Global Existence and Decay Estimates of Energy of Solutions for a New Class of 
 p
 -Laplacian Heat Equations with Logarithmic Nonlinearity

The present research paper is related to the analytical studies of 
 
 p
 
 -Laplacian heat equations with respect to logarithmic nonlinearity in the source terms, where by using an efficient technique and according to some sufficient conditions, we get the global existence and decay estimates of solutions.


A Brief History and Contribution
Consider the following nonlinear p-Laplacian problem: equation with logarithmic nonlinearity: where Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, the function u 0 is given initial data and exponent p verify 2 < p<∞, if n ≤ p, 2 < p < np n − p , if n > p: In the last few decades, the researchers have shown significant interest in polynomial nonlinear terms in different areas, such as edge detection, viscoelasticity, engineering, electromagnetic, electrochemistry, cosmology, signal processing material science, turbulence, diffusion, physics, and acoustics. Many other problems in applied sciences are also modeled by linear and nonlinear evolutionary partial differential equations [1][2][3][4][5][6][7][8][9][10][11][12][13]. Various dynamical systems in physics and engineering are also modeled by using evolutionary differential equations. Many researchers have contributed a lot to provide an outstanding history of the evolutionary differential partial equations related to pðxÞ-Laplacian such as [13][14][15][16][17].
The majority of problems in science are nonlinear, and it is not easy to find its analytical solutions. The physical problems are mostly designed by using higher nonlinear partial differential equations (PDEs). It is found to be very difficult to find the exact or analytical solutions for such problems.
However, in the last several centuries, many scientists have made significant progress and adopted different techniques to study the analytical side of the nonlinear PDEs. Through recent years and in the literature on nonlinear PDEs, logarithmic nonlinearity has received much interest from mathematicians and physicists. If we read in recent research, we notice that logarithmic nonlinearity has been entered into nonrelativistic wave equations that describe spinning particles that move in an external electromagnetic field and in the relativistic wave equation for spinless particles (see, for example, [2,4,18,19]). In addition to what we mentioned above, this type of nonlinearity is used in various branches of physics such as optics, nuclear physics, geophysics, and inflationary cosmology (to read about this in detail, see [18][19][20][21][22][23][24][25][26][27][28][29][30][31]). Given all the basic previous meanings in physics, the study of universal solutions of this type of nonlinear logarithms is of great interest on the part of mathematicians.
Recently, Wu and Xue in [32] gave the uniformly proof of energy decay of the solution using the multiplier method of the following problem: Moreover, the author in [33] studied the exponential and polynomial decay rate of solutions for seminar problem (3) by applying the inequality of Nakao.
On the another handle, for a Laplacian parabolic equation related to the logarithmic in the right-hand side, the authors in [24] gave the analytical side of the following problem: Then, in [27], Nhan and Truong studied the global existence, decay together with the blow up the solutions of the following problem: where p > 2: In addition, in [25], Cao and Liu gave for 1 < p < 2, the blow up and global boundedness results of problem (5).
Most recently, in [14], Piskin et al. studied the p-Laplacian hyperbolic case Motivated by the last mentioned papers, especially [14], in this current research, we consider problem (1) with the presence of nonlinear diffusion Δ p = div ðj∇uj p−2 ∇uÞ, logarithmic nonlinearity juj p−2 u ln juj together with a damping term which is an extension of the previous recent analytical study in [14], where the authors considered the hyperbolic case without damping terms. Our goal is to exploit a potential well method for problem (1) in order to obtain global existence and decay estimate of solutions. More precisely, we give the global existence and decay estimates of solutions under some sufficient conditions.

Preliminaries
In this section, we put the definitions and lemmas that we need in the rest of the paper: for 1 < p < ∞: We denote the positive constants by C and C i (i = 1, 2, ⋯).
for every q ∈ ½1,∞ if n ≤ p, and 1 ≤ q ≤ np/ðn − pÞ if n > p: We choose constant B q,p related only on Ω, p and q: Denote B p,p by B p : where C > 0, and we have the following: (ii) For n > 1 and p < n,q ∈ ½r, ðnp/n − pÞ if r ≤ np/n − p and q ∈ ½r, ðnp/n − pÞ if r ≤ np/n − p

Result of the Global Existence
We give in this section the proof of the global existence for (1). First, putting the following functionals: Hence, (21) and (22) give and we have As in [35], the potential depth of the well is given as Hence, two sets can be assigned, the first stable W and the second V unstable by where Proof.
(i) From gðλÞ which we get 3 Journal of Function Spaces According to kuk p p ≠ 0, we find lim λ→∞ gðλÞ = −∞, and lim λ→0 gðλÞ = 0: (ii) From the derivative of gðλÞ, we get There exists a unique λ * verify ðd/dλÞJðλuÞj λ=λ * , by taking Of course, we note that the recent property is the result of the following: Thus, we have the desired results such that Proof. According to inequality of logarithmic Sobolev, it can be found Selecting μ = p in (34) gives Thus, we have (i) If 0 < kuk p < l, then IðuÞ > 0 using the last inequality (ii) Suppose that IðuÞ < 0: This is due to (35), and it (iii) Similar to the proof of (ii), we prove (iii) As for functional J, it represents the Nehari manifold Using Lemma 7 in order to prove that ℵ is an unempty set, consider that if u ∈ ℵ, we obtain We use (23). Further, it proves that J is coercive with respect to ℵ: In addition, if we give Ω 1 and Ω 2 such that Journal of Function Spaces From Remark 5, we can get that where ζ > 0: Under Lemma 6, we get Choosing ζ < p 2 /n, we obtain By using Young's inequality together with (41), we get where ε > 0 and β = ð1 − αÞðp + ζÞ/p − αðp + ζÞ > 1: As u ∈ ℵ, by (22) and (44), we get Select ε < 1. Then, combining (38) and (44), we find Hence, the coercivity of J on ℵ.

Lemma 9.
(i) The depth of the potential well is given by where L p is given as in Lemma 2 (iii) There exists a positive function u ∈ ℵ, verify JðuÞ = d Proof.
(i) According to Lemma 7, it implies that for every u ∈ W 1,p 0 ðΩÞ \ f0g, there exists a λ * , verify Iðλ * uÞ = 0, that is λ * u ∈ ℵ: Using (47) gives From Lemma 7, the maximizer of JðλuÞ is exact λ * , such that By the combination of (50) and (49), we find So that, as u ∈ W 1,p 0 ðΩÞ \ f0g, we have d ≠ 0: And if u ∈ ℵ by (30), we obtain that λ * is the only critical point in ð0, ∞Þ of the mapping gðλÞ: Therefore, for any u ∈ ℵ: Then, By (51) By using (50) and (54), we get According to (i), we find that d ≥ K: Hence, we have fju k jg ∞ k ⊂ u ∈ ℵ is also a minimizing sequence for J due to ju k j ⊂ u ∈ ℵ and Jðju k jÞ = Jðu k Þ: For this, we can suppose that u k > 0 a.e. Ω for any k ∈ u ∈ ℵ: From it, we note that J is coercive on u ∈ ℵ; in other words, fu k g ∞ k is bounded in W 1,p 0 ðΩÞ: Since W 1,p 0 ðΩÞ°L p ðΩ Þ is compact embedding, ∃u is a function and a subsequence of fu k g ∞ k , still given by fu k g ∞ k , such that Hence, u ≥ 0 on Ω and We apply Lebesgue dominated convergence theorem and weak lower semicontinuity.
As u k ∈ u ∈ ℵ, we have u k ∈ W 1,p 0 ðΩÞ \ f0g and Iðu k Þ which implies According to Lemma 8, we have kuk p ≠ 0 converge strongly in L p ðΩÞ; that is to say, that u ∈ W 1,p 0 ðΩÞ \ f0g: Moreover, using weak lower continuity, we find As a final stage of proof (iii), we prove that IðuÞ = 0: If this is false, we get IðuÞ < 0 ; hence, by Lemma 7, ∃λ * < 1 which verifying Iðλ * uÞ = 0: Further, we find And it produces a stark contrast. Meaning that the proof of Lemma 9 has ended. Definition 10. We say that function uðtÞ represents a weak solution to problem (1) on Ω × ½0, TÞ, if satisfies Lemma 11. Let u 0 ∈ W 1,p 0 ðΩÞ \ f0g and l = e ðn+p 2 Þ/p 2 ðp 2 /nL p Þ n/p 2 : Suppose that 0 < Eð0Þ < l p /p 2 < d: such that T is the maximum time of existence of uðtÞ: Proof.
According to (64), (b) is impossible, that is, Iðu 0 Þ = 0 and kΔðu 0 Þk ≠ 0. But it is ∃Jðu 0 Þ ≥ d if 0 < d = inf u∈ℵ JðuÞ: From this, we have a stark contrast, uðtÞ ∈ W is obtained for ∀t ∈ 0, TÞ: (ii) In the same way, we prove case (ii) Proof. Consider the orthogonal basis fw j g ∞ j=1 of the "separable" space W 1,p 0 ðΩÞ which is orthonormal in L 2 ðΩÞ: Let the following subspace V m on the finite dimensional where the projections of the initial data be defined by for all j = 1, 2, ⋯, m: Now, we can see the approximated solutions of (1) as in the following form of the approximate problem in V m It produces an ordinary differential equation system (ODE) made up of unknown functions h m j ðtÞ. Starting from the standard theory of existence, there are functions According to (85), we get Setting c 2 = εββ 2 > 0 and integrating (97) yield Finally, by (85), we obtain (87). This is the end of the proof.

Conclusion
As mentioned earlier in the introduction, the majority of problems in science are nonlinear and their analytical solutions are not easy to find, and most physical problems mostly use higher nonlinear partial differential equations (PDEs). It has been found to be extremely difficult to find accurate or analytical solutions to such problems. However, in the past several centuries, many scientists have made great progress and adopted various techniques to study the analytical side of these famous problems, and nonlinear logarithmic has also received much attention from physicists and mathematicians. Log nonlinearity was introduced into the relativistic wave equation describing spinning particles moving in an external electromagnetic field and in the relativistic wave equation (see, for example, [1-3, 6, 14, 18, 19, 29, 36, 37]); in this contribution, under some sufficient initial and boundary conditions, we have studied the analytical side of p -Laplacian heat equations with respect to logarithmic nonlinearity in the right-hand side, where the global existence and decay estimates of weak solutions are proved. In the next work, we extend our recent work to the coupled system for this important problem. Also, some numerical examples will be given in order to ensure the theory study by using some famous algorithms which are presented in [38,39].

Data Availability
No data were used to support the study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.