Mathematical Behavior of Solutions of the Kirchhoff Type Equation with Logarithmic Nonlinearity

{ 2γ + 2k ≤ p ≤ 2(n−2) n−4 , n > 4, 2γ + 2k ≤ p ≤ ∞, n ≤ 4, where Ω ⊂ R (n ≥ 1) is a bounded domain with smooth boundary ∂Ω. 1.1. Wave equation with logarithmic term Studies of logarithmic nonlinearity have a long history in physics as it occurs naturally in different areas of physics such as inflation cosmology, supersymmetric field theories, quantum mechanics and nuclear physics [4,8]. In [6], Cazenave and Haraux studied the existence of the solution following equation


Wave equation with logarithmic term
Studies of logarithmic nonlinearity have a long history in physics as it occurs naturally in different areas of physics such as inflation cosmology, supersymmetric field theories, quantum mechanics and nuclear physics [4,8].
In [6], Cazenave and Haraux studied the existence of the solution following equation in R 3 . Nowadays, there are much more works related to logarithmic nonlinearity in the literature, we refer the interested readers to [7,12,16,17,22,25] and papers cited there in. Al-Gharabli and Messaoudi [1,2] studied the following equation where k > 0 and h (s) = s. They proved the local-global existence and the exponential decay rate of the solutions. Then Peyravi [15], studied stability and instability at infinity of solutions to a logarithmic wave equation in an bounded domain Ω ⊂ R 3 with h (s) = k 0 + k 1 |s| m−1 .

Kirchhoff type equation
In 1876, Kirchhoff [3] introduced Kirchhoff type equation in order to study the nonlinear vibrations of an elastic string. Kirchhoff model to described the transverse oscillations of stretched string, with local or nonlocal flexural rigidity [24]. So that, Kirchhoff model was very important for many applications in mechanics, elastic theory and other areas of mathematical physics [3]. It is worth noting that there have been mang interesting study of the with initial and boundary value problems for Kirchhoff type equation, for details on Kirchhoff type equation, we refer to see the works [9,13,19,20,23] The original equation is where t > 0 and 0 < x < L. E is the Young modulus, p is the mass density, h is the cross-section area, P 0 is the initial axial tension, δ is the resistance modulus, f is the external force.

Kirchoff type equation with logarithmic term
Then, some authors discussed the following Kirchhoff type equation which including more general function M and dissipative term (see [10,11,14]). Nowadays, the studies have intensified about analysis of solutions for a class of Kirchhoff equation with logarithmic source term. We refer to work of see [5,21]. In 2019, Yang et. al [21] considered the following equation where M (s) = α + βs γ , γ > 0, α ≥ 1, β > 0. They studied local existence, finite time blow up and asymptotic behavior of solutions in cases subcritical energy and critical energy. And also, they proved finite time blow up solutions in case arbitrary high energy. Motivated by the above studies, we established the global existence and decay estimates of the solution for the problem (1.1). The results can be also viewed as a improved proof to the global existence theorem in [18].
The rest of our work is organized as follows. In section 2, we gave some notations and lemmas which will be used throughout this paper. In section 3, we proved the global existence of the solutions of the problem. The decay estimates result are presented in section 4.
We define the energy functional E(t) of the problem (1.1) by is a nonincreasing function for t ≥ 0 and Proof. By multiplying the equation in (1.1) by u t and integrating on Ω, we have and Let c > 0, γ ∈ L 1 (0, T ; R + ) and assume that the function w :

Global existence
In this section, we prove the global existence of solution of the problem (1.1) . Now, we define the following functionals and and The potential well depth is defined as and the outer space of the potential well Now, we establish some properties of the J(u) and I(u).
Then from Lemma 3.1 we can deduce λ * = λ 1 . Again from Lemma 3.1 of property (iii) it shows that I (λ * u) = I (λ 1 u) = 0, which means λ * u ∈ N. By the definition of d we get This complete our proof for (i). ii) By virtue of I (u) = 0, definition of I (u) and embedding theorems we obtain (3.16) From the definition of d, we have u ∈ N. By (3.15) and I (u) = 0, we get we take 2γ ≤ p − 2k and since β 1 > 1, β 2 > 0 and k is a positive constant. Combining of (3.14) and (3.16), we can see clearly that Lemma 3.4. Let u (t) be a weak solution problem of (1.1) and u 0 (t) ∈ H 2 0 (Ω) , u 1 (t) ∈ L 2 (Ω). Suppose that E (0) < d.
We shall prove I (u (t)) > 0 for 0 < t < T. We will use conradiction and we suppose that; there is a t 1 ∈ (0, T ) such that I (u (t 1 )) < 0. Observe by the continuity of I (u (t)) in t that there exists a t * ∈ (0, T ) such that I (u (t * )) = 0.Then by (3.10), we get which is a contradiction.
Proof. Let {w j } ∞ j=1 be an orthogonal basis of the "separable" space H 2 0 (Ω) which is orthonormal in and let the projections of the initial data on the finite dimensional subspace V m be given by for j = 1, 2, ..., m.
We look for the approximate solutions This leads to a system of ordinary differantial equations for unknown functions h m j (t). According to the standard existence theory for ordinary differantial equation, one can obtain functions h j : [0, t m ) → R, j = 1, 2, ..., m, which satisfy (3.17) in a maximal interval [0, t m ) , 0 < t m ≤ T. Next, we show that t m = T and that the local solution is uniformly bounded independent of m and t. For this purpose, let us replace w by u m t in (3.17) and integrate by parts, we have Integrating (3.18) from 0 to t, and using of (3.4), we obtain and initial data, for choosing large m and 0 ≤ t < ∞, we get u m (0) ∈ W. By (3.21) and an argument similar to Lemma 3.4, by choosing large m and 0 ≤ t < ∞, we have u m (t) ∈ W. Therefore, by virtue of (3.21) and (3.1) we get where 0 ≤ t < ∞ and choosing k is a positive constant smallest enough and p ≥ 2γ + 2k. For a sufficiently large m and 0 ≤ t < ∞, (3.22) gives Then we obtain By using of Sobolev embedding inequality, (3.21) and (3.22) we get , so that we obtain |u m | p+1 , is uniformly bounded in L ∞ 0, ∞; L p+1 (Ω) .
Then integrating (3.17) with respect to t , for 0 ≤ t < ∞ we have Therefore, up to a subsequence, we may pass to the limit in (3.23), and get a weak solution (u) to problem (1.1) with the above regularity. On the other hand, initial data conditions in (3.17) we may conclude (u (x, 0)) = (u 0 ) in H 2 0 and (u t (x, 0)) = (u 1 ) in L 2 (Ω) ∩ L q+1 (Ω) . ✷