] are not suitable for researching (1.1). In this paper, we need to construct some auxiliary functions that are completely different from those in [12] to study the blow-up solution of (1.1). We give sufficient conditions to guarantee that the solution of (1.1) blows up at a finite time t * . The upper bounds of the blow-up time t * and the upper estimates of the blow-up rate are also obtained.
For convenience, in this paper we use a comma to represent partial derivative and adopt the summation convention, for example,

The main result and its proof
Due to the need to study the blow-up solution of (1.1), we define the following two constants: where u 0 is the initial value of (1.1). For the same purpose, we also construct two auxiliary functions as follows: which implies that the function H has an inverse function H -1 . The following Theorem 2.1 is the main result of the blow-up solution to (1.1).
We first study the first case. By (2.1), we derive Then, we study the second case. With (2.5), we have Finally, we study the third case. Using of the boundary condition of (1.1), we obtain It follows from parabolic maximum principles and (2.22)-(2.24) that the minimum value from which we get the following differential inequality: At the pointx ∈ Ω where u 0 (x) = M 0 , we integrate (2.25) from 0 to t to get It follows from (2.26) that u must blow up at some finite time t * . In fact, assuming that the solution u does not blow up, we have, for any t > 0, Letting t → +∞ in (2.27), we derive which contradicts (2.6). Hence, u must blow up at some finite time t * . Furthermore, letting t → t * in (2.26), we obtain At each fixed point x ∈ Ω, we integrate (2.25) from t toť (0 < t <ť < t * ) and use (2.4) to deduce The proof is complete.

Conclusions
In this paper, we combine parabolic maximum principles with the differential inequality techniques to study the blow-up solution of problem (1.1). The key to our research is constructing two auxiliary functions (2.3) and (2.4). With their help, we obtain sufficient conditions for the existence of the blow-up solution of problem (1.1). In addition, we also give an upper bound on the blow-up time and an upper estimate of the blow-up rate.