Global solution for wave equation involving the fractional Laplacian with logarithmic nonlinearity

1.   Introduction

It is well known that the fractional integro-differentiation operation can be considered as an extension of the differentiation operations. It is well known that the idea of fractional differentiation as an extension of the concept of derivatives to the non-integer value arose almost together with the concept of differentiation. The first mention of this idea appears in the correspondence of G. W. Leibniz and the Marquis de l'Hospital in $1695$; see ^[1]. It was then developed by L. Euler, where the expression gives meaning even for non-integer values. The explicit advanced calculation was given in many references. If one replaces the classical Laplacian operator by fractional Laplacian, it will be motivated by the need to represent anomalous waves. The main mathematical models are the fractional Laplacians that have special symmetry and in-variance properties.

The basic evolution equation is

In principl, the fractional wave is linear, in which $s$ is some interpolation power of the Laplacian and one can conduct harmonic analysis, then one can generate the semi-groups. The researchers who performed the analysis were not inclined to analysis; the evolution associated with the fractional operator was carried out in stochastic processes because it was discovered that the typical approach to Brownian motion, this type of equations, was not a relevant model for many processes where there is a lack of convergence. Intense work in stochastic processes for several decades, but not in PDE analysis until tens years ago, initiated around pr. Caffarelli but only in the linear case in which we can return to the nonlinear case, while forgetting the probabilities.

In recent years, fractional Laplacian operators and related equations have an increasingly wide utilization in many important fields. In connection with the intensive development of industry, the electric power industry, the theory of nonlinear oscillations, automatic control, and optimal processes, the theory of damped partial differential equations is being developed, and its methods are actively used to solve problems in various fields of natural science and technology, especially when it comes with the fractional Laplacian. For more recent results involving the fractional Laplacian, for example ^{[2,3,4,5,6,7,8]} and the references therein. Recently, the fractional hyperbolic problems with continuous non-linearities have been studied by many researchers. For example, the authors in ^[5] studied the initial-boundary value problem of degenerate Kirchhoff-type for $y \in \Gamma_1, t \in \mathbb{R}_+$

where $\Gamma_1 \subset {\mathbb R}^{n}, 1\leq n$ is a bounded domain with Lipshcitz boundary $\partial \Gamma_1$, $[w]$ is the Gagliardo semi-norm of $w$, $r\in (0, 1), 1\leq \theta < \frac{2^\star_r}{2}, 2^\star_r = \frac{2n}{(n-2r)}, p \in (2\theta-1, 2^\star_r-1]$ and $[w]_r$ is the Gagliardo semi-norm of $w$ defined by

By using the Galerkin method, the global existence/nonexistence is obtained for solutions of (1.1) under certain conditions. Furthermore, in the work ^[9], it is proposed the following damped equation for $y \in \Gamma, t \in \mathbb{R}_+$

where $2 < \alpha < 2\theta < p < 2^\star < r$. Under some natural assumptions, the authors obtained the global existence, vacuum isolating, asymptotic behavior, and blow-up of solutions for (1.2) by combining the Galerkin method with potential wells theory (see ^{[10,11,12,13,14,15]}). In ^[9], Lin et al. studied the initial-boundary value problem of the Kirchhoff wave equation for $y \in \Gamma, t \in \mathbb{R}_+$

Regarding the Galerkin method's explanation with related logarithmic nonlinearities, we can review ^{[16,17,18,19]}.

In the present paper, we consider IBVP involving the fractional Laplacian non-linearity. To begin with, let $w = w(y, t)$, $\Gamma_1 \subset {\mathbb R}^{n}, n\geq1$ with Lipschitz boundary $\partial \Gamma_1$ and $\Gamma_2 = \mathbb{R}^n \setminus \Gamma_1, t > 0$

where $r\in(0, 1)$. The parameter $p$ satisfies

The rest of the paper is organized as follows. In Sections 2 and 3, we introduce our problem and recall necessary definitions and properties of the fractional Sobolev spaces. In Section 4, we study the global existence of solutions for our main problem (1.4).

2.   Auxiliary results and function spaces

In this section, we first recall some necessary definitions and properties of the fractional Sobolev spaces, see ^[20].

The fractional Laplacian of order $r$, $(-\Delta_y)^r$ of the function $w$ is defined by

We define the fractional-order Sobolev space by

with the norm

Set

then $H^{r}_0(\Gamma_1)$ is a closed linear subspace of $H^{r}(\Gamma_1)$, where

The space $H^{r}_0(\Gamma_1)$ is a Hilbert space with

Let $\Gamma_1$ be a bounded domain, then

1) The embedding $H^{r}_0(\Gamma_1)\hookrightarrow L^{p}(\Gamma_1)$ is compact $\forall 1\leq p < 2^\star_r$;

2) The embedding $H^{r}_0(\Gamma_1)\hookrightarrow L^{2^\star_r}(\Gamma_1)$ is continuous.

For any $1\leq s\leq 2^\star_r$, $\exists C_0 = C_0(n, s, r > 0)$ such that $\forall v\in H^{r}_0(\Gamma_1)$

For any $s\in [1, 2^*_r]$ and any bounded sequence $\{u_j\}_{j = 1}^{\infty}$ in $H^{r}_0(\Gamma_1)$ there exists $v$ in $L^s(\mathbb{R}^n)$, with $v = 0$ a.e. in $\mathbb{R}^n \setminus\Gamma_1$, such that up to a sub-sequence, still written as $\{v_j\}_{j = 1}^{\infty}$,

We have the following property for any $h$ positive number,

Let $\theta\in(0, 1)$, and $p_\theta\in ]p_0, p_1[\subset [1, +\infty[$ with $\frac{1}{p_\theta} = \frac{1-\theta}{p_0}+\frac{\theta}{p_1}$, we have the following inequality,

3.   Important theories and properties

We denote by $Z = H^{r}_0(\Gamma_1)\setminus \lbrace 0\rbrace$. The associate energy $\mathcal{E}$ of (1.4) is defined by

with the functional $\mathcal{K}\in (Z, \mathbb{R})$ associated with problem (1.4) is given by

the Nehari functional $\mathcal{J}\in (Z, \mathbb{R})$, defined by,

where $< ., . >$ denotes the dual pairing between $H^{r}_0(\Gamma_1)$ and $\left(H^{r}_0(\Gamma_1)\right)^{'}$.

We define the following two groups:

and

and

Now, define $d$ as

which characterized as

For any $\alpha$ satisfying $p < p+\alpha\leq 2_{r}^{\star}$, we put

where $k$ is the optimal embedding constant of

i.e.,

Lemma 3.1. Let $w\in H^{r}_0(\Gamma_1)\setminus \lbrace 0\rbrace$ and $p < p+\alpha\leq 2_{r}^{\star}, r\in (0, 1)$. We have

1) If $0 < \|w\|_{H^{r}_0(\Gamma_1)}\leq r(\alpha)$, then $\mathcal{J}(w) > 0$.

2) If $\mathcal{J}(w)\leq 0$, then $\|w\|_{H^{r}_0(\Gamma_1)} > r(\alpha), \alpha > 0$.

Proof. Let $w\in Z$, according to (2.9), we have

From the definition of $\mathcal{J}$, we obtain

by the definition of $k$, we know

then

1) If $0 < \|w\|_{H^{r}_0(\Gamma_1)}\leq r(\alpha)$, then $\mathcal{J}(w) > 0$;

2) If $\mathcal{J}(w)\leq 0$, implies that $\|w\|_{H^{r}_0(\Gamma_1)} > r(\alpha)$.

We put

and

and

Lemma 3.2. Let $\alpha\in(0, 2^\star_r-p]$. We have

Proof. ● Since $r$ and $R$ are two continuous functions on a compact $[0, 2^\star_r-p]$, the $r^\star$ and $r_\star$ exists.

● Let $r(\alpha)\leq R(\alpha), \forall \alpha\in(0, 2^\star_r-p]$ and let $w\in H^{r}_0(\Gamma_1)$, then $w\in L^p(\Gamma_1)\cap L^{p+\alpha}(\Gamma_1)$, by Holder's inequality we have

then

and

we obtain

so

Corollary 3.3. Let $w\in H^{r}_0(\Gamma_1)\setminus \lbrace 0\rbrace$ and $p < p+\alpha\leq 2_{s}^{\star}$. We have

1) If $0 < \|w\|_{H^{r}_0(\Gamma_1)}\leq r_\star$, then $\mathcal{J}(w) > 0$.

2) If $\mathcal{J}(w)\leq 0$, then $\|w\|_{H^{r}_0(\Gamma_1)} > r_\star$.

Lemma 3.4. Let $\alpha\in(0, 2^\star_r-p]$, we have

Proof. Let $w\in \mathcal{N}$, we have $w\in H^{r}_0(\Gamma_1)\setminus \lbrace 0\rbrace$ and $\mathcal{J}(w) = 0$, thus by Corollary 3.3, we obtain $\|w\|_{H^{r}_0(\Gamma_1)}\geq r_\star$, then

Let $w\in H^{r}_0(\Gamma_1)$, if $\mathcal{J}(w) < 0$, then there exists a $\mu^{\star}\in(0, 1)$ such that

Let $w\in H^{r}_0(\Gamma_1)$, if $\mathcal{J}(w) < 0$, then

Lemma 3.5. We have

1) $d = \inf\limits_{w\in Z}\sup\limits_{\mu > 0}\mathcal{K}(\mu w)$ has a positive lower bound, namely

2) There exists function $w\in\mathcal{N}$, such that $\mathcal{K}(w) = d$.

Proof. 1) Let $w\in H^{r}_0(\Gamma_1)$, we have

by $\mu^\star w\in \mathcal{N}$, thus

Combining (3.9) and (3.10), we obtain

On the other hand, if $w\in\mathcal{N}$, we obtain the only critical point in $(0, +\infty)$ of the mapping $\mathcal{K}$ is $\mu^\star = 1$. Thus,

for $w\in \mathcal{N}$. Hence

we have $\mathcal{J}(\mu^\star w) = 0$. This implies

yields

2) Let $\lbrace w_k\rbrace_{k = 1}^{\infty}\subset\mathcal{N}$ be minimizing sequence for $\mathcal{K}$ such that

On the other hand, we have $\lbrace w_k\rbrace_{k = 1}^{\infty}$ is bounded in $H^{r}_0(\Gamma_1)$. Since $H^{r}_0(\Gamma_1)\hookrightarrow L^2(\Gamma_1)$ is compact, there exists a function $w$ and a sub-sequence of $\lbrace w_k\rbrace_{k = 1}^{\infty}$, still denoted by $\lbrace w_k\rbrace_{k = 1}^{\infty}$, such that

we claim that

this implies

and

we can obtain

We can choose now a constant $h > 0$ such that

Then we can infer that there must be a constant $C_\star > 0$ such that

by $\log(z)\leq \frac{1}{h}z^h(h, z > 0)$, then

Then, for any $t \in[0, +\infty)$, we have

We conclude that

On the other hand, we have

We deduce

By $w_k\in \mathcal{N}$ we have $w_k\in H^{r}_0(\Gamma_1)$ and $\mathcal{J}(w_k) = 0$, then we obtain

Hence

It remains to show that $\mathcal{J}(w) = 0$. Arguing by contradiction, if this is not true then we have $\mathcal{J}(w) < 0$, there exists a positive constant $\mu^\star$ such that $\mu^\star < 1$ and satisfying $\mathcal{J}(\mu^\star w) = 0$. Therefore, by definition of d, we obtain

but this is a contradiction.

4.   Global existence of solutions

Here, we state and prove the result regarding the global existence of solutions.

Definition 4.1. A function

with

is said to be a global (weak) solution of (1.4), if

and

$t\in \mathbb{R}^{\star}_{+}$

If a (weak) global solution $w \in C(0, \infty; H^{r}_0(\Gamma_1))$, it is said that $w$ is a strong global solution of (1.4).

Theorem 4.2. Let $w_0\in H^{r}_0(\Gamma_1)$ and $w_{1}\in L^{2}(\Gamma_{1})$, suppose that $\mathcal{E}(t = 0) < d$, and $\mathcal{J}(w_0) > 0$. Then problem (1.4) admits a global solution $w\in L^\infty(0, \infty, H^{r}_0(\Gamma_1))$, with $w_{t}\in L^\infty(0, \infty, L^{2}(\Gamma_1))$ and $w\in \mathcal{W}, \forall t\in \mathbb{R}^{\star }_{+}$.

Proof. We will use the Galerkin method. For this end, we divide the proof into next steps

Step 1: By ^[21] there exists a sequence $(u_{j})_{j}\subset C_{0}^{\infty}(\Gamma_{1})$ of eigenfunctions of the fractional Laplace operator $(-\Delta_y)^{r}$, which is an orthonormal basis of $L^{2}(\Gamma_{1})$ and an orthogonal basis of $H^{r}_0(\Gamma_1)$.

Let $\lbrace V_m\rbrace_{m\in\mathbb{N}}$ be the Galerkin space of the separable Banach space $H^{r}_0(\Gamma_1)$, i.e.,

with $\left\lbrace u_j\right\rbrace _{j = 1}^n$ is an orthonormal basis in $L^2(\Gamma_1)$.

Let $w_0\in H^{r}_0(\Gamma_1)$, then we can find $w_{0n}\in V_n$. We shall find the approximate solutions of the following equality:

satisfying

Substituting $w_n$ into (1.4), we obtain

Owing to well-known standard ODE theory, we can see that (4.1) drives to a system of ODEs in $t$ that admits a local solution $0\leq w_n(t), 0\leq t \leq T_n$.

Step 2 : By multiplication of problem (1.4) by $g_{j}^{n'}$, summing for $j$, we obtain

integrating the above equation with respect to $\tau$, we obtain $\forall t\in \mathbb{R}^\star_+$

$\forall t\in \mathbb{R}^\star_+$, we obtain

so

where

For $n$ large enough, we can obtain $\mathcal{E}_n(t = 0) < d$ and then $\mathcal{E}(t = 0) < d$.

Then by (4.2), we have

According to $w_0\in \mathcal{W}$, we can find that $\partial_t w_{m}\in\mathcal{W}$. Next, for $t\in [0, T_n]$, we will prove that $w_n \in\mathcal{W}$. Indeed, if not the case, there exist $t_2\in(0, T_n]$ such that $w_n(t_2) = 0$ and $\mathcal{J}(w_n(t_2)) = 0$, then $w(t_2)\in \mathcal{N}$. Then $\mathcal{K}(w_n(t_2))\geq d = \inf\limits_{w\in\mathcal{N}}\mathcal{K}(w)$, which contradicts (4.3). Then, for sufficiently large $n$ and $\forall t\in[0, T_n]$, we have $w_n \in\mathcal{W}$.

By (4.3), then $w_n\in\mathcal{W}$ and

Then for $t\in [0, T_n]$ and $n$ large enough, we have

which gives, $\forall 0\leq t \leq T_n$

So $T_n = +\infty$. Then we know (4.4) and $w_n\in\mathcal{W}, \forall 0 \leq t < +\infty$.

Then

we can get

We can choose now a constant $h > 0$ such that

Then we can infer that there must be a constant $C_\star > 0$ such that

by $\log(z)\leq \frac{1}{h}z^h, \ (h, z > 0)$, then

Then, for sufficiently large $n$ and for any $0 \leq t < +\infty$, we have

Step 3: We see that there must be a function $w = w\in L^\infty(0, \infty, H^{r}_0(\Gamma_1))$ with $\partial_t w\in L^\infty(0, \infty, L^{2}(\Gamma_1))$, $\xi \in L^2(0, \infty, L^{\frac{q}{q-1}}(\Gamma_1))$ and a subsequence of $\lbrace w_i\rbrace_{i = 1}^{n}$, as $n\rightarrow \infty$, such that,

As in ^[9], the injection

is compact. We know

Then it follows from the convergence of $w_n$ and $w_{nt}$ that

then

Therefore, by $\mathcal{E}(t)\leq \mathcal{E}(t = 0)$ for a.e. $0 \leq t < +\infty$ and $w_0\in\mathcal{W}$, it is easy to prove $w\in\mathcal{W}$ for $0 \leq t < +\infty$.

Lemma 4.3. Let $w$ be a weak solution of (1.4), where $T$ is the maximum existence time. Thus

1) If $\mathcal{E}(t = 0) < d$, $w_0\in\mathcal{W}$, then $w\in\mathcal{W}$, for $0\leq t < T$;

2) If $\mathcal{E}(t = 0)\geq d$, $(w_0, w_1)\geq 0$, then $w\in\mathcal{V}$, and for $0\leq t < T$, supply that $w_0\in \mathcal{V}$.

Proof. Let $T$ be the maximal existence time of the weak solution of $w$. We have

1) Case of $\mathcal{E}(t = 0) < d$. By contradiction, if not the case, then for $0 < t_0 < T$ we have $\mathcal{J}(w) < 0, \forall 0\leq t < t_0$ and $\mathcal{J}(t_0) = 0$.

By claim 2) of Corollary 3.3, we have $\|w\|_{H^{r}_0(\Gamma_1)}\geq r^{\star} > 0$, for $0\leq t < t_0$, then $w(t_0)\neq 0$. Thus $w(t_0)\in\mathcal{N}$ and $\mathcal{K}(w(t_0))\geq 0$, which contradicts $\mathcal{K}(w(t_0))\leq \mathcal{E}(t_0) < \mathcal{E}(t = 0) < d$.

2) Case of $\mathcal{E}(t = 0)\geq d$ and $(w_0, w_1)\geq 0$. By contradiction, if not the case, then there exists surely a $0 < t_0 < T$ such that $\mathcal{J}(w) < 0$, for $0 \leq t < t_1$ and $\mathcal{J}(t_1) = 0$.

By Corollary 3.3, the claim 2), we have $\|w\|_{H^{r}_0(\Gamma_1)}\geq r^{\star} > 0$, for $0\leq t < t_0$, then $w(t_1)\neq 0$. Thus $w(t_0)\in\mathcal{N}$ and $\mathcal{K}(w(t_0))\geq 0$. By definition of $\mathcal{E}(t)$, we have

then $\mathcal{K}(w(t_1))\leq d$ and $\|w(t_1)\|^{2}_{2} = 0$. We first introduce an auxiliary function

and

and

by (4.6), we have,

Hence, we obtain

then $\mathcal{M'}(t)$ is strictly increasing for $\leq t < t_1$. As $\mathcal{M'}(0)\geq 0$, we have $\mathcal{M'}(t) = 2(\partial_t w, w)\geq \mathcal{M'}(0) > 0$, which conflicts with $\|w(t_1)\|^{2}_{2} = 0$.

Lemma 4.4. Let $(w_0, w_1)\in H^r_0(\Gamma_1)\times L^2(\Gamma_1)$, suppose that $0 < \mathcal{E}(t = 0) < d$. Then

1) If $w_0\in\mathcal{W}$, then $w\in\mathcal{W}$, for $0\leq t < T$,

2) If $w_0\in\mathcal{V}$, then $w\in\mathcal{V}$, for $0 \leq t < T$.

Proof. Let $T$ be the maximal existence time of the weak solution of $w$. We have

1) We claim that $w\in \mathcal{W}, \forall 0\leq t < T$. By contradiction, if not, then there must exist a $t_0\in(0, T)$ such that $w(t_0)\in \partial\mathcal{W}$, and then we have $\mathcal{J}(w) = 0$. $\mathcal{K}(w_0)\geq d$ contradicts the assumption (4.8).

5.   Conclusions and challenges

The present article is concerned with the wave equation involving the fractional Laplacian with logarithmic nonlinearity. With the aid of techniques from variational methods, we proved the global existence of weak solutions by the Galerkin approximation argument.

5.1.   Different formulas for fractional Laplacian operator

It is more difficult situation in the case where the fractional Laplacian is nonlinear. We assume that the space variable $y \in \mathbb{R}^n$ and the fractional exponent are $0 < s < 1$. The next version is equivalent.

1) The first pseudo-differential operator given by the Fourier transform

2) Singular integral operator

with this definition, it is the inverse of the Riesz integral operator $(-\Delta_y)^s u$. This one has a kernel $C_1\vert \xi - y\vert ^{n+2s}$, which is not integrable.

$3)$ The $\beta$-harmonic extension: find the solution of the $(n+1)$ problem

If we put $\beta = 2s$, we obtain

when $s = 1/2, i.e., (\beta = 1)$, the extended function $v$ is harmonic and the operator is the Dirichlet-Neumann map on the space $\mathbb{R}^n$.

These three alternatives can be studied in probability and PDEs.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by Researchers Supporting Project number (RSPD2024R736), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare there are no conflicts of interest.