On the Role of Diffusion Behaviors in Stability Criterion for p-Laplace Dynamical Equations with Infinite Delay and Partial Fuzzy Parameters under Dirichlet Boundary Value

By the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space W1,p 0 (Ω), a global asymptotical stability criterion for p-Laplace partial differential equations with partial fuzzy parameters is derived under Dirichlet boundary condition, which gives a positive answer to an open problem proposed in some related literatures. Different from many previous related literatures, the nonlinear p-Laplace diffusion item plays its role in the new criterion though the nonlinear p-Laplace presents great difficulties. Moreover, numerical examples illustrate that our new stability criterion can judge what the previous criteria cannot do.


Introduction
Very recently, time-delay -Laplace ( > 1) dynamical equations have attracted rapidly growing interest, because the nonlinear Laplace diffusion dynamical equations admit many physics and engineering background [1][2][3][4][5][6], such as Cohen-Grossberg neural networks and recurrent neural networks.In real world, diffusion phenomena cannot be unavoidable.Particularly, 2-Laplace ( = 2) is called the linear Laplace, and the diffusion phenomenon is always simulated by linear Laplace diffusion for simplicity ( [7][8][9][10] and their references therein).However, diffusion behavior is so complicated that the nonlinear reaction-diffusion models were considered in many other recent literatures [1][2][3][4][5][6][11][12][13].Even the nonlinear -Laplace diffusion ( > 1) is considered in simulating some diffusion behaviors [1][2][3][4][5][6].But the previous related literature lost sight of the role of the nonlinear diffusion in their stability criteria.As pointed out in [2], the problem how the -Laplace diffusion item plays a role in stability criteria remains open and challenging.And such a situation motivates our present study.Besides, fuzzy logic theory has been shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear systems.Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear systems using some local linear subsystems [14][15][16].Motivated by some ideas and methods of [17][18][19][20][21], we obtain a global asymptotical stability criterion for fuzzy T-S -Laplace partial differential equations with Dirichlet boundary value by the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space  1, 0 (Ω).And in the obtained criterion, the nonlinear -Laplace diffusion item plays a positive role.

Model Description and Preliminaries
Let us consider a class of fuzzy Takagi-Sugeno (T-S) -Laplace partial differential equations described as follows. where . .,   (V  ( − ())))  are the activation functions of the neurons.And the second and third equations of (1) represent the initial condition and the Dirichlet boundary condition, respectively.

Main Result
Before giving the main result of this paper, we have to present the following Lemma via some variational methods in the Sobolev space  [22][23][24] for details).
Lemma 2. Let  = diag( 1 ,  2 , . . .,   ) be a positive definite matrix, and let V be a solution of the fuzzy system (2).Then One has where and  is a positive scalar, satisfying  > .
Proof.Since V is a solution of system (2), it follows by Gauss formula and the Dirichlet boundary condition that 1/ ))  (see, e.g., [22]).
Theorem 4. Suppose that  =  1 / 2 > 1, where  1 is an even number while  2 is an odd number.If, in addition, there exist a positive definite matrix  = diag( 1 ,  2 , . . .,   ) and two positive scalars ,  such that the following inequalities hold: Proof.Define the Lyapunov-Krasovskii functional as follows: where Evaluating the time derivation of  1 () along the trajectory of the fuzzy system (2), we can get by Lemma 2 where we denote V  = V( − (), ) for convenience.
Then we can get by (H1), (H2) and the restrictive conditions on the parameter It follows by (H3) and Lemma 1 that So we conclude from (10) that It follows by the standard Lyapunov functional theory that the null solution of the fuzzy system (2) is globally asymptotically stable.
Remark 5.In many previous related literatures (see, e.g., [1][2][3][4][5][6]), the nonlinear -Laplace ( > 2) diffusion terms were omitted in the deductions, which results in that their stability criteria do not contain the diffusion terms.In other words, the diffusion terms do not affect their results.In addition, when  = 2, 2-Laplace is the linear Laplace, and there are many papers (see, e.g., [25][26][27][28][29]) in which the Laplace diffusion item plays its role in their stability criteria, for the linear Laplace PDEs can be considered in the special Hilbert space  1 (Ω) that can be orthogonally decomposed into the direct sum of infinitely many eigenfunction spaces.However, the nonlinear -Laplace ( > 1,  ̸ = 2) brings great difficulties, for the nonlinear -Laplace PDEs should be considered in the frame of Sobolev space  1, (Ω) that is only a reflexive Banach space.Indeed, owing to the great difficulties, the authors only provide in [4] the stability criterion in which the nonlinear -Laplace items play roles in the case of 1 <  < 2. However, in this paper, the nonlinear -Laplace diffusion terms play a positive role in our Theorem 4 for the case of  > 2 or  > 1, which also gives a positive answer to the open problem proposed in [2] to some extent.Besides, we will provide a numerical example where our Theorem 4 works whereas [2, Corollary 15] do not (see, Example 2).Remark 6. Particularly when  = 2, Theorem 4 provides a global asymptotical stability criterion for the familiar reaction-diffusion fuzzy CGNNs with infinite delay.Even in this particular case, the result is also good thanks to the infinite allowable upper bounds of time delays.

Numerical Example
Example 1.Consider the -Laplace fuzzy T-S dynamic equations as follows.

Conclusions
In this paper, the global asymptotical stability criterion of the nonlinear -Laplace fuzzy T-S dynamical equations with infinite delay was derived by the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space  1, 0 (Ω).The -Laplace diffusion item plays its role in our stability criterion while the stability criteria obtained in many previous literatures did not contain the diffusion terms.In fact, when  = 2, 2-Laplace is the linear Laplace, and there are many papers (see, e.g., [25][26][27][28][29]) in which the Laplace diffusion item plays its role in their stability criteria, for the linear Laplace PDEs can be considered in the special Hilbert space  1 (Ω) that can be orthogonally decomposed into the direct sum of infinitely many eigenfunction spaces.However, the nonlinear -Laplace ( > 1,  ̸ = 2) brings great difficulties, for the nonlinear -Laplace PDEs should be considered in the frame of Sobolev space  1, (Ω) that is only a reflexive Banach space.Indeed, owing to the great difficulties, the authors only provide in [4] the stability criterion in which the nonlinear -Laplace items play roles in the case of 1 <  < 2. Now in this paper, we present the stability criterion in which the nonlinear -Laplace items play roles in the case of  > 2 or  > 1.Moreover, numerical example shows the effectiveness of the proposed methods.Since the non-linear -Laplace dynamical equations have many physics and engineering background, including the famous Cohen-Grossberg neural networks, a further profound study is very interesting in mathematical theories, methods, and even practice.Up to now, we do not know how the -Laplace ( > 1) diffusion where   (V  ) and   (V  ) represent an amplification function at time  and an appropriately behaved function at time .  = ( ()  ) × and   = ( ()  ) × are connection matrices.The time-varying delays are () ∈ [0, +∞).
item plays a role in -stability criteria.This problem remains open and challenging.

Table 1 :
Comparisons between [2, Corollary 15] and Theorem 4.Remark 7. Since we consider the role of the nonlinear -Laplace diffusion item in the stability criterion, Example 2 illustrates that our Theorem 4 can judge what [2, Corollary 15] cannot do.