On a logarithmic wave equation with nonlinear dynamical boundary conditions: local existence and blow-up

This paper deals with a hyperbolic-type equation with a logarithmic source term and dynamic boundary condition. Given convenient initial data, we obtained the local existence of a weak solution. Local existence results of solutions are obtained using the Faedo-Galerkin method and the Schauder ﬁxed-point theorem. Additionally, under suitable assumptions on initial data, the lower bound time of the blow-up result is investigated.


Introduction
In this paper, we study the problem of wave equation with logarithmic nonlinearity and dynamic boundary condition where ⊂ R n , n ≥ 1 is a regular, bounded domain with a boundary ∂ = 0 ∪ 1 , 0 ∩ 1 =, where 0 and 1 are measurable over ∂ , endowed with the (n -1) dimensional Lebesgue measures λ n-1 ( 0 ) and λ n-1 ( 1 ).Additionally, λ n-1 ( 0 ) and λ n-1 ( 1 ) are assumed to be positive throughout paper.k ≥ 2 and p ≥ 2 are positive constants to be chosen later.
Dynamic boundary problems are widely applied in many mathematical models, such hydro logic filtration process, thermoelasticity, diffusion phenomenon, and hydrodynamics [2,15,[25][26][27].A dynamic boundary condition has been introduced by a group of physicists to underline the fact that the kinetics of the process, i.e. the term ∂u ∂n becomes more visible in some boundary conditions [18,24].This type of option is characterized by the interaction of the components of the system with the walls (i.e., within ) [7].Since the paper by Lions [29] has been introduced in 1969, evolution equations with dynamical boundary conditions (first order equations in time) have been studied well.Later, mathematicians and physicists studied it for a long time and achieved creative success; see [3,6,9,17,19,20,22,28,31] and references therein.
In [31], the author considered the problem (1) without logarithmic source term for ∂ ∂n u(x, t) = -|u t | k-2 u t + |u| p-2 u boundary condition and proved the local and global existence under suitable condition.When 2 < p < k, the solutions exist globally for arbitrary initial data.For k < p, solutions blow up.Later, Zhang and Hu [36] considered the blow-up of the solution under the condition E(0) < d when the initial data are in the unstable set.
In [12], they established blow-up results of the solution for a finite time at a critical energy level or high-energy level for the same problem.
Let us go back and look at a wave equation with logarithmic nonlinearity associated with problem (1).In [8], Cazenave and Haraux considered the following equation for the Cauchy problem u ttu = ku ln |u|. ( They studied deeply the existence and uniqueness of the solutions using different techniques.As far as is known, this type of problem has been employed in various areas of physics, such as geophysics, nuclear physics, and optics; see in Bialynicki-Birula and Mycielski [4,5].Moreover, there are many research points devoted to the given problem in different models of hyperbolic wave equation with logarithmic source term [10,13,14,16,21,23,33].Ma and Fang [32] considered problem (2) with strong damping term.They proved decay estimates and blow-up result under the null Dirichlet boundary condition.
In [11], Cui and Chai considered the following equation with acoustic boundary condition.They obtained local existence and uniqueness using the semigroup theory.As far as is known, not many works are related to the logarithmic wave equation with a dynamic boundary condition.According to the studies mentioned above, our work aims to expand the result of wave equation with logarithmic nonlinearity and dynamic boundary conditions.The rest of the work is arranged as follows: In Sect. 2 gives notations and lemmas to illustrate our paper path.Sections 3-4 state the local existence result and potential well of (1).In the last part, we established blow-up result for a lower bound time.

Preliminaries
First, we denote and (u| 0 is in the trace sense).Let T > 0 be a real number and X be a Banach space endowed with norm • x .L p (0, T; X) indicates the space of functions h, which are L p over (0, T) with values in X, which are measurable with h x ∈ L p (0, T).We set the Banach space endowed with the norm .
L ∞ (0, T; X) denotes the space of functions h : (0, T) → X, which are measurable with h x ∈ L ∞ (0, T).We set the Banach space endowed with the norm .
We know that if X and Y are Banach spaces such that X is continuous embedding to Y , then L p (0, T; X) → L p (0, T; Y ) for 1 ≤ p ≤ ∞.
We define the total energy function as By the definition of E(t) on H 1 1 ( ), the initial energy can be considered Lemma 1 [1] (Trace-Sobolev Embedding inequality).Let H 1 0 ( ) → L p ( 1 ) for 2 ≤ p < κ hold, where So that, there is a constant C p that is the smallest nonnegative number, satisfying Proposition 2 Suppose that Lemma 1 holds, we define Lemma 3 E(t) is a nonincreasing function for 0 ≤ s ≤ t ≤ T and Proof By multiplying equation (1) by u t and integrating on , we have By integrating of (7) over (s, t), we have equality (6).
Lemma 4 Let ϑ be a positive number.Then, the inequality holds for A > 0.

Existence of local solution
We will apply the Faedo-Galerkin technique and the Schauder fixed-point theorem.
Theorem 5 There exists T > 0, such that problem (1) has a unique local weak solution u of (1) on (0, T) × .Therefore, and the energy identity ) is decreasing in the first variable.Now, we will give some existence result and lemma used for the proof of Theorem 5. To define the function and show that the fixed point exists, we introduce the following problem: Let the solution v of problem (9) be v = ζ (u).We can see that v corresponds to u and ζ : X T → X T .Lemma 6 Let 2 ≤ p ≤ κ and κ κ-p+1 < k.Assume that u ∈ H 1 0 ( ) and u 1 ∈ L k ( ) hold.Then, there exists a unique weak solution u of (9) on (0, T) × .Therefore, endowed with the norm and the energy identity holds for 0 ≤ s ≤ t ≤ T.
To see the first step of the proof of Lemma 6, we will use the following proposition.The proposition was proved similar to [35].We have some results in [35] as follows: Proposition 7 Let 2 ≤ p ≤ κ and κ κ-p+1 < k.Assume that u ∈ H 1 0 ( ) and u 1 ∈ L k ( ) hold.Then, there is T > 0 and a unique solution v for (9) problem on (0, T) such that, i.e.
and the energy identity holds for 0 ≤ s ≤ t ≤ T. Now, we can state the proof of Lemma 6.
Proof Let {w j } ∞ j=1 be a sequence of linearly independent vectors in X = {u ∈ H 1 0 ( ) : u| 1 ∈ L k ( 1 )} whose finite linear combinations are dense in X.In the event, using the Grahm-Schmidt orthogonalization method, we can conclude {w j } ∞ j=1 to be orthonormal in L 2 ( ) ∩ L 2 ( 1 ).Using some technical mathematical result, we can clearly see that ) and in L 2 ( ).Moreover, there exist u 0m , u 1m ∈ [w 1 , w 2 , . . ., w m ] where w 1 , w 2 , . . ., w m are the span of the vectors such that According to their multiplicity of we denote by {λ i } the related eigenvalues to w 1 , w 2 , . . ., w m .For all m ≥ 1, we will seek an approximate solution (m functions γ im ) such that satisfying the following Cauchy problem where t ≥ 0. In (15), for the first term, we obtain Similarly, For the fourth term, we get Then, we insert ( 16)-( 18) in ( 15) so that (15) yields the following Cauchy problem for a linear ordinary differential equation for unknown functions γ m i (t) for i = 1, 2, . . ., m; where for t ∈ [0, T].Then the problem above has a unique local solution γ m i ∈ C 2 [0, T] for all i, which satisfies a unique v m defined by (14) and satisfies (15).Now, taking w i = v mt in equation ( 15) and then integrating over [0, t], 0 < t < t m and by parts, for each m ≥ 1.
To estimate the last term on the right-hand side of ( 21), set v m ∈ H 1 (0, t m ; H 1 0 ( )) and by the trace theorem; v m ∈ H 1 (0, t m ; L k ( 1 )).Applying the Young and the trace Sobolev inequalities, we conclude that since 1 is bounded.To estimate (22), we focus on the first term We define Then, the use of Lemma 4 gives where inf s∈(0,1) By the Sobolev embedding Case n = 1, 2 proof is similar.So that, for taking t = T, we conclude that |u(s 24), ( 25) into (22), we conclude that Replacing ( 26) into ( 21), we can write where C is a positive constant independent of m.Since the elementary estimate 1 a-1 (28) for C 1 , C 2 ≥ 0 and a > 1, ( 27) can be written as where we have that v m (t) is bounded in L ∞ (0, T; H 1 0 ( )).Consequently, it follows from ( 29) and (30) that Using a standard procedure of the Aubin-Lions lemma [30,34], we deduce that , where η 1 = v t and v(0) = v 0 .Now, we suppose that η 2 = v t a.e. in (0, T) × 1 .It is clear that, since the weak limit of v mt on (0, T) × ∂ is equal to η 2 on (0, T) × 1 and to 0 on (0, T) × 0 , and since u = 0 on (0, T) × 0 , the assumption is that the weal limit of v mt on (0, T) × ∂ is the distribution time derivative of v on (0, T) × ∂ .Therefore, up to subsequence, we can pass to limit in (15) and find a weak solution (9) applying argument similar to that given in [35] (see Proposition 1).Uniqueness proof is given by contradiction, claiming two distinct solutions exist.Say w and v have the same initial data.Subtracting both two equations and testing result by w tv t , we conclude that From the following inequality ≤ 0 which satisfies wv = 0. Therefore, (9) satisfies a unique weak solution.
Now, we can deal with the proof of Theorem 5.
Proof To obtain the proof, we apply the contraction mapping theorem.For T > 0, we denote the convex closed subset of Y T as We define where r 2 = 1 2 ( u 1 2 + ∇u 0 2 ).Thanks to Lemma 6, for any u ∈ B r (X T ), we can introduce , which is the unique solution of (9).We can see that v corresponds to u and ζ : X T → X T .Our aim is to get that ζ is a contraction map, which implies ζ (B r (X T )) ⊂ B r (X T ) for any T > 0. Using energy identity for all t ∈ (0, T], we have Then by The last term on the right-hand side of inequality (34) can be estimated using the Holder inequality and similar calculations as for ( 23) and ( 25), By taking t = T and using the inequality (28), we have Because of the inequality for X, Y ≥ 0, where a is a positive constant, (36) yields that Now, we insert (38) into ( 35) and obtain the following inequality So that, we have Using inequality (37) and (40), we have Combining ( 39) and (41), we have By choosing T small enough and r large enough, we derive that ζ (u) ∈ B r (X T ) and T = T(r 2 0 , k, p, , 1 ) is a decreasing with respect to the first variable.Next, we will verify that ζ is a contraction mapping continuous on B r (X T ) and . Then, by using Lemma 6, the energy functional can be written for problem (42) such that for 0 ≤ t ≤ T. We denote the basic inequality for x ≥ 2, a 1 , a 2 ∈ R such that For estimating the last integral on the left-hand side of (43), we apply the basic inequality by taking b = k when k ≥ 2 and b = k k-1 when 1 < k < 2. So that, (43) becomes for k ≥ 2, and Now, we need to estimate the logarithmic term in (45).If we set From the mean value theorem, we have where 0 < ϑ < 1.From Lemma 4, we conclude that Inserting (47) into (45), we obtain We choose κ 0 ∈ (p, κ) such that Using (49), we can define l ∈ (0, 1) such that where l < κ 0 p-2 .Using (37) and the Holder inequality, we can write the first term of the integral term of (48) as Since l(p -2) < κ 0 , by the trace Sobolev embedding and definition of r, we obtain Applying the Holder inequality, we conclude that Thanks to (40) and r 0 ≤ r, (52) yields Using (54), we can define l 1 ∈ (0, 1) such that where l 1 < κ 1 p-2+ε .Using calculations similar to (50)-(52), we obtain where ε > 0 constant.Using the trace Sobolev embedding and the Holder inequality in time and (36), we have where κ 3 ∈ (p, κ).

Potential well
In this section, we will demonstrate the global existence of the proofs of solution (1).
We defined some useful functionals total energy function as Then, combining (64), (65), and definition of E(u) gives and The potential well depth is defined as and the outer space of the potential well ii) There exists λ * > 0 satisfying d dλ J(λ * u) = 0 such that Proof i) Take J(λu), By virtue of u p p, 1 , we see that lim λ→0 J(λu) = 0, lim λ→∞ J(λu) = -∞.ii) Now, taking the derivative of J(λu) with respect to λ, we have which gives that d dλ J(λu) > 0 on (0, λ * ) and d dλ J(λu) < 0 on (λ * , ∞).Thus, we have the desired results such that where N is the Nehari manifold given by and d has a positive lower bound, namely, , where C is defined as a positive constant.
Proof i) By (64), thanks to definitions of the Nehari manifold and d, it satisfies d ≥ 0. So that, our purpose is to prove that there is a positive function such that J(u) = d.We define {u i } ∞ i=1 ⊂ N as a minimizing sequence for J.So that, we conclude that It is clearly that, {|u i |} ∞ i=1 ⊂ N a minimizing sequence for J. Now, we suppose that u i > 0 in for all i ∈ N.
We also obtain that J is coercive on u ∈ N satisfying {u i } ∞ i=1 and is bounded in H 1 0 ( ).Since H 1 0 ( ) → L p+α ( 1 ) is compact embedding, there is a function u and a subsequence of Then, we get u ≥ 0 a.e. in .Moreover, using the dominated convergence theorem, weak lower semicontinuity and definition of J(u), I(u) and N gives Since x -y ln x ≤ 1 ey for x, y > 0 and the trace Sobolev embbedding theorem, we have where C * is the best Sobolev constant, which means Therefore, we conclude that Using the dominated convergence theorem, we have which means that u = 0. Last, we show that I(u) = 0. Indeed, if it is not true, we get I(u) < 0. So, thanks to Lemma 8, we have a positive constant λ * < 1 implying that I(λ * u) = 0. Therefore, it follows that . it is a contradiction.

Lower bound for blow-up time
In this part, we prove a lower bound for blow-up time of problem (1).First, we give lemma, which will play a role of the proof of Theorem 11.
Moreover, we have a contradiction.So, we get (u, u t ) ∈ V for all t ≥ 0.
It is easy to see that there is a time T 1 such that the solution goes to the infinity with lim t→T 1 H(t) = ∞.Thus, we have a lower bound for T 1 given by .
This completed the proof.

Conclusion
This work proves the existence of the result for a hyperbolic-type equation with logarithmic nonlinearity and dynamical boundary condition.This result is modern for these types of problems, and it can be generalized to many problems in the coming literature.

Funding
There is no applicable fund.

Lemma 9 i
) The depth of potential well depth defined by d Then d is a positive function such that