Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2021View this Special Issue
Blow-Up for a Stochastic Viscoelastic Lamé Equation with Logarithmic Nonlinearity
In this paper, we consider an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and logarithmic nonlinear source terms. We proved a blow-up result for the solution with decreasing kernel.
In recent years, stochastic partial differential equations in a separable Hilbert space have been studied by many authors, and various results on the existence, uniqueness, stability, blow-up, and other quantitative and qualitative properties of solutions have been established.
In this work, we consider the following problem of stochastic wave equation: where is a bounded domain in , , with a smooth boundary ; are the Lamé constants which satisfy , ; is a positive function, ; the constant is a small nonnegative real number; and is the set of square integrable function on equipped with the inner product and its norm .
is an infinite dimensional Wiener process, is valued progressively measurable, and is a positive constant which measures the strength of noise.
It is common to observe a wave motion as a physical phenomenon which is mathematically modeled by a partial differential equation of hyperbolic type. Much has been written about such equations regarding their widespread applications to engineering and sciences. However, for more realistic models, the random fluctuation had been taken into consideration which led to introduced stochastic wave equation in 1960’s. Several examples of linear stochastic wave propagation and applications can be found in . Mueller  was the first who investigate the existence of explosive solutions for some stochastic wave equation. Motivated by Mueller , Chow  was interested by knowing how does a random perturbation affect the solution behavior for a wave equation with a polynomial nonlinearity. He was concerned with the existence of local and global solutions of the stochastic equation: where the initial data and are given functions and the nonlinear terms and are assumed to be polynomials in . Four years later, he  established an energy inequality and the exponential bound for a linear stochastic equation and gave the existence theorem for a unique global solution for the randomly perturbed wave equation:
In 2009, Chow  studied the problem of explosive solutions for a class of nonlinear stochastic wave equation in a domain for ,
We can mention some other works such as Cheng et al.  who studied the existence of a global solution and blow-up solutions for the nonlinear stochastic viscoelastic wave equation with nonlinear damping and source terms:
The authors proved that finite time blow-up with nonnegative probability is explosive or it is explosive in energy sense for .
Moreover, Kim et al.  considered the stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms:
They showed the existence of a global solution and blow-up in finite time.
Recently, Yang et al.  treated the following stochastic nonlinear viscoelastic wave equation:
They established the existence of global solution and asymptotic stability of the solution by using some properties of the convex function.
However, it was noticed that the logarithmic nonlinearity appears naturally in many branches of physics such as nuclear physics, optics, and geophysics (see [9, 10]). These specific applications in physics and other fields attract a lot of mathematical scientists to work with such problems. In the deterministic case, Al-Gharabli  investigated the stability of the solution of a viscoelastic plate equation with a logarithmic nonlinearity source term for the following problem: where is a bounded domain with a smooth boundary . The vector is the unit outer normal to , and is the nondecreasing nonnegative function.
Mezouar et al.  treated a more general problem where they considered the following nonlinear viscoelastic Kirchhoff equation with a time-varying delay term:
The paper is organized as follows: in Section 2, we introduce some basic definitions, necessary assumptions, and lemmas that are helpful in proving our main result. Section 3 is devoted to show the blow-up of the solution of our problem.
Let be a complete probability space for which a filtration of increasing sub fields is given and be a continuous Wiener random field in this space with a mean zero and the covariance operator satisfying
is defined by where is a sequence of real-valued standard Brownian motions mutually independent on the probability space , are the eigenvalues of , and are the corresponding eigenvectors. That is,
Note stands for expectation with respect to probability measure . Let be the set of -valued processes with the norm where denotes the adjoint operator of and which is equivalent to . For any process , we can define the stochastic integral with respect to the -Wiener process as which is a martingale. For more details about the infinite dimension Wiener process and stochastic integral, we refer to Da Prato and Zabczyk (pp. 90-96, ).
To state and prove our result, we need some assumptions.
A1. Assume that is a nonincreasing function satisfying and there exist tow nonnegative constants and such that
Theorem 1. Assume that (A1) and (A3) hold. If and, then there exists a solution in whichholds (1) on the interval in the sense of distributions over for almost all a test function such that
We define the energy associated to the solution of system (1) by where
We rewrite (1) as an equivalent Itô’s system which can be written as the integral equation
Lemma 2  (Sobolev-Poincaré’s inequality). Let be a number with or Then there exists a constant such that
Lemma 3 . For , we have
Proof. We can apply the Itô’s formula to (21) for each after integrating the above equation over to get By using integration by parts, we get By applying Lemma 3, we have We have By replacing (29)–(32) in (28) and multiplying equation (28) by , we arrive at (27).
We prove our main result for ; we purpose where
Lemma 5. Let be a solution of system (21) with initial data . Then, we have
Proof. Using the Itô’s formula and by following the same way as our discussions in Lemma 4 with taking the expectations, we obtain (37).
We multiply the second equation in (22) by and integrate the result over , and we take expectation; we obtain (38).
We set As is a positive decreasing function so Consequently,
Lemma 6. Let be a solution of system (21). Assume that (A1) holds. Then, there exists a positive constant such that where .
The last inequality is getting from (A1).
Case 7. If , then .
By applying Lemma 2, we obtain , then
Case 8. If , then .
Hence, Consequently, we obtain (41).
We are ready to state and prove our main result for . For this purpose, we define where and is a very small constant determined later.
Theorem 9. Assume (A1) and (A2) hold. Let be a solution of system (21) with initial data satisfying where is a nonnegative constant and is given in (35). If , then there exists a positive time such that where and is given later.
A direct differentiation of gives
Recalling (39) and (19), (51) leads to
By using Young’s and Hölder’s inequalities, we get
As , then so by using Young’s and Hölder’s inequality; we obtain
where and are constants.
We consider the following partition of : We have By (40), (47), and , we have Therefore, From (57), (58), and (59), we get As is increasing positive nonnegative function and by recalling (46), we get Taking into account (61) in (55), we find Substituting (62) into (54), we get where
Using Lemma 6, we arrive at Once is fixed, we pick small enough so that It implies that where and which is positive from (A2).
From (A1), (19), and Lemma 2, we have Now we add and subtract in (66), and using (67), we find where .
Using (60), we obtain where is the minimum of the coefficients of , ,