Global existence and decay of solutions for p-biharmonic parabolic equation with logarithmic nonlinearity

: In this paper, we study the initial boundary value problem for a p-biharmonic parabolic equation with logarithmic nonlinearity. By using the potential wells method and logarithmic Sobolev inequality, we obtain the existence of the unique global weak solution. In addition, we also obtain decay polynomially of solutions.

They obtained the global existence and blow-up of solutions. Also, they discussed the upper bound of blow-up time under suitable conditions. Nhan and Truong [4] studied the following nonlinear pseudo-parabolic equation They obtained results as regard the existence or non-existence of global solutions. Also, He et al., [5] proved the decay and the finite time blow-up for weak solutions of the equation. Cao and Liu [6] studied the following nonlinear evolution equation with logarithmic source u t − ∆u t − div |∇u| p−2 ∇u − k∆u t = |u| p−2 u ln |u| .
They established the existence of global weak solutions. Moreover, they considered global boundedness and blowing-up at ∞.
Wang and Liu [7] considered the following p-biharmonic parabolic equation with the logarithmic nonlinearity u t + ∆ |∆u| p−2 ∆u = |u| q−2 u ln |u| They studied existence of weak solutions by potential well method, blow up at finite time by concative method.
It is necessary to note that prence of the logarithmic nonlinearity causes some difficulties in deploying the potantial well method. In order to handle this situation we need the following logarithmic Sobolev inequality which was introduced by ([4, 21,22]).

Proposition 1.
Let u be any function in H 1 (R n ) and µ > 0 be any number. Then where
Let us introduce the energy functional J and Nehari functional I defined on X 0 as follow and By (3) and (4), we get be the Nehari manifold. Thus, we may define d is positive and is obtained by some u ∈ N . Then it is obvious that From [4], we know d ≥ M.
The local existence of the weak solutions can be obtained via the standard parabolic theory. It is easy to obtain the following equality Lemma 1. Let u ∈ X 0 . Then we possess , decreasing on (λ * , +∞) and attains the maximum at λ * ; Proof. For u ∈ X 0 , by the definition of j, we get It is clear that (i) holds due to Ω |u| q dx ̸ = 0. We have Hence, there exists a is increasing on (0, λ * ), decreasing on (λ * , +∞) and attains the maximum at λ * . So (iii) holds. The last property, (iv), is only a simple corallary of the fact that Thus, I(λu) > 0 for 0 < λ < λ * , I(λu) < 0 for λ * < λ < +∞ and I(λ * u) = 0. So (iv) holds. The proof is complete.
Next we denote R := p 2 e nL p n/p 2 .
Proof. By the definition of I(u), we get Choosing µ = p, we have property (iii) we can argue similarly the proof of (ii). The proof of lemma is complete.
Here, the constant C depends on n, p, q and r.

Main results
Now as in ( [4]), we introduce the follows sets: Definition 1. (Maximal Existence Time). Assume that u be weak solutions of problem (1). We define the maximal existence time T max as follows Then (i) If T max < ∞, we say that u blows up in finite time and T max is the blow-up time; (ii) If T max = ∞, we say that u is global.
Let u 0m be an element of V m such that as m → ∞. We construct the following approximate solution u m (x, t) of the problem (1) where the coefficients a mj (1 ≤ j ≤ m) satisfy the ordinary differential equations for i ∈ {1, 2, ..., m}, with the initial condition a mj (0) = a mj , j ∈ {1, 2, ..., m}.
We multiply both sides of (11) by a ′ mi , sum for i = 1, ..., m and integrating with respect to time variable on [0, t], we get t 0 ∥u ms (s)∥ 2 where T max is the maximal existence time of solution u m (t). We shall prove that T max = +∞. From (9), (13) and the continuity of J, we obtain J(u m (0)) → J(u 0m ), as m → ∞, Thanks to J(u 0 ) < d and the continuity of functional J, it follows from (14) that And therefore, from (13), we obtain t 0 ∥u ms (s)∥ 2 for sufficiently large m. Next, we will study for sufficiently large m. We assume that (16) does not process and think that there exists a sufficiently small time t 0 such that u m (t 0 ) / ∈ W + 1 . Then, by continuity of u m (t 0 ) ∈ ∂W + 1 . So, we get and Nevertheless, by definition of d, we see that (17) could not consist by (15) while if (18) holds then, we get which also contradicts with (15). Moreover, we have (16), i.e., J(u m (t)) < d, and I(u m (t)) > 0, for any t ∈ [0, T max ), for sufficiently large m. Then, from (5), we obtain and Since u m (x, t) ∈ W + 1 for m large enough, it follows from (5) that J(u m ) ≥ 0 for s large enough. So, by (15) it follows for m large enough t 0 ∥u ms (s)∥ 2 By (20), we know that T max = +∞.
The above inequality implies that the solution u decays polynomially.