Stable anisotropic capillary hypersurfaces in a wedge

1.   Introduction

Recent research on nonlinear propagation of sound in the case of high amplitude waves has shown that there is a literature on well-grounded partial differential models. (see, e.g., ^{[1,5,7,9,10,11,12,13,16,17,18,20,21,23,24,25,26,27,28,29,30,39,49]}). This highly active field of research is being carried out by a wide range of applications such as the medical and industrial use of high intensity ultrasound in lithotripsy, thermotherapy, ultrasound cleaning and ultrasound chemistry. The classical models of nonlinear acoustics are Kuznetsov's equation, the Westervelt equation, and the KZK (Kokhlov-Zabolotskaya-Kuznetsov) equation. For mathematics. Existence and singularity analysis of several types of initial boundary value problems of this second nonlinear order in evolutionary PDEs, we refer (see ^{[19,22,31,32,33,34,35,36,37,38,40,41,42,43,44,45,46,47,48,50,51]}). Focusing on the study of sound wave propagation, it should be noted that the MGT equation is one of the nonlinear sound equations describing the propagation of sound waves in gases and liquids. The behavior of sound waves depends strongly on the average property of scattering, scattering, and nonlinear effects. Arises from high-frequency ultrasound (HFU) modeling see (^[16,25,41]). The original derivation dates back to ^[19]. This model is realized through the third order hyperbolic equation

the unknown function $u = u(x, t)$ denotes the scalar acoustic velocity, $c$ denotes the speed of sound and $\tau$ denotes the thermal relaxation. Besides, the coefficient $b = \beta c^{2}$ is related to the diffusively of the sound with $\beta \in \left(0, \tau \right]$. In ^[19]$,$ W Chen and A Palmieri studied the blow-up result for the semilinear Moore- Gibson-Thompson equation with nonlinearity of derivative type in the conservative case defined as following

This paper is related to the following works (see ^[27,46]). Now when we talk about the $\left(MGT\right)$ equation with memory term, we have I. Lasieka and X.Wang in ^[29] studied the exponential decay of energy of the temporally third order (Moore-Gibson-Thompson) equation with a memory term as follow

where $\tau, \alpha, b, c^{2}$ are physical parameters and $A$ is a positive self-adjoint operator on a Hilbert space $H.$ The convolution term $\int_{0}^{t}g(t-s)Aw\left(s\right) ds$ reflects the memory effects of materials due to viscoelasticity. In ^[13] I. Lasieka and X. Wang studied the general decay of solution of same problem above. Moore-Gibson-Thompson equation with nonlocal condition is a new posed problem. Existence and uniqueness of the generalized solution are established by using Galerkin method. This problems can be encountered in many scientific domains and many engineering models, see previous works (^{[20,22,31,32,33,34,35,36,37,42,43,47,48]}). Mesloub and Mesloub in ^[40] have applied the Galerkin method to a higher dimension mixed nonlocal problem for a Boussinesq equation. While, S. Boulaaras, A. Zaraï and A. Draifia investigated the Moore-Gibson-Thompson equation with integral condition in ^[17]. In motivate by these outcomes, we improve the existence and uniqueness by Galerkin method of the Fourth-Order Equation of Moore-Gibson-Thompson Type with source term and integral condition, this problem was cited by the work of F. Dell'Oro and V. Pata in ^[24].

We define the problem as follow

The convolution term $\int_{0}^{t}h(t-s)\Delta u(s)ds$ reflects the memory effect of materials due to vicoelasticity, $F$ is a given function and $h$ is the relaxation function satisfying

(H1) $h \in C^{1}(\mathbb{R} _{+}, \mathbb{R}_{+})$ is a non-increasing function satisfying

where $H(\infty) = \int_{0}^{\infty}h\left(s\right) ds > 0$, $H(t) = \int_{0}^{t}h\left(s\right) ds$ and $h^{\prime \prime } > 0, h''' < 0$.

(H2) $\exists\zeta > 0$ satisfying

The impartial of this manuscript is to consider the following nonlocal mixed boundary value problem for the Moore-Gibson-Thompson (MGT) equation for all $(x; t)\in Q_{T} = (0, T)$, where $\Omega \subset \mathbb{R} ^{n}$ is a bounded domain with sufficiently smooth boundary $\partial \Omega.$ solution of the posed problem.

We divide this paper into the following: In the second part, some definitions and appropriate spaces have been given. Then, we use the Galerkin's method to prove the existence, and in the fourth part we demonstrate the uniqueness.

2.   Preliminaries

Let $V\left(Q_{T}\right)$ and $W\left(Q_{T}\right)$ be the set spaces defined respectively by

and

where

Consider the equation

where

and $(., .)_{L^{2}\left(Q_{T}\right) }$ defend for the inner product in $L^{2}\left(Q_{T}\right), \ u$ is supposed to be a solution of (1.1) and $v\in W\left(Q_{T}\right).$ Upon using (2.1) and (1.1), we find

Now, we give two useful inequalities:

● Gronwall inequality: If for any $t\in I$, we have

where $h\left(t\right)$ and $y\left(t\right)$ are two nonnegative integrable functions on the interval $I$ with $h\left(t\right)$ non decreasing and $c$ is constant, then

● Trace inequality: When $w\in W_{1}^{2}\left(\Omega \right),$ we have

where $\Omega \ $is a bounded domain in $\mathbb{R} ^{n}\ $with smooth boundary $\partial \Omega, \ $ and $l\left(\varepsilon \right) \ $is a positive constant.

Definition 1. If a function $u\in V\left(Q_{T}\right)$ satisfies Eq (2.1) for each $v\in W\left(Q_{T}\right)$ is called a generalized solution of problem (1.1).

3.   Solvability of the problem

Here, by using Galerkin's method, we give the existence of problem (1.1).

Theorem 1. If $u_{0}, u_{1}, u_{2}\in W_{2}^{1}(\Omega)$, $u_{3}\in L^{2}(\Omega)$ and $F\in L^{2}(Q_{T})$, then there is at least one generalized solution in $V\left(Q_{T}\right)$ to problem (1.1).

Proof. Let $\left\{ Z_{k}\left(x\right) \right\} _{k\geq 1}$ be a fundamental system in $W_{2}^{1}(\Omega)$, such that

First, we will find an approximate solution of the problem (1.1) in the form

where the constants $C_{k}\left(t\right)$ are defined by the conditions

and can be determined from the relations

Invoking to (3.1) in (3.3) gives for $\ l = 1, ..., N.$

From (3.4) it follows that

Let

Then (3.5) can be written as

A differentiation with respect to $t$ (two times), yields

Thus for every $n$ there exists a function $u^{N}\left(x\right)$ satisfying (3.3).

Now, we will demonstrate that the sequence $u^{N}$ is bounded. To do this, we multiply each equation of (3.3) by the appropriate $C_{k}^{\prime }\left(t\right)$ summing over $k$ from $1$ to $N$ then integrating the resultant equality with respect to $t$ from 0 to $\tau,$ with $\tau \leq T,$ yields

after a simplification of the LHS of (3.9), we get

Taking into account the equalities (3.10)-(3.14) in (3.9), we obtain

Now, multiplying each equation of (3.3) by the appropriate $C_{k}^{\prime \prime }\left(t\right), \ $add them up from $1$ to $N$ and them integrate with respect to $t$ from $0$ to $\tau$, with $\tau \leq T$, we obtain

With the same reasoning in (3.9), we find

Upon using (3.17)-(3.21) into (3.16), we have

Now, multiplying each equation of (3.3) by the appropriate $C_{k}^{\prime \prime \prime }\left(t\right), \ $add them up from $1$ to $N$ and them integrate with respect to $t$ from $0$ to $\tau$, with $\tau \leq T$, we obtain

With the same reasoning in (3.9), we find

A substitution of equalities (3.24)-(3.29) in (3.23), gives

Multiplying (3.15) by $\lambda _{1}$, (3.22) by $\lambda _{2}$, and (3.30) by $\lambda _{3}$ such as $(\lambda_{1}+\lambda_{2} < \lambda_{3})$, we get

We can estimate all the terms in the RHS of (3.31) as follows

and

Substituting (3.32)-(3.83) into (3.31) and make use of the following inequality

where

we have

where

Choosing $\varepsilon _{7}, \ \varepsilon _{8}, \ \varepsilon _{9}, \ \varepsilon _{10}, \ \varepsilon _{11}, \ \varepsilon _{13}$, $\varepsilon _{14}, \varepsilon _{15}, \ \varepsilon _{16}, \ \varepsilon _{17}$ and $\varepsilon _{18}$ sufficiently large

the relation (3.84) reduces to

where

Applying the Gronwall inequality to (3.87) and then integrate from $0$ to $\tau$ appears that

We deduce from (3.89) that

where

Therefore the sequence $\left\{ u^{N}\right\} _{N\geq 1}$ is bounded in $V\left(Q_{T}\right),$ and we can extract from it a subsequence for which we use the same notation which converges weakly in $V\left(Q_{T}\right) \ $ to a limit function $u\left(x, t\right)$ we have to show that $u\left(x, t\right)$ is a generalized solution of (1.1). Since $u^{N}\left(x, t\right) \rightarrow u\left(x, t\right)$ in $L^{2}\left(Q_{T}\right)$ and $u^{N}(x, 0)\rightarrow \zeta \left(x\right)$ in $L^{2}(\Omega)$, then $u(x, 0) = \zeta \left(x\right).$

Now to prove that (2.1) holds, we multiply each of the relations (3.5) by a function $p_{l}\left(t\right) \in W_{2}^{1}(0, T), \ p_{l}\left(t\right) = 0,$ then add up the obtained equalities ranging from $l = 1\ $to $l = N,$ and integrate over $t$ on $(0, T).\ $If we let $\eta ^{N} = \sum\limits_{k = 1}^{N}p_{k}\left(t\right) Z_{k}\left(x\right),$ then we have

for all $\eta ^{N}$ of the form $\sum\limits_{k = 1}^{N}p_{l}\left(t\right) Z_{k}\left(x\right).$

Since

and

therefore we have

Thus, the limit function $u$ satisfies (2.1) for every $\eta ^{N} = \sum\limits_{k = 1}^{N}p_{l}\left(t\right) Z_{k}\left(x\right).$ We denote by $\mathbb{Q}_{N}$ the totality of all functions of the form $\eta ^{N} = \sum\limits_{k = 1}^{N}p_{l}\left(t\right) Z_{k}\left(x\right),$ with $p_{l}\left(t\right) \in W_{2}^{1}(0, T),$ $p_{l}\left(t\right) = 0.$

But $\cup _{l = 1}^{N} \mathbb{Q}_{N}$ is dense in $W\left(Q_{T}\right)$, then relation (2.1) holds for all $u$ $\in W\left(Q_{T}\right).\ $Thus we have shown that the limit function $u\left(x, t\right)$ is a generalized solution of problem (1.1) in $V\left(Q_{T}\right).$

4.   Uniqueness of solution

Theorem 2. The problem (1.1) cannot have more than one generalized solution in $V\left(Q_{T}\right).$

Proof. Suppose that there exist two different generalized solutions $u_{1}\in V\left(Q_{T}\right)$ and $u_{2}\in V\left(Q_{T}\right) \ $for the problem (1.1). Then, $U = u_{1}-u_{2}$ solves

and (2.1) gives

where

Consider the function

It is obvious that $v\in W\left(Q_{T}\right)$ and $v_{t}\left(x, t\right) = -U\left(x, t\right)$ for all $t\in \left[0, \tau \right].\ $Integration by parts in the left hand side of (4.2) gives

Plugging (4.4)-(4.10) into (4.2) we get

Now since

then

Using the trace inequality, the RHS of (4.11) can be estimated as follows

and

and

Combining the relations (4.13)-(4.16) and (4.11) we get

Next, multiplying the differential equation in (4.1) by $U_{ttt}$ and integrating over $Q_{\tau } = \Omega \times (0, \tau), \ $we obtain

An integration by parts in (4.18) yields

Substitution (4.19)-(4.25) into (4.18) we get the equality

The right hand side of (4.26) can be bounded as follows

So, combining inequalities (4.27)-(4.38) and equality (4.26) we obtain

Adding side to side (4.17) and (4.39), we obtain

Now to deal with the last term on the right hand side of (4.40) , we define the function $\theta \left(x, t\right)$ by the relation

Hence using (4.12) it follows that

and

And make use of the following inequality

Let

choosing $\varepsilon _{1}^{\prime }, \ \varepsilon _{2}^{\prime },$ $\varepsilon _{3}^{\prime },$ $\varepsilon _{4}^{\prime }$, $\varepsilon _{5}^{\prime }, \ \varepsilon _{6}^{\prime },$ $\varepsilon _{7}^{\prime },$ $\varepsilon _{8}^{\prime }$ and $\varepsilon _{9}^{\prime }$ sufficiently large

Since $\tau$ is arbitrary we get that $\alpha_{1}: = \frac{\varrho }{2}-2\tau \bigg(h_{0}+\varepsilon \frac{\left(\varrho +\delta +\gamma+h_{0} \right)}{2}\bigg) > 0, \ $thus inequality (4.40) takes the form

where

We obtain

where

Further, applying Gronwall's lemma to (4.53), we deduce that

where $\alpha_{2}: = \frac{\varrho }{4h_{0}+2\varepsilon \left(\varrho +\delta +\gamma+h_{0} \right) }$.

Proceeding in the same way for the intervals $\tau \in \left[(m-1)\alpha_{2}, m\alpha_{2}\right] \ $ to cover the whole interval $\left[0, T\right],$ and thus proving that $U(x, \tau) = 0$, for all $\tau$ in $\left[0, T\right].\ $Thus, the uniqueness is proved.

5.   Conclusions

Study of sound wave propagation, it should be noted that the Moore-Gibson-Thomson equation is one of the nonlinear sound equations that describes the propagation of sound waves in gases and liquids. The behavior of sound waves depends strongly on the average scattering, scattering and nonlinear effects. Arises from high-frequency ultrasound (HFU) modeling (see ^[16,25,41]). In this work, we have studied the solvability of the nonlocal mixed boundary value problem for the fourth order of Moore-Gibson-Thompson equation with source and memory terms. Galerkin's method was the main used tool for proving the solvability of the given non local problem. In the next work, we will try to using the same method with Hall-MHD equations which are nonlinear partial differential equation that arises in hydrodynamics and some physical applications (see for example ^[2,3,4,6]) by using some famous algorithms (see ^[8,14,15]).

Acknowledgments

The fourth author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P-2/1/42).

Conflict of interest

This work does not have any conflicts of interest.