A class of fourth-order parabolic equation with logarithmic nonlinearity

In this paper, we study a class of fourth-order parabolic equation with the logarithmic nonlinearity. By using the potential well method, we obtain the existence of the unique global weak solution. In addition, we also obtain results of decay and blow-up in the finite time for the weak solution.

They obtained the existence, uniqueness and blow-up of solutions. Liu and Liu [4] considered the following equation: They combine the potential well method, the classical Galerkin method and the energy method to give a threshold result for the global existence and non-existence of signchanging weak solutions to the problem. The relevant equations have also been studied in [5,6].
In this paper, we study a fourth-order parabolic equation with the logarithmic nonlinearity. The second-order parabolic equation with the logarithmic nonlinearity is diffusely studied. Chen, Luo and Liu [7] studied the heat equation with the logarithmic nonlinearity. Ji, Yin and Cao [8] established the existence of positive periodic solutions and discussed the instability of such solutions for the semilinear pseudo-parabolic equation with the logarithmic source. Nahn and Truong [9] studied the following nonlinear equation: u tu t -div |∇u| p-2 ∇u = |u| p-2 u log |u| . (1.3) They obtained results as regards the existence or non-existence of global weak solutions. He, Gao and Wang [10] considered the following equation: u tu t -div |∇u| p-2 ∇u = |u| q-2 u log |u| , (1.4) where 2 < p < q < p(1 + 2 n ), they proved the decay and the finite time blow-up for weak solutions.
In this paper, we prove the existence of the unique global weak solution of the problem (1.1) based on the potential well method. In addition, we also obtain some properties of the solutions. This paper is organized as follows: in Sect. 2, we introduce some lemmas. In Sect. 3, we mainly introduce the existence of the unique local weak solution of the problem (1.1). In Sect. 4, under some conditions, we obtain the existence of the unique global weak solution of the problem (1.1). Meanwhile, we find that the solution is decaying. In the last section, we prove the blow-up theorem.

Lemma 2.2
There exists a u > 0 with u ∈ N such that J(u) = d.

Global existence and decay estimates
Now as in [9], we introduce the following sets: Definition 4.1 (Maximal existence time) Let u(t) be a solution of problem (1.1). We define the maximal existence time T max as follows: Then: (i) if T max < +∞, we say that u(t) blows up in finite time and T max is the blow-up time; (ii) if T max = +∞, we say that u(t) is global. Furthermore, if u 0 ∈ W + 1 , the solution u(t) decays exponentially.
Proof We will consider the following two cases. First we address the case of the initial data u 0 ∈ W + 1 . where T max is the maximal existence time of solution u m (x, t). We will prove that T max = ∞. By for sufficiently large m. Next, we will study u m (t) ∈ W + 1 , t ∈ [0, T max ), (4.5) for sufficiently large m. We assume that (4.5) does not hold and think that there exists a smallest time t 0 such that u m (t 0 ) / ∈ W + 1 . Then, we have u m (t 0 ) ∈ ∂W + 1 . So, we have J u m (t 0 ) = d, (4.6) or I u m (t 0 ) = 0.
Using (4.8) and combining with the Poincaré inequality, we have where C 21 and C 22 are the Poincaré constants. By (4.4) and (4.8), we have Now we address the case of the initial data u 0 ∈ W + 2 .
Since u 0 ∈ W + 1 , similar to the first case, we obtain u(t) ∈ W + 1 for any t ∈ [0, ∞). By (2.3) and (4.1), we obtain By I(u(t)) > 0, (2.4) and Lemma 2.1, there exists a λ 3 > 1 such that I(λ 3 u(t)) = 0. We have (4.14) Using (4.13) and (4.14), we have which implies that (4.15) It follows from (2.2) that (4.16) Using (4.15) and (4.16), we have which implies that Proof Since u 0 ∈ W -1 , it follows from the local existence that there exists a unique local weak solution u(t) of the problem (1.1) such that Next, we prove u(t) ∈ W -1 for t ∈ [0, T max ]. We assume u(t) leaves W + 1 at time t = t 1 , then there exists a sequence {t n } such that I(u(t n )) ≤ 0 as t n → t -1 . It follows from lower semicontinuity of L 2 norm that I u(t 1 ) ≤ lim inf n→∞ I u(t n ) ≤ 0. (5.2) We have I(u(t 1 )) = 0 according to u(t 1 ) / ∈ W + 1 . By (2.4) and (5.1), we have which is a contradiction. So, u(t) ∈ W -1 for t ∈ [0, T max ]. Next, we will study that u(t) blows up in finite time by contradiction. Thus, we assume u(t) is global. We contract a function Φ : [0, ∞) → R + , and This is a contradiction to our assumption.