Abstract
The present research paper is related to the analytical studies of -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.
1. A Brief History and Contribution
Consider the following nonlinear -Laplacian problem: equation with logarithmic nonlinearity: where is a bounded domain with smooth boundary the function is given initial data and exponent verify
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 [1ā13]. 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 -Laplacian such as [13ā17].
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 [18ā31]). 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 In addition, in [25], Cao and Liu gave for , the blow up and global boundedness results of problem (5).
Most recently, in [14], Piskin et al. studied the -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 , logarithmic nonlinearity 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.
2. Preliminaries
In this section, we put the definitions and lemmas that we need in the rest of the paper: for We denote the positive constants by and ().
We give the function of energy by
Lemma 1. is a nonincreasing function, for
Proof. Multiplying equation (1) by and using the integration on we have Thus,
Lemma 2 (see [5, 14]). Let be any function . Then, for, where
Remark 3. Let and by defining for we can write
Lemma 4 (see [27]). Let . Therefore, we can easy give the following result: such as
Remark 5. According to Lemma 4, we have
Lemma 6 (see [34]). (i)For all function we havefor every if and if We choose constant related only on and Denote by (ii)For every with we getwhere , and we have the following: (i)For (ii)For and if and if (iii)For (iv)For
3. 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 and the second unstable by
Lemma 7. Let be all function and let . Hence, we have (i)(ii)where
Proof. (i)From which we getAccording to we find , and (ii)From the derivative of , we getThere exists a unique verify , by taking Of course, we note that the recent property is the result of the following: Thus, we have the desired results such that
Lemma 8. For every and we get (i)If then (ii)If , then (iii)If , then
Proof. According to inequality of logarithmic Sobolev, it can be found Selecting in (34) gives Thus, we have (i)If then using the last inequality(ii)Suppose that This is due to (35), and it(iii)Similar to the proof of (ii), we prove (iii)As for functional , it represents the Nehari manifold Using Lemma 7 in order to prove that is an unempty set, consider that if , we obtain We use (23). Further, it proves that is coercive with respect to In addition, if we give and such that From Remark 5, we can get that where Under Lemma 6, we get where Choosing , we obtain By using Youngās inequality together with (41), we get where and As , by (22) and (44), we get Select . Then, combining (38) and (44), we find Hence, the coercivity of on .
Lemma 9. (i)The depth of the potential well is given by(ii) admis a positive lower bound, given bywhere is given as in Lemma 2(iii)There exists a positive function , verify
Proof. (i)According to Lemma 7, it implies that for every there exists a , verify that is Using (47) givesFrom Lemma 7, the maximizer of is exact , such that
By the combination of (50) and (49), we find
So that, as we have And if by (30), we obtain that is the only critical point in of the mapping Therefore,
for any Then,
By (51) and (53), (i) is obtained.
(ii)From Lemma 7, , we get Lemma 8 givesBy using (50) and (54), we get
According to (i), we find that (iii)Consider the minimize sequence for , verifyHence, we have is also a minimizing sequence for due to and For this, we can suppose that a.e. for any
From it, we note that is coercive on ; in other words, is bounded in Since is compact embedding, is a function and a subsequence of still given by such that
Hence, on and
We apply Lebesgue dominated convergence theorem and weak lower semicontinuity.
As , we have and which implies
According to Lemma 8, we have converge strongly in ; that is to say, that Moreover, using weak lower continuity, we find
As a final stage of proof (iii), we prove that If this is false, we get hence, by Lemma 7, which verifying Further, we find
And it produces a stark contrast. Meaning that the proof of Lemma 9 has ended.
Definition 10. We say that function represents a weak solution to problem (1) on if satisfies
Lemma 11. Let and Suppose that (i)If then for (ii)If then for ,such that is the maximum time of existence of
Proof. (i)We put is the maximum time of existence of solution . From (24) combined with (47), we findThen, we have for every . If it is false, hence verify , we get either and or (b) .
According to (64), (b) is impossible, that is, and . But it is if From this, we have a stark contrast, is obtained for (ii)In the same way, we prove case (ii)
Theorem 12. Consider . If and or Therefore, problem (1) admits a weak global solution
Proof. Consider the orthogonal basis of the āseparableā space which is orthonormal in Let the following subspace on the finite dimensional
where the projections of the initial data be defined by
for all
Now, we can see the approximated solutions of (1) as in the following form
of the approximate problem in It produces an ordinary differential equation system (ODE) made up of unknown functions . Starting from the standard theory of existence, there are functions
which verify (68) in a maximal interval Next, we prove that and that the local solution is uniformly bounded independent of and . For this purpose, let us replace by in (68) and integrate by parts, we get
such as
Integrating (70) from to and using (24), we obtain
According to (68), with , we find We select large enough; we find
Hence, by (23), we have
By we select large enough and we find By (24) and Lemma 11, by picking large enough and we get Further, according to (24) and (21), we obtain
where By choosing large enough and (76), we get
According to Remark 5, we find
where is pick satisfying as and as and and
Applying the embedding theorem, Lemma 6 and Youngās inequality, gives from (78):
Therefore, we choose for , where with
Using (79) and (76), for we find
Hence, we get
Using the integration on (68), we get for Further, after passing through the limit in (ref 4030), we arrive at the weak solution to the problem (ref 300). According to the initial data in (ref 300), we conclude that in .
4. Decay of Solution
In this section, by using the Lyapunov functional, we show the decay of solution to (1).
First, we define the Lyapunov functional by where . We will prove the equivalence between and .
Lemma 13. For small enough, we have
where
We find by choosing small enough.
Theorem 14. Let . Assume further , where hence, satisfies
Proof. A differentiation of and equation (1) gives Adding and subtracting into (88) (), we obtain Using the inequality of logarithmic Sobolev together with () gives Noting that and using (21) and Theorem 12, we find By satisfying we guarantee then, we obtain Hence, inequality (95) becomes According to (85), we get Setting and integrating (97) yield Finally, by (85), we obtain (87). This is the end of the proof.
5. 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, [1ā3, 6, 14, 18, 19, 29, 36, 37]); in this contribution, under some sufficient initial and boundary conditions, we have studied the analytical side of -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.
Acknowledgments
The fourth author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a research group program under grant (R.G.P-2/1/42).